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		<title>free range statistics - R</title>
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				<title>Revisiting depression incidence by county and vote for Trump by @ellis2013nz</title>
					<description>&lt;p&gt;Just before Christmas &lt;a href=&quot;/blog/2024/12/23/depression-and-vote&quot;&gt;I blogged about the positive correlation between depression incidence in US counties and their vote for Trump in the 2024 presidential election&lt;/a&gt;. In addition to my casual interest in the topic, I used it as a case study in multilevel modelling while adjusting for spatial correlation. I explicitly said that I didn’t think it likely that the depression-vote relationship was a causal link; I suspected that most likely, some underlying variable that caused depression was also related to voting behaviour.&lt;/p&gt;

&lt;p&gt;I’m coming back to the issue because on reflection, I have three bits of unfinished business:&lt;/p&gt;

&lt;ol&gt;
  &lt;li&gt;I had a nagging thought that with 50+ counties per state (and hence some degrees of freedom to spare), I perhaps should have allowed random slopes for depression incidence in each state, rather than just random intercepts&lt;/li&gt;
  &lt;li&gt;I thought my spatial statistics workaround, of just adding a rubbery mat to space to suck up the spatial autocorrelation between observations, perhaps was a bit slap-dash and I should be as a matter of course modelling the spatial autocorrelation explicitly&lt;/li&gt;
  &lt;li&gt;An alert reader, Jonathan Spring, pointed out that in the USA there are &lt;a href=&quot;https://disq.us/url?url=https%3A%2F%2Fpmc.ncbi.nlm.nih.gov%2Farticles%2FPMC6390869%2F%23%3A%7E%3Atext%3DStudies%2520that%2520have%2520explored%2520the%2Copposed%2520to%2520African%2520Americans%252C%2520whose%3AIlHBLjpwdFZl-Dn9J9CLQ5OH2gc&amp;amp;cuid=3714645&quot;&gt;very marked racial differences between depression diagnoses&lt;/a&gt; and suggested that perhaps in my model the depression incidence was standing in as a proxy for “whiteness”.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;Of these, the first two felt like bits of probably-immaterial-to-the-question methodological details that I should polish up, whereas number 3 seemed likely to be the explanation of the whole phenomenon. In other words, race is likely a &lt;a href=&quot;/blog/2023/06/04/causality-sims&quot;&gt;confounder&lt;/a&gt; of the depression-voting relationship, as per this directed acyclic graph:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286c-dag.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-dag.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Which means that if we want to actually understand the causal relationship of depression on voting we would need to control for race in the regression. Now, there are other things we’d need to do too; in particular to identify and control for the various “other factors”. I’m not up for that right now - this is someone else’s job - but I’m interested enough to go part-way into things to at least check out the degree to which race makes the depression relationship go away.&lt;/p&gt;

&lt;p&gt;This is a long post and parts of it are likely to be of interest only to people concerned with the minutiae of multilevel modelling with spatial autocorrelation. If you just want to see what including race in the model does to the depression effect on voting for Trump, you can skip down to the final section.&lt;/p&gt;

&lt;p&gt;All the code for this post assumes that the code from the previous post has already been run. If you want a version of the whole script that just runs, it’s &lt;a href=&quot;https://github.com/ellisp/blog-source/blob/master/_working/0286c-depression-and-race.R&quot;&gt;here in the source repository&lt;/a&gt; of the whole blog.&lt;/p&gt;

&lt;p&gt;But here’s the code for just that DAG diagram:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;GGally&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggdag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;patchwork&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kableExtra&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# you need to have run the code from the previous blog first, line below will &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# work only for those who are running this from a clone of my whole repository:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;combined2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;source&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0286-voting-and-depression.R&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-----------diagram--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dag&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dagify&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Vote&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Race&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Depression&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Other factors&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Depression&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Race&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Other factors&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;125&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggdag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;link&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;node&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_dag_blank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_node&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lightgreen&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;19&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;19&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;19&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;17&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_edges&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lightblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lightblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_width&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;edge_linetype&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_dag_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;OK, on to fixing up the loose ends with my modelling approach.&lt;/p&gt;

&lt;h2 id=&quot;move-from-gam-to-gamm&quot;&gt;Move from &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam&lt;/code&gt; to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gamm&lt;/code&gt;&lt;/h2&gt;

&lt;p&gt;I decided that the simplest material improvement to my approach to the spatial autocorrelation would be to explicitly model the spatial co-variance of the residuals. One way of doing this that is the minimal change from my approach so far would be to move the model-fitting from &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;mgcv::gam()&lt;/code&gt; to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;mgcv::gamm()&lt;/code&gt;. &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gamm()&lt;/code&gt; fits models by iterating between calls to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;nlme::lme()&lt;/code&gt; and a generalized additive model until convergence. Basically this gives us the ability to use the correlation structures for residuals and mixed random and fixed effects of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;lme()&lt;/code&gt; while still using the splines and response distributions of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam()&lt;/code&gt;. This is what I need as I want to continue modelling my response with a quasi-binomial family, and I am probably going to want to keep my “rubber sheet” nuisance effect over the US space modelled with a two-dimensional spline (&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(x, y)&lt;/code&gt;), even while using the correlation features of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;lme()&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;So the first thing I do is create a &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;model6b&lt;/code&gt;, as similar as possible to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;model6&lt;/code&gt; that was the best and final model from the last blog post, but just changes the estimation method. So here it is, a straight transfer from &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam()&lt;/code&gt; to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gamm()&lt;/code&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------moving from gam to gamm--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# start with the same model as our final one in the last post, but estimated differently:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6b&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;re&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we can&apos;t compare the AIC of models created with gam and gamm, see&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://stats.stackexchange.com/questions/70512/huge-%CE%94aic-between-gam-and-gamm-models&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# some differences eg effective degrees of freedom less in the GAMM. But the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# main conclusions (significance of cpe) the same&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6b&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;… which gives us these two different results:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; summary(model6)

Family: quasibinomial 
Link function: logit 

Formula:
per_gop ~ cpe + s(x, y) + s(state_name, bs = &quot;re&quot;)

Parametric coefficients:
            Estimate Std. Error t value Pr(&amp;gt;|t|)    
(Intercept)  -2.7972    20.7488  -0.135    0.893    
cpe          15.5908     0.6778  23.001   &amp;lt;2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                edf Ref.df     F p-value    
s(x,y)        28.39  28.94 7.995  &amp;lt;2e-16 ***
s(state_name) 49.00  49.00 7.949  &amp;lt;2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.386   Deviance explained = 44.8%
GCV = 3138.5  Scale est. = 3170.7    n = 3107
&amp;gt; summary(model6b$gam)

Family: quasibinomial 
Link function: logit 

Formula:
per_gop ~ cpe + s(x, y) + s(state_name, bs = &quot;re&quot;)

Parametric coefficients:
            Estimate Std. Error t value Pr(&amp;gt;|t|)    
(Intercept)  -2.8878     0.1481  -19.50   &amp;lt;2e-16 ***
cpe          15.0166     0.6271   23.95   &amp;lt;2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                edf Ref.df     F p-value    
s(x,y)        25.41  25.41 6.438  &amp;lt;2e-16 ***
s(state_name) 39.91  49.00 7.154  &amp;lt;2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.382   
  Scale est. = 3005.3    n = 3107
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;There’s some differences coming from the different estimation methods. The &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gamm()&lt;/code&gt; model uses less effective degrees of freedom for both the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(x,y)&lt;/code&gt; rubber mat and the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(state_name, bs=&quot;re&quot;)&lt;/code&gt; state level random intercept. The fixed effect coefficient for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cpe&lt;/code&gt; (‘crude prevalence estimate’ of county-level depression) is a little different - 15.0 versus 15.6. But we can see we’re fitting the same model and getting substantively the same results.&lt;/p&gt;

&lt;p&gt;Next small change I make is to move the state level random intercept from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam&lt;/code&gt; specification into &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;lme&lt;/code&gt;. Again, this is an identical model, just changing how the fitting is done:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# move the state random effect into the things to be estimated by nlme:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6c&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# fixed coefficients are identical to 6b, but fit was much faster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6c&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# not shown&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This is a big performance speed up (which we’re going to need as the models start getting more complex) for materially the same results.&lt;/p&gt;

&lt;h2 id=&quot;random-slopes&quot;&gt;Random slopes&lt;/h2&gt;

&lt;p&gt;Now I’m ready to take advantage of the move to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gamm&lt;/code&gt; syntax and I let the slopes, not just the intercepts, vary for each state:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------random slopes--------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# AIC is 200 less so worth having the random intercepts&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6c&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;…which gives us these results. Note that the Akaike Information Criterion has decreased by 200 from adding the random slopes, suggesting the model is overall worth the increased complexity. The cpe (depression) coefficient has decreased in size but is still quite large and definitely significant.&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; AIC(model7$lme, model6c$lme)
            df      AIC
model7$lme   9 8605.692
model6c$lme  7 8806.902
&amp;gt; summary(model7$lme)
Linear mixed-effects model fit by maximum likelihood
  Data: data 
       AIC      BIC    logLik
  8605.692 8660.064 -4293.846

Random effects:
 Formula: ~Xr - 1 | g
 Structure: pdIdnot
             Xr1      Xr2      Xr3      Xr4      Xr5      Xr6      Xr7
StdDev: 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986
             Xr8      Xr9     Xr10     Xr11     Xr12     Xr13     Xr14
StdDev: 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986
            Xr15     Xr16     Xr17     Xr18     Xr19     Xr20     Xr21
StdDev: 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986
            Xr22     Xr23     Xr24     Xr25     Xr26     Xr27
StdDev: 4.440986 4.440986 4.440986 4.440986 4.440986 4.440986

 Formula: ~1 + cpe | state_name %in% g
 Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr  
(Intercept)  3.294624 (Intr)
cpe         15.833056 -0.985
Residual    50.963709       

Variance function:
 Structure: fixed weights
 Formula: ~invwt 
Fixed effects:  list(fixed) 
                 Value Std.Error   DF   t-value p-value
X(Intercept) -2.099504 0.5204783 3054 -4.033796  0.0001
Xcpe         10.482779 2.5041023 3054  4.186243  0.0000
Xs(x,y)Fx1    0.221830 0.1339667 3054  1.655859  0.0979
Xs(x,y)Fx2    0.134028 0.1735957 3054  0.772070  0.4401
 Correlation: 
           X(Int) Xcpe   X(,)F1
Xcpe       -0.985              
Xs(x,y)Fx1 -0.083  0.084       
Xs(x,y)Fx2 -0.121  0.126  0.568

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-9.65917765  0.05092882  0.32541129  0.62277480  6.27537901 

Number of Observations: 3107
Number of Groups: 
                g state_name %in% g 
                1                50 
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;We can use this model to make a new version of the final chart from the previous blog post:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;preds7&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_ribbon&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crude prevalence estimate of depression&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage vote for Trump in 2024 election&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Grey ribbons are 95% confidence intervals from quasibinomial generalized additive model with spatial effect and state-level random intercept effect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Counties with more depression voted more for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286c-random-slopes.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-random-slopes.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;It doesn’t look much different. So my hunch on this aspect was right; there’s enough data to justify letting the depression-vote slope vary in each state (ie we get a better model) but it doesn’t change the substantive conclusion.&lt;/p&gt;

&lt;h2 id=&quot;better-spatial-autocorrelation&quot;&gt;Better spatial autocorrelation&lt;/h2&gt;

&lt;p&gt;Now I’m ready to improve the way I’m handling spatial autocorrelation. As I mentioned last time, my approach to this is to model the centroid of each county with an &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(x, y)&lt;/code&gt; two-dimensional spline, a sort of rubber mat overlaid over the USA to soak up the things counties have in common with their neighbouring counties. I still think this is much better than nothing, but admit there is still a problem that even after doing this the residuals of counties that are close together will still be correlated. Which means that each observation is not worth as much as if it were truly independent, which means that my inferences are over-confident in their precision.&lt;/p&gt;

&lt;p&gt;First I wanted to check out if this hunch was correct. The standard way to look for spatial autocorrelation is via a variogram. I couldn’t get the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Variogram()&lt;/code&gt; function in &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;nlme&lt;/code&gt; to return sensible results from the models that had been fit with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gamm()&lt;/code&gt; so I had to make one more explicitly with the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;variogram()&lt;/code&gt; function in &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gstat&lt;/code&gt;:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gstat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;residuals&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coordinates&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sp_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variogram&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sp_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variogram for pairs of counties&apos; residuals&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xlab&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Distance between pairs of counties&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;… which gets me this:&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-variogram.png&quot; width=&quot;100%&quot; /&gt;&lt;/p&gt;

&lt;p&gt;A variogram works by calculating the distances between each pair of observations, binning these and then assessing how related the pairs of observations (in this case, residuals after the model fitting) are in each bin. It doesn’t do this by a correlation coefficient but by another measure that I haven’t got my head around but is described as &lt;a href=&quot;https://www.kgs.ku.edu/Tis/surf3/s3krig2.html&quot;&gt;‘half the variance of the differences between all possible points spaced a constant distance apart.’&lt;/a&gt;. The important thing is that a score of zero means perfect correlation, and the higher the numbers are the more independent the pairs of observations contained in the bin are. So to interpret the chart above we can say that the counties that are about 20 degrees (of latitude/longitude, ignoring curvature of the earth) apart or closer have some degree of correlation with eachother; once they get to that far apart this measure more or less stablises.&lt;/p&gt;

&lt;p&gt;I don’t know why spatial statistics doesn’t just use a good-ole correlation coefficient for this job, but presume there are interesting historical reasons. To check that I wasn’t mangling things, I decided to “roll my own” spatial correlation measure, with:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# let&apos;s roll our own on a similar concept to see what&apos;s happening&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# find the counties&apos; distance from each other country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_pairs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;from&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fx&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;to&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tx&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;((&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fx&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fy&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ty&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res7&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;residuals&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_pairs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from_res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;from&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to_res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;to&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cut&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;24&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;48&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;108&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from_res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to_res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distance&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation between pairs of counties&apos; residuals&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Distance between two counties&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation in residuals from model7&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of county-pairs&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gives me this chart, which is more like the sort of thing we use in time series analysis.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286c-spatial-correlation.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-spatial-correlation.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’m pleased it tells a similar story to the variogram (which is more of a black box to me) - the correlation between pairs of counties’ residuals is above zero for counties that are within 6 degrees/units of eachother, and stablises at a small negative number from about 20 units apart onwards.&lt;/p&gt;

&lt;p&gt;My conclusion from this is that yes, there is still some spatial autocorrelation that should be taken into account. Particularly for those counties very close together (around 2 degree/units apart or less).&lt;/p&gt;

&lt;p&gt;Now, there is a tricky problem with the &lt;em&gt;shape&lt;/em&gt; of spatial autocorrelation. Even if you’re prepared to assume (as I am in this case) that it is symmetrical north-south and east-west (not always the case when it depends on eg wind), there are differing shapes of decay in the level of correlation, as a function of distance. Experts can apparently/allegedly judge which method is best by looking at the shape of the variogram, but I feel safest by trying all five correlation structures available to me and choosing the one with the lowest AIC.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# o help us decide the shape of the spatial autocorrelation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corGaus&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corLin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corRatio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corSpher&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;err&quot;&gt;\&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model7&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gets me these results:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; sapply(model8, \(m){round(AIC(m$lme), -1)})
[1] 8230 8600 8590 8330 8580
&amp;gt; AIC(model7$lme)
[1] 8605.692
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;We see that all the models that adjust for spatial correlation are better than model7 (which doesn’t), but the models using &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;corExp&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;corRatio&lt;/code&gt; are much better than the other three. The &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;corExp&lt;/code&gt; model (&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;model8[[1]]&lt;/code&gt;) is our new best model so far.&lt;/p&gt;

&lt;p&gt;If we look at the t statistics for the coefficients of these five models, we see that for the first time our best model has a “non-significant” slope for cpe (ie depression), 1.695. Now, I’m not going to remove a variable from a complex mixed effects model like this because of a non-significant t statistic, but it is definitely note-worthy that the depression effect has become less important once we let it vary by state and do the best correction for spatial autocorrelation:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; sapply(model8, \(m){summary(m$gam)$p.t})
                 [,1]      [,2]      [,3]      [,4]      [,5]
(Intercept) -2.031622 -4.036009 -3.987122 -3.494438 -3.952872
cpe          1.695218  4.206850  4.193251  3.746362  4.167077
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Also noteworthy though is that the other spatial correlation shapes still leave depression with significant t statistic.&lt;/p&gt;

&lt;p&gt;Here’s the full report from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;lme&lt;/code&gt; part of the best fit so far:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; summary(model8[[1]]$lme)
Linear mixed-effects model fit by maximum likelihood
  Data: data 
       AIC      BIC    logLik
  8231.487 8291.901 -4105.744

Random effects:
 Formula: ~Xr - 1 | g
 Structure: pdIdnot
            Xr1     Xr2     Xr3     Xr4     Xr5     Xr6     Xr7     Xr8
StdDev: 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243
            Xr9    Xr10    Xr11    Xr12    Xr13    Xr14    Xr15    Xr16
StdDev: 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243
           Xr17    Xr18    Xr19    Xr20    Xr21    Xr22    Xr23    Xr24
StdDev: 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243 6.68243
           Xr25    Xr26    Xr27
StdDev: 6.68243 6.68243 6.68243

 Formula: ~1 + cpe | state_name %in% g
 Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr  
(Intercept)  2.702224 (Intr)
cpe         12.092468 -0.99 
Residual    66.289425       

Correlation Structure: Exponential spatial correlation
 Formula: ~x + y | g/state_name 
 Parameter estimate(s):
    range 
0.7289188 
Variance function:
 Structure: fixed weights
 Formula: ~invwt 
Fixed effects:  list(fixed) 
                 Value Std.Error   DF    t-value p-value
X(Intercept) -0.889436 0.4390918 3054 -2.0256268  0.0429
Xcpe          3.370112 1.9919865 3054  1.6918346  0.0908
Xs(x,y)Fx1    0.165268 0.1447102 3054  1.1420604  0.2535
Xs(x,y)Fx2    0.000906 0.1909150 3054  0.0047431  0.9962
 Correlation: 
           X(Int) Xcpe   X(,)F1
Xcpe       -0.988              
Xs(x,y)Fx1 -0.065  0.066       
Xs(x,y)Fx2 -0.131  0.141  0.510

Standardized Within-Group Residuals:
       Min         Q1        Med         Q3        Max 
-5.2880800  0.2592565  0.5963373  1.0331947  4.6052452 

Number of Observations: 3107
Number of Groups: 
                g state_name %in% g 
                1                50 
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;I believe that  &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Formula: ~Xr - 1 | g&lt;/code&gt; bit refers to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(x,y)&lt;/code&gt; term from the GAM when brought into the linearised LME fit. The range of 0.73 refers to the exponential spatial correlation.&lt;/p&gt;

&lt;h2 id=&quot;race&quot;&gt;Race&lt;/h2&gt;

&lt;p&gt;Finally I’m ready to look at the question that’s of most substantive interest - does incuding race in the model make the “depression effect” go away, suggesting that depression is acting as a proxy for whiteness. I sourced data on county characteristics of various sorts from the UN Census Bureau. I considered four candidate variables:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;white_alone&lt;/code&gt; which is the proportion of the county that describe themselves as white and not other race&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;white_all&lt;/code&gt; which is the proportion of the county that describe themselves as white, whether or not they also are of another race as well&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;hispanic&lt;/code&gt; proportion of county that describe themselves as hispanic&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;hispanic_multi&lt;/code&gt; proportion of county that describe themselves as hispanic and more races in addition&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;I deliberately left out the proportion of African-Americans in each county, assuming it would be very collinear with some combination of the others. If were seriously interested in how race worked in this election I would probably have included it anyway.&lt;/p&gt;

&lt;p&gt;Looking at these four variables (without peeking at their relationship to vote) gives me this pairs plot:&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286c-pairs.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-pairs.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;which convinces me I should save a degree of freedom and drop the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;white_all&lt;/code&gt; variable from my modelling as containing virtually no extra information from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;white_alone&lt;/code&gt; variable. Here’s the code to import that data and draw the pairs plot:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;o&quot;&gt;---------------------&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;on&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;race&apos;&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# county characteristis from US Census Bureau&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# see https://www2.census.gov/programs-surveys/popest/technical-documentation/file-layouts/2020-2023/CC-EST2023-ALLDATA.pdf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for metadata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cc-est2023-alldata.csv&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://www2.census.gov/programs-surveys/popest/datasets/2020-2023/counties/asrh/cc-est2023-alldata.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# key columns:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# TOT_POP total population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# WA_MALE &quot;White alone&quot; male&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# WAC_MALE &quot;white alone or in combination&quot; male&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# H_MALE Hispanic male&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# HTOM_MALE Hispanic AND more races male&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;race&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# just 2023 and TOTAL age group:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;YEAR&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AGEGRP&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WA_MALE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WA_FEMALE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TOT_POP&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WAC_MALE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WAC_FEMALE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TOT_POP&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;H_MALE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;H_FEMALE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TOT_POP&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;HTOM_MALE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;HTOM_FEMALE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TOT_POP&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;STATE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;COUNTY&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CTYNAME&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;race&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CTYNAME&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggpairs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Next I wanted to look at the relationship of these race variables to the logit of vote for Trump, to check if a linearity assumption was going to be reasonable. That got me this chart, which seemed linear enough for me (for my purposes):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286c-exp-v-response.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-exp-v-response.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;…produced with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;race&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# visual check that the counties joined correctly:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CTYNAME&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check for linearity of relationships to the response variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;logit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;logit_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;logit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;logit_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`
         Depression incidence`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Proportion only white`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Proportion only hispanic`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Proportion hispanic plus another`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;logit_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;logit_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;free_x&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Explanatory variable value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;logit of vote for Trump, 2024&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;OK so now I fit a whole bunch of different models to be confident that individual decisions from me weren’t going to be leading to my final conclusions. I fit models with many of the combinations of using a rubber mat to deal with spatial autocorrelation, explicitly modelling the spatial autocorrelation, random or fixed slopes (ie varying by state) for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cpe&lt;/code&gt; (depression), random or fixed slopes (ie varying by state) for hte race variables. Then I calculated the AIC of all these models and the t statistics for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cpe&lt;/code&gt; (depression), which gets me this table, sorted with the best models (lowest AIC) at the top:&lt;/p&gt;

&lt;table class=&quot;table&quot; style=&quot;margin-left: auto; margin-right: auto;&quot;&gt;
 &lt;thead&gt;
  &lt;tr&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; model &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; AIC &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; rubber_mat &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; random_cpe_slope &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; race &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; random_race_slope &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; SAC_fix &lt;/th&gt;
   &lt;th style=&quot;text-align:right;&quot;&gt; cpe_t_stat &lt;/th&gt;
  &lt;/tr&gt;
 &lt;/thead&gt;
&lt;tbody&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 15 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 5676.498 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; -1.612598 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 13 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 5904.910 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; -1.900880 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 10 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 6063.537 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; -1.581859 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 14 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 6241.311 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; -4.612077 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 11 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 6416.801 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; -2.996010 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 12 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 7136.013 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 5.347793 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 8 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 8231.487 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1.695218 &lt;/td&gt;
  &lt;/tr&gt;
  &lt;tr&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 9 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 8246.613 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 0 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 1 &lt;/td&gt;
   &lt;td style=&quot;text-align:right;&quot;&gt; 2.162823 &lt;/td&gt;
  &lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;

&lt;p&gt;Lots of interesting things here, but most important:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;The best models are definitely those that include race&lt;/li&gt;
  &lt;li&gt;The best model of all is also the most complex - random slopes for both race and depression, rubber mat, plus addressing the spatial autocorrelation (SAC) explicitly with an exponential correlation structure&lt;/li&gt;
  &lt;li&gt;The best models all have a &lt;em&gt;negative&lt;/em&gt; sign for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cpe&lt;/code&gt;, indicating that after controlling for race, it is the counties with &lt;em&gt;less&lt;/em&gt; depression that were more likely to vote for Trump - a compelling change in narrative from looking at the data without controlling for race. But in the best model of all, depression isn’t very important.&lt;/li&gt;
  &lt;li&gt;One model that I describe in the code below as “definitely illegitimate because it makes no attempt at all to correct for spatial autocorrelation” is the one that gives the biggest positive effect (as in t statistics) for the depression impact on voting for Trump&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;I regard this as good evidence that in fact incidence of depression by county wasn’t important in driving the vote for Trump, but that race composition of counties probably was (or at least ‘more likely’ was). Of course, how that mechanism works, and whether ‘whiteness’ here is in fact standing in for something else, is beyond the scope of this blog post (already far too long) to explain.&lt;/p&gt;

&lt;p&gt;Here’s the code that fit all those candidate models described above and generates the table:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# no rubber mat, no race, but does have spatial autocorrelation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model9&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# no rubber mat, but race:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model10&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# no random slopes or rubber mat, but race:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model11&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# no random slope, rubber mat or spatial autocorrelation fix at all. this model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# is definitely illegitimate in that it makes no effort to fix for spatial issues.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model12&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# rubber mat, race, random slope for cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model13&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# rubber mat, race, only random intercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model14&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# fullest model so far PLUS giving random slopes to the nuisance variables&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# rubber mat, race, random slope for cpe &amp;amp; random slope for the two main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# race variables:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model15&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# compare the no-rubber mat models (expect #15 to be best, with random slopes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for CPE and race - the most flexibility)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model9&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model10&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model11&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model12&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model13&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model14&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model15&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rubber_mat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random_cpe_slope&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;race&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random_race_slope&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;SAC_fix&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe_p_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model9&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model10&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model11&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model12&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model13&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model14&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model15&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.pv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe_t_stat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model9&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model10&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model11&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model12&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model13&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model14&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model15&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p.t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;cpe&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe_p_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe_p_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for space decided not to show this column, can just use t stat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe_p_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;knitr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;format&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;html&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;kable_styling&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;writeClipboard&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Here’s the key results from the overall winning model:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; summary(model15$lme)
Linear mixed-effects model fit by maximum likelihood
  Data: data 
       AIC      BIC    logLik
  5676.498 5797.326 -2818.249

Random effects:
 Formula: ~Xr - 1 | g
 Structure: pdIdnot
             Xr1      Xr2      Xr3      Xr4      Xr5      Xr6      Xr7
StdDev: 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787
             Xr8      Xr9     Xr10     Xr11     Xr12     Xr13     Xr14
StdDev: 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787
            Xr15     Xr16     Xr17     Xr18     Xr19     Xr20     Xr21
StdDev: 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787
            Xr22     Xr23     Xr24     Xr25     Xr26     Xr27
StdDev: 4.947787 4.947787 4.947787 4.947787 4.947787 4.947787

 Formula: ~1 + cpe + white_alone + hispanic | state_name %in% g
 Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr                
(Intercept)  1.503864 (Intr) cpe    wht_ln
cpe          6.544726 -0.761              
white_alone  1.055536 -0.437 -0.188       
hispanic     1.719388 -0.134 -0.152  0.180
Residual    43.651012                     

Correlation Structure: Exponential spatial correlation
 Formula: ~x + y | g/state_name 
 Parameter estimate(s):
    range 
0.8315562 
Variance function:
 Structure: fixed weights
 Formula: ~invwt 
Fixed effects:  list(fixed) 
                    Value Std.Error   DF   t-value p-value
X(Intercept)    -2.395289  0.271082 3051 -8.836033  0.0000
Xcpe            -1.862160  1.155263 3051 -1.611892  0.1071
Xwhite_alone     3.678207  0.191161 3051 19.241375  0.0000
Xhispanic       -0.544371  0.327396 3051 -1.662729  0.0965
Xhispanic_multi  1.751862  4.307237 3051  0.406725  0.6842
Xs(x,y)Fx1       0.202288  0.093536 3051  2.162664  0.0306
Xs(x,y)Fx2       0.429865  0.131173 3051  3.277087  0.0011
 Correlation: 
                X(Int) Xcpe   Xwht_l Xhspnc Xhspn_ X(,)F1
Xcpe            -0.761                                   
Xwhite_alone    -0.461 -0.164                            
Xhispanic       -0.178 -0.053  0.161                     
Xhispanic_multi -0.002 -0.114  0.101 -0.278              
Xs(x,y)Fx1      -0.051  0.030  0.028 -0.032  0.157       
Xs(x,y)Fx2      -0.121  0.101  0.083  0.000 -0.081  0.350

Standardized Within-Group Residuals:
       Min         Q1        Med         Q3        Max 
-6.1786290  0.2312745  0.4830700  0.8228585  5.5847101 

Number of Observations: 3107
Number of Groups: 
                g state_name %in% g 
                1                50 
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;Looking at that table of fixed effects, we see the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;white_alone&lt;/code&gt; variable is definitely significant, as is the rubber mat &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(x,y)&lt;/code&gt; spatial correlation absorber. So we could tentatively conclude that this suggests that whiteness strongly contributed to vote for Trump; that depression incidence didn’t contribute anywhere near as much; and there was a strong spatial correlation not explained by either whiteness or depression.&lt;/p&gt;

&lt;p&gt;Looking for a way to summarise all this I came up with this chart of the partial relationship of whiteness and of depression incidence to vote for Trump; &lt;em&gt;after&lt;/em&gt; controlling for the other things in the final model:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286c-partials.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286c-partials.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That was produced with this code, which involved fitting a new model to produce the residuals after controlling for race but not controlling for depression (once combination that hadn’t yet been done in the frenzy of model-fitting above):&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#--------------------partial charts-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# model 8[[1]] is the full model except for race. we also need a full model except&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for depression (cpe). then we will use the residuals from each for some charts&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model17&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gamm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic_multi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;random&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hispanic&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correlation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;corExp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;form&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Difference between this and model 15 is no depression variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model17&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model15&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after_cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;residuals&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;white_alone&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after_cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suffix&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;m&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e-6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage of county that is &apos;white&apos; as its only race&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Residual vote for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;After controlling for counties&apos; depression incidence&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Partial relationship of &apos;whiteness&apos; and Trump vote&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Total votes, 2024:&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after_race&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;residuals&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model17&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after_race&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suffix&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;m&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e-6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Incidence of diagnosed depression in each country&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Residual vote for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;After controlling for counties&apos; racial composition&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Partial relationship of depression incidence and Trump vote&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Total votes, 2024:&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: voting from tonmcg, depression from CDC, race from US Census Bureau; analysis by freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s it for today. I hope that’s of interest for someone, either as a messy but realistic modelling strategy case study, or for those interested in the specific issue of depression and the 2024 election.&lt;/p&gt;
</description>
				<pubDate>Fri, 03 Jan 2025 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2025/01/03/depression-and-vote-again</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2025/01/03/depression-and-vote-again</guid>
			</item>
		
			<item>
				<title>Depression incidence by county and vote for Trump by @ellis2013nz</title>
					<description>&lt;p&gt;A &lt;a href=&quot;https://bsky.app/profile/mchinn.bsky.social/post/3ldwkfv7uz22v&quot;&gt;skeet floated across my Bluesky feed&lt;/a&gt; that looked at the cross-sectional relationship between incidence of depression in 2020 and voting for Trump in the 2024 Presidential election. The data in the skeet and immediate blog post was at state level, but the hypothesis of interest in an article that sparked all this was an individual one (are depressed people voting for Trump). I don’t have the individual level microdata that might help explore the actual hypothesis, but I was surprised to see that the state-level regression had a significant evidence of an effect, and wondered if this would continue at the county level, which still has relatively accessible data.&lt;/p&gt;

&lt;p&gt;This also led me down an interesting but familiar rabbit hole of multilevel modelling in the presence of a spatial correlation nuisance.&lt;/p&gt;

&lt;h2 id=&quot;ordinary-least-squares&quot;&gt;Ordinary Least Squares&lt;/h2&gt;

&lt;p&gt;Well, it turns out depression at the county level does correlate with voting for Trump, as we can see with this first, simple chart which shows the expected vote based on a model fit with ordinary least squares (OLS):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-ols.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-ols.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’ll be fitting some more fancy models and getting better charts further down, but the basic message is the same as in this one - counties with higher incidence of depression had a higher proportion of their vote going to Trump than was the case with counties with lower levels of depression.&lt;/p&gt;

&lt;p&gt;Before I say anything else or show any code, let’s get straight that this is very possibly not a causal link. In fact there are at least three things that are plausibly happening here:&lt;/p&gt;

&lt;ol&gt;
  &lt;li&gt;People who are more depressed were more likely to vote for Trump (or less likely to turn up to vote against him, which given voluntary voting, has a similar result although for importantly different reasons)&lt;/li&gt;
  &lt;li&gt;People (who may themselves be not depressed) who are in areas with lots of depressed people around them were more likely to vote for Trump (eg because voters think “Trump will be able to do something about all the depressed people around here”)&lt;/li&gt;
  &lt;li&gt;Some underlying factor (eg economic, social or cultural conditions) that leads to some areas having higher rates of depression also leads to higher votes for Trump, through some other mechanism&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;My expectation is that #3 is the more likely explanation, but I personally don’t actually have evidence to choose between them. Nor are these hypotheses mutually exclusive; two or all of them might be true at once.&lt;/p&gt;

&lt;p&gt;OK here’s the code that gets that data and produces the first chart and a simple model with a statistically significant effect:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readxl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mgcv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# county level prevalence of depression at (have to hit the &apos;download&apos; button)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://stacks.cdc.gov/view/cdc/129404&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_excel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cdc_129404_DS1.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;skip&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;2024_US_County_Level_Presidential_Results.csv&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://raw.githubusercontent.com/tonmcg/US_County_Level_Election_Results_08-24/refs/heads/master/2024_US_County_Level_Presidential_Results.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;votes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;2024_US_County_Level_Presidential_Results.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;votes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;inner_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;CountyFIPS code&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Crude Prevalence Estimate`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Age-adjusted Prevalence Estimate`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# what was missed in this join?&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;votes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;anti_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;CountyFIPS code&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# 37 counties in Alaska, 9 and Connecticut and 7 in DC. Will ignore these&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for my purposes.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#========================modelling==================&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#----------Ordinary Least Squares------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# note several things could be happening here:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# - depressed people makes you vote for Trump&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# - being around depressed people makes you vote for Trump&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# - some underlying condition (eg economic) both leads to higher depression&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#   and more likely to vote for Trump. This seems the most likely.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: data from tonmcg and CDC; analysis by freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw chart:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crude prevalence estimate of depression&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage vote for Trump in 2024 election&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Line is ordinary least squares fit to all county data together&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Counties with more depression voted more for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Excerpt from the results:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;Coefficients:
            Estimate Std. Error t value Pr(&amp;gt;|t|)    
(Intercept)  0.40002    0.01861   21.49   &amp;lt;2e-16 ***
cpe          1.27547    0.08712   14.64   &amp;lt;2e-16 ***
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;h2 id=&quot;generalized-linear-model-with-a-quasibinomial-response&quot;&gt;Generalized linear model with a quasibinomial response&lt;/h2&gt;
&lt;p&gt;Now, I wanted to improve this for all sorts of reasons, although I suspected it was actually good enough for pragmatic purposes - case proven really, that counties with more depressed people voted more for Trump. Proven enough to justify the further work with additional data needed to explore why. But I had some statistical loose ends to tidy up. First of which is that vote is a proportion, and it feels icky to use OLS (which can send values above 1 and below 0) to model a proportion when we have generalized linear models (GLMs) designed for the job and easily available.&lt;/p&gt;

&lt;p&gt;I didn’t want to use a straight logistic regression because the county-level data is far more dispersed than would be expected if it really were individuals making their own voting decisions. But a GLM with a quasi-binomial response keeps the link function used in logistic regression and the relationship of the mean and variance, while allowing the variance to be larger (or smaller) by some consistent ratio. Here’s what I get with that GLM:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-glm.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-glm.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;…created with this code. Note that we now have started to weight counties based on their overall number of voters, which makes particular sense for a binomial or similar family response. I suspect this is one of the key issues driving the line vertically down, compared to the OLS version. The other key difference of course is that now it is slightly curved, as the ‘linear’ in a GLM is on the link scale, not the scale the response is originally observed on and used for this chart.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------Quasibinomial GLM----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw chart:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_ribbon&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crude prevalence estimate of depression&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage vote for Trump in 2024 election&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Line is generalized linear model with quasibinomial response, fit to all county data together&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Counties with more depression voted more for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Here’s an excerpt from that summary of model2:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;Coefficients:
            Estimate Std. Error t value Pr(&amp;gt;|t|)    
(Intercept) -1.82178    0.06823  -26.70   &amp;lt;2e-16 ***
cpe          9.25748    0.34153   27.11   &amp;lt;2e-16 ***
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;We see &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cpe&lt;/code&gt; (crude prevalence estimate, ie not the age-standardised one) has very definitely “significant” evidence that it isn’t zero, with a point estimate of 9.3 and a standard error of only 0.3.&lt;/p&gt;

&lt;h2 id=&quot;introducing-a-state-effect&quot;&gt;Introducing a state effect&lt;/h2&gt;
&lt;p&gt;One thing that all the world knows is how spatially-based are US politics. Everything is thought of in terms of states, in particular, and smaller areas sometimes. It follows naturally from the ways that US politics is discussed that we should use a multi-level ie mixed-effects model with state as a random intercept, when looking at something like the overall relationship between depression and voting. In other words, we have to let the vote for Trump in any state vary for all the state-specific things that aren’t in our model, and see if after doing that we still get an overall  (constant nation-wide) relationship between depression and voting in the counties within each state.&lt;/p&gt;

&lt;p&gt;I often reach to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;lme4&lt;/code&gt; library by Bates, Bolker and Walker as my starting point for mixed effects models and this is the results in this case:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-glmer.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-glmer.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Note we have a bunch of parallel (on link scale) lines, one per state, with their height varying by the state random effect. I love the effect of this chart, but unfortunately &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;lme4::glmer&lt;/code&gt; which is used in this case doesn’t let us use a quasibinomial family response; we have to use a binomial response which forces the variance to equal &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;p(1-p)/n&lt;/code&gt;, not just be proportional to it. The net result is that the confidence intervals are much narrower than is justified.&lt;/p&gt;

&lt;p&gt;An alternative way to fit a similar model is the the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam()&lt;/code&gt; function from the amazing &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;mgcv&lt;/code&gt; library by Simon Wood. There’s &lt;a href=&quot;https://fromthebottomoftheheap.net/2021/02/02/random-effects-in-gams/&quot;&gt;a great discussion of this&lt;/a&gt; on Gavin Simpson’s blog. By specifying a spline around a categorical factor like &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;s(state_factor, bs = &apos;re&apos;)&lt;/code&gt; (‘re’ stands for random effect) we can use &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam()&lt;/code&gt; to add random intercepts while using the full range of families available to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam()&lt;/code&gt; including quasibinomial. That gives us this alternative version of the last model; this time with much fatter (and realistic) confidence intervals!:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-gam.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-gam.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The confidence intervals are now very fat. But there’s still no doubt about the significance of the evidence of the relationship of the crude prevalence of depression on voting behaviour, even after allowing a random state-level intercept.&lt;/p&gt;

&lt;p&gt;Here’s the code for both the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;glmer&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam&lt;/code&gt; versions of this random state effect model:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------------With state random effect with lmer4::glmer--------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lme4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glmer&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;binomial&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# note can&apos;t use quasibinomial family with glmer so we aren;t really dealing&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# properly with the overdispersion. what to do about that? Confidence intervals&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# will be too narrow. Various alternatives posisble.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_ribbon&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crude prevalence estimate of depression&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage vote for Trump in 2024 election&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Lines are logistic regression with state-level random intercept effect (confidence intervals are too narrow)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Counties with more depression voted more for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------state random effect with mgcv::gam--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# gam lets us have a random effect and a wider range of families&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# see https://fromthebottomoftheheap.net/2021/02/02/random-effects-in-gams/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# must be a factor to use as a random effect in gam():&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;re&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# note standard error for cpe is much higher 0.6224, compared to 0 .0107&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds5&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds5&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_ribbon&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crude prevalence estimate of depression&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage vote for Trump in 2024 election&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Lines are quasibinomial generalized additive model with state-level random intercept&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Counties with more depression voted more for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Excerpt from the gam results (the better of the two, with fatter confidence intervals but still very significantly different from zero):&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;per_gop ~ cpe + s(state_name, bs = &quot;re&quot;)

Parametric coefficients:
            Estimate Std. Error t value Pr(&amp;gt;|t|)    
(Intercept)  -3.2466     0.2802  -11.59   &amp;lt;2e-16 ***
cpe          16.3333     0.6224   26.24   &amp;lt;2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                edf Ref.df     F p-value    
s(state_name) 48.44     49 20.54  &amp;lt;2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.345   Deviance explained = 40.3%
GCV = 3333.9  Scale est. = 3387.6    n = 3107
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;h2 id=&quot;allowing-for-spatial-autocorrelation&quot;&gt;Allowing for spatial autocorrelation&lt;/h2&gt;
&lt;p&gt;OK we’re not quite there yet. The final thing at the back of my mind is that so far, we have treated counties as though they are independent, equally valuable observations. This is pretty common in these types of cross-sectional regression, but it’s not fair; it exaggerates the value of each county data point. The values of depression and Trump vote in two adjacent counties are, frankly, not equally valuable independent observations. Once you have one, you have a good chance of predicting the values in the county next door. By failing to account for this, we are treating our data as more valuable than it is, which translates to over-confidence in our inferences and confidence intervals that are too narrow.&lt;/p&gt;

&lt;p&gt;How to fix this? Well, there are numerous ways, but my favourite pragmatic correction is to add a fixed effect that is a sort of rubber sheet laid across the geography of interest. This can be done with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gam()&lt;/code&gt; by including in the formula something like &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;+ s(x, y)&lt;/code&gt;, where x and y are the centrepoints of the regions from where we have sourced observations.&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;Note - the other ways of correcting for this all seem to me to be &lt;em&gt;way&lt;/em&gt; more complicated than this. Perhaps they are better! One thing I’m sure, it’s much better to do it my way than to ignore spatial autocorrelation altogether, which seems to be the near-universal approach. How often do you see an adjustment for spatial autocorrelation in a regression of US counties or states?&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Now, even my pragmatic method is going to be a bother, because we need to get spatial data that isn’t available in our sources so far. The next chunk of code downloads shapefiles of all the US counties, calculates the centroids of each, and puts it into shape to join to our existing data.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------gam, spatial, state effect--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# each county isn&apos;t really an independent data point, as counties next to eachother&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# probably have lots in common. A great thing about gam is that not only can we&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# have a quasibinomial family, we can do gam core business of adding in splines,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# including a two dimensional &quot;rubber mat&quot; that effectively knocks out our &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# spatial correlation problem for us.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# but first we need to know the centroids of all the counties:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cb_2023_us_county_500k.zip&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;https://www2.census.gov/geo/tiger/GENZ2023/shp/cb_2023_us_county_500k.zip&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cb_2023_us_county_500k.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unzip&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_read&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cb_2023_us_county_500k.shp&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_cent&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;counties&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_centroid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_coordinates&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_cent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_cent&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_cent&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# combine the two digit state code with the 3 digit county code:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_fips&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;STATEFP&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;COUNTYFP&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check that we have successfully re-created the country_fips on same basis&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# as our voting and depression data:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_cent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;NAME&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;county_cent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;county_fips&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check the county centres are where we expect. Note Alaska still missing&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# (because voting data is not by country so lost on the first join)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_map&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Centres of counties after merging data&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This gives us this plot to check that we haven’t (for example) mangled states, or latitude and longitude:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-counties-map.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-counties-map.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Very nice. Now we can fit the model, with this code&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;model6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gam&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;re&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;quasibinomial&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_votes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the spatial rubber mat that is correcting for spatial correlation for us;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# scheme=1 is what makes it draw a perspective plot rather than contour or&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# heatmap):&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scheme&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Higher vote for GOP&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gives us this ‘rubber mat’ that I’ve included. The bit in the bottom left is Hawaii.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-rubber-sheet.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-rubber-sheet.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Basically this is going to absorb the variance that can be explained by mere proximity of one county to another. Which means that what is left to be explained by either the state random effect, or the relationship between depression and voting, is a fairer estimate.&lt;/p&gt;

&lt;p&gt;Now, when we try to visualise this, the lines ar emuch squigglier than our previous charts. So I’ve chosen to put this into a faceted plot with a panel for each state. Here’s the end result:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0286-gam-spatial.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0286-gam-spatial.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;… created with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;preds6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;predict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;combined2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds6&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;preds6&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se.fit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fit&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.96&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cpe&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;per_gop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_ribbon&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crude prevalence estimate of depression&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Percentage vote for Trump in 2024 election&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Grey ribbons are 95% confidence intervals from quasibinomial generalized additive model with spatial effect and state-level random intercept effect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Counties with more depression voted more for Trump&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;state_name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, I think this is the best model I’ve fit to this data. The prevalence of depression (&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cpe&lt;/code&gt;) is still very much a significant (definitely non-zero) effect, and the large magnitude of this is comparable to the simpler models used so far. But we’ve got confidence that this isn’t just an artefact of the spatial closeness of counties, or of other non-included state level effects. So the extra work is worth it.&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;Parametric coefficients:
            Estimate Std. Error t value Pr(&amp;gt;|t|)    
(Intercept)  -2.7972    20.7488  -0.135    0.893    
cpe          15.5908     0.6778  23.001   &amp;lt;2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                edf Ref.df     F p-value    
s(x,y)        28.39  28.94 7.995  &amp;lt;2e-16 ***
s(state_name) 49.00  49.00 7.949  &amp;lt;2e-16 ***
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;That’s all folks. I guess the lesson here (not that these blogs are lessons) is really just to allow for spatial effects: both categorical ones that fit into a mixed effects / multilevel paradigm (eg random intercept at state level); and those that are more subtle spatial autocorrelation. Really, this should be done much more.&lt;/p&gt;

&lt;p&gt;And &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;mgcv::gam()&lt;/code&gt; is a great tool to address both of these at once.&lt;/p&gt;
</description>
				<pubDate>Mon, 23 Dec 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/12/23/depression-and-vote</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/12/23/depression-and-vote</guid>
			</item>
		
			<item>
				<title>Death rates by cause of death by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;overview&quot;&gt;Overview&lt;/h2&gt;
&lt;p&gt;OK, so the USA health system is very much in the news. One thing that has been getting a lot of coverage, not least because it was explicitly referred to in a written statement from the guy who seems to have murdered UnitedHealthcare CEO Brian Thompson, is the low life expectancy in the USA compared to the amount spent in that country on health services. So I have seen several variants on this chart going around:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-life-exp-scatter.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-life-exp-scatter.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;It’s more common for the charts you see to focus on the upper right quadrant of that chart, relatively wealthy countries, in which case the USA is an even more prominent outlier. I like to see the whole world as context, not least because it shows other countries (including a couple in my working life patch, Tuvalu and Nauru) have as large a disappointing gap in life expectancy from what you might hope given income, spend on health, etc. There’s also some countries - like Sri Lanka, Thailand and Bangladesh - with relatively high life expectancy given income and health spend.&lt;/p&gt;

&lt;p&gt;Now, it only takes a moment’s reflection to realise that this isn’t just a matter of spend quality, effectiveness and efficiency of the health system. Life expectancy is heavily determined by other things. Most obviously:&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;if people can’t afford enough food, they die earlier for reasons related to malnutrition&lt;/li&gt;
  &lt;li&gt;if people eat too much and too unhealthy types of food, they die earlier from diseases related to obesity (diabetes, heart attacks, etc)&lt;/li&gt;
  &lt;li&gt;if housing is poor they are more likely to get diseases like tuberculosis&lt;/li&gt;
  &lt;li&gt;when water and sanitation is unsafe, people (including children, with a big impact on life expectancy) will die of cholera, diarrhea, typhoid, etc.&lt;/li&gt;
  &lt;li&gt;if high numbers of young people commit suicide, die in accidents or are murdered it will directly and substantially reduce life expectancy.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;These are all “public health” issues, but the causes and solutions are (mostly) not to be found in health expenditure but in general economic and social conditions. I say ‘mostly’ because of course health interventions can indeed make a difference to people with diabetes, mental health problems contemplating suicide, people who have been assaulted but picked up by the emergency services before dying, etc. But these are very much “ambulance at the bottom of the cliff” measures rather than a fence at the top.&lt;/p&gt;

&lt;p&gt;Of course all this is well known in the right circles, and in fact an article had come out some weeks before Thompson’s murder showing how the low life expectancy in the USA was materially impacted by deaths associated with &lt;a href=&quot;https://www.aamcresearchinstitute.org/our-work/data-snapshot/narrowing-gap&quot;&gt;alcohol, drugs and firearms&lt;/a&gt;:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“In 2022, there were more than 48,000 firearm-related deaths (more than half were recorded as suicides…); nearly 108,000 drug-related deaths; and more than 51,000 alcohol-induced deaths.While these numbers made up a small fraction of the nearly 3.3 million deaths in 2022 in the United States, they disproportionately affected children and younger adults — and, as a result, lowered the U.S. life expectancy at birth. If these deaths were eliminated (and other causes of death remained the same), life expectancy at birth would increase by 1.6 years.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;So remembering that when I read the ‘manifesto’ of Thompson’s alleged killer prompted me to look into death rates by cause of death myself. There is good data on this from the OECD, and not just for OECD countries. Here is my best chart I came up with illustrating the issues for the USA:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-us-1719-lim.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-us-1719-lim.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;As you can see, the horizontal axis in that  chart shows the difference in the age-standardised death rate between the USA and a set of comparison countries. The comparison countries I chose are those with a higher life expectancy than the USA that also had cause of death data available in the OECD database. In effect, these are the countries the USA might try to catch up to, if it were seeking to improve its life expecancy.&lt;/p&gt;

&lt;p&gt;What we see in this chart is that indeed, the cause of death that is most disproportionately high in the USA is ‘external’ - assaults, accidents and suicides. With relation to cancer - one of the diseases where the health system is most important in terms of treatment prolonging life - the USA has unusually low death rates. (Also for the mysterious ‘symptoms, signs, ill-defined causes’ cause of death, which unfortunately seems to be a measurement anomaly that is much more present in a small number of countries than others).&lt;/p&gt;

&lt;p&gt;The second most relatively high cause of death for the US compared to its comparator countries is diseases of the nervous system, which includes Alzheimer’s. Then we have circulatory problems (eg heart attacks) and respiratory (influenza, pneumonia etc).&lt;/p&gt;

&lt;p&gt;I drew lots more charts, some of which are highlighted in the rest of this blog, but if you’re interested in them systematically here are PDFs that include the chart for each country available for different time periods: &lt;a href=&quot;https://freerangestats.info/img/0285-all-countries-2017-2019.pdf&quot;&gt;2017 to 2019&lt;/a&gt; and for &lt;a href=&quot;https://freerangestats.info/img/0285-all-countries-2021.pdf&quot;&gt;2021&lt;/a&gt;. Adobe Acrobat on full screen mode (crtl-l) works well to flick through these. I considered making a Shiny app to let people explore this but decided the PDFs actually work better.&lt;/p&gt;

&lt;h2 id=&quot;more-detail&quot;&gt;More detail&lt;/h2&gt;

&lt;p&gt;OK that was the no-code summary but here is some more detail including how I drew those charts, some interesting variants including for other countries, and a bit of an exploration of which countries suffer most from which causes of death.&lt;/p&gt;

&lt;h3 id=&quot;drawing-the-life-expectancy-scatter-plot&quot;&gt;Drawing the life expectancy scatter plot&lt;/h3&gt;

&lt;p&gt;First, here is the code for the life expectancy chart. You can see I am drawing the data from the World Bank’s World Development Indicators. There’s quite a lag in reporting both these figures which comes from the complexity of collecting the data and analysing them; 2021 was the year that had the most complete data for a large number of countries.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#===============Setup==================&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rsdmx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;janitor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggbiplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WDI&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggrepel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; Collapse a character vector&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param ... passed through to \code{paste()}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @details&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; this is just a convenience wrapper around paste()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pastec&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;collapse&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;, &apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#===============Part 1 - life expectancy compared to health spend============&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------------life expectancy--&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# had to explore a bit to find the indicators to use&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# w &amp;lt;- WDIsearch(&quot;health&quot;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# View(w)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;WDI&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicator&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;health_spend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;SH.XPD.CHEX.PC.CD&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;SP.DYN.LE00.IN&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# find the &apos;countries&apos; that are really groupings of countries         &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ZW&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ZM&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ZA&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;XK&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
             &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;^[XZ]&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;EU&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V3&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V4&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;OE&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distinct&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# remove regions and groupings (which have ISO2 code beginning with Z or X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# but leave in 4 countries with real codes beginning Z or X)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# and also knock out some specific things like EU, OECD average, etc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;((&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ZW&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ZM&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ZA&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;XK&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;^[XZ]&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;iso2c&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;EU&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V3&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;V4&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;OE&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;drop_na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# originally i picked the latest year for each country, but in so many&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# cases it was 2021 I decided to just force it to be 2021&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# going ot use this model to identify &apos;outlier&apos; countries worth labelling&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;health_spend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# add the residuals of hte model and label any countries not from 2021 (this&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# label change was more necessary when I had multiple years in the data, but&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# I&apos;ve left it in for future reference)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;residuals&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_yr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{country}, {year}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Draw scatter plot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;health_spend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lm&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lightgreen&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# label some interesting countries:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;health_spend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8500&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;84&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;55&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-10&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_yr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# annoying ggrepel couldn&apos;t put Norway in a good position automatically&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# so I am going to move it ad hoc:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Norway&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;health_spend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.189&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;health_spend&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;125&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Health expenditure per capita (US dollars, logarithmic scale)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Life expectancy&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Health expenditure is associated with higher life expectancy&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;USA is a high income outlier, but many poorer countries also have poor life expectancy given health spend per person.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: data for 2021 from the World Development Indicators; analysis by freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Probably the only noteworthy thing in that code was that I explicitly fit a model of expected life expectancy given a level of health spending, purely to use the residuals from that model to identify outliers to draw on the chart. An interesting example of a model servicing a visualisation rather than the other way around.&lt;/p&gt;

&lt;h3 id=&quot;wrangling-cause-of-death-data-and-metadata-and-drawing-the-cause-of-death-plots&quot;&gt;Wrangling cause of death data and metadata and drawing the cause of death plots&lt;/h3&gt;

&lt;p&gt;Next, how did I get the data for the cause of death from the OECD? It’s all available in OECD.Stat, and you can use the &lt;a href=&quot;https://data-explorer.oecd.org/vis?_ga=2.241262853.682686041.1734140631-789640812.1734140631&amp;amp;tm=cause%20of%20death&amp;amp;pg=0&amp;amp;snb=13&amp;amp;df[ds]=dsDisseminateFinalDMZ&amp;amp;df[id]=DSD_HEALTH_STAT%40DF_COM&amp;amp;df[ag]=OECD.ELS.HD&amp;amp;df[vs]=1.0&amp;amp;dq=.A......_T.STANDARD....&amp;amp;pd=2015%2C&amp;amp;to[TIME_PERIOD]=false&quot;&gt;Data Explorer&lt;/a&gt; to find the data you want. Once there you can use the ‘Developer API’ button to get a link to both the metadata and the data itself. We need the metadata in this case because the data download just has codes for cause of death, and we want the full names.&lt;/p&gt;

&lt;p&gt;The meta data is extensive and comes in the form of a complex nested XML object, in SDMX format. We use this at my work but I personally am not an expert so there was a bit of fiddling to find and extract the codelists that I wanted.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#===============Part 2 - comparing various cause of death numbers=========&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------metadata for causes of death------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;md&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://sdmx.oecd.org/public/rest/dataflow/OECD.ELS.HD/DSD_HEALTH_STAT@DF_COM/1.0?references=all&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# English names for each of the code lists&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;md&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codelists&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codelists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;err&quot;&gt;\&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;en&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# noting number 7 is Cause of death and 2 is area ie country&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That last command extracts from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;md&lt;/code&gt; S4 object (hold your breath): the English element from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Name&lt;/code&gt; slot from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;codelists&lt;/code&gt; slot from the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;codelists&lt;/code&gt; slot (yes, that’s nested &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;codelists&lt;/code&gt; slots). It gives us this list of the codelists available:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; sapply(md@codelists@codelists, \(x)x@Name$en)
 [1] &quot;Codelist for age&quot;                          
 [2] &quot;Codelist for reference areas and geocoding&quot;
 [3] &quot;Codelist for concept \&quot;Sex\&quot;&quot;              
 [4] &quot;Codelist for Unit of measure&quot;              
 [5] &quot;Codelist for calculation methodology&quot;      
 [6] &quot;Codelist for cancer site&quot;                  
 [7] &quot;Cause of death&quot;                            
 [8] &quot;Codelist for diseases&quot;                     
 [9] &quot;Gestation period threshold&quot;                
[10] &quot;Health status&quot;                             
[11] &quot;Codelist for health status measures&quot;       
[12] &quot;Socio-economic status&quot;                     
[13] &quot;Decimals&quot;                                  
[14] &quot;Code list for concept \&quot;Frequency\&quot;&quot;       
[15] &quot;Observation Status&quot;                        
[16] &quot;Code list for the Unit Multiplier&quot; 
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;I only need numbers 2 and 7, which I extracted in this hacky way:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; convenience function for extracting codelists from sdmx metadata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; This is not very robust but does the job for today&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract_codes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;metadata&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;description_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;description&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;md&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codelists&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codelists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Code&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lookup&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;code&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;err&quot;&gt;\&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;description&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;err&quot;&gt;\&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unlist&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lookup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;description_name&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lookup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_codes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract_codes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;md&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cause_of_death&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;area_codes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract_codes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;md&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;country&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# country is still a list within a list so need to extract further&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;err&quot;&gt;\&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;en&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]}))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gives me code-to-description lookup tables for both the cause of death and area (ie country) codes.&lt;/p&gt;

&lt;p&gt;Now, the cause of death codes are hierarchical and some of them are parents of others. For example, tuberculosis is a subset of ‘Certain infectious and parasitic diseases’ and if you include them both you are in a sense double-counting. I decided for my purposes I only wanted the highest level codes. This must be encoded in the SDMX somewhere (because the OECD Data Explorer shows it correctly), but I couldn’t find it in the metadata. For example, the snippet below just returns all NAs for the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;parentCode&lt;/code&gt; slot in the metadata that I would have thought should indicate a parent when one existed (like for Tuberculosis).&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# the codeelists Code should have parentCode to show the hierarchical nature&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# of the cause of death codes, but it doesn&apos;t seem to be there&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;md&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codelists&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;codelists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Code&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;err&quot;&gt;\&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;@&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;parentCode&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;In the end I had to make by hand a vector of the highest level codes, by printing to the console all the codes we had and eliminating the ones that the Data Explorer showed me were children of others:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# you are meant to be able to get parent codes from this but they all look to be&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# NA (see above( so I couldn&apos;t see how to do this. So instead I&apos;ve made a list&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# by hand of all those at the top level of hte classification&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;high_level_cod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Certain infectious and parasitic diseases&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Neoplasms&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the blood and blood-forming organs&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Endocrine, nutritional and metabolic diseases&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Mental and behavioural disorders&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the nervous system&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the respiratory system&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the circulatory system&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the digestive system&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the skin and subcutaneous tissue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the musculoskeletal system and connective tissue&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the genitourinary system&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Certain conditions originating in the perinatal period&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Congenital malformations, deformations and chromosomal abnormalities&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Symptoms, signs, ill-defined causes&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;External causes of mortality&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Codes for special purposes: COVID-19&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;OK, so now we can import the data, and join it to the codelist looks I made earlier:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# The actual death rates (standardised by age)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://sdmx.oecd.org/public/rest/data/OECD.ELS.HD,DSD_HEALTH_STAT@DF_COM,1.0/.A..DT_10P5HB.._T...STANDARD....?startPeriod=2015&amp;amp;dimensionAtObservation=AllDimensions&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_codes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;death_cause&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;code&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;area_codes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ref_area&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;code&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;It took me a while but I realised eventually I had a challenge that not all countries had data for all years. The first few versions of charts I drew were misleading in comparing (for example) countries with data only up to 2019 with countries like the USA that had data up to 2021, or Australia which has data to 2022. In a stable time when cause of death doesn’t change much year by year this might not matter, but that doesn’t describe the 2020s! My mistake made it look like the USA was worse relatively speaking in terms of Covid-19 deaths than they actually are. Here’s some summaries of how many countries have data each year:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; # A problem, its different sets of countries for each year
&amp;gt; count(death_rates, time_period)
# A tibble: 8 × 2
  time_period     n
  &amp;lt;chr&amp;gt;       &amp;lt;int&amp;gt;
1 2015         2144
2 2016         2154
3 2017         2059
4 2018         2057
5 2019         2047
6 2020         1908
7 2021         1414
8 2022          439
&amp;gt; 
&amp;gt; death_rates |&amp;gt;
+   count(country, time_period) |&amp;gt;
+   count(country, sort = TRUE, name = &quot;number_years&quot;) |&amp;gt;
+   count(number_years, name = &quot;number_countries&quot;)
# A tibble: 6 × 2
  number_years number_countries
         &amp;lt;int&amp;gt;            &amp;lt;int&amp;gt;
1            2                3
2            4                1
3            5                3
4            6               11
5            7               19
6            8                9
&amp;gt; 
&amp;gt; death_rates |&amp;gt;
+   count(country, time_period) |&amp;gt;
+   count(time_period, sort = TRUE, name = &quot;number_countries&quot;) 
# A tibble: 8 × 2
  time_period number_countries
  &amp;lt;chr&amp;gt;                  &amp;lt;int&amp;gt;
1 2015                      45
2 2016                      45
3 2017                      43
4 2018                      43
5 2019                      43
6 2020                      39
7 2021                      29
8 2022                       9
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;Some countries have as few as 2 years of data (noting that I set 2015 as the starting point myself); most have 6 or 7.&lt;/p&gt;

&lt;p&gt;OK, so once I’d recovered from that fishhook, and had a bit of trial and error with drawing charts, I eventually put all my code doing this into this single-analysis (ie not portable, only for use in this script) function. You can see it takes arguments for the years being used, which countries to compare to, and different ways of colouring the bars. I only ended up using the version that colours them according to the all-country average value because that seemed to me to give extra information for minimal extra visual clutter; but the alternative is ok too.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; Draw a bar chart for one country showing its death rates in comparison to an average&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NULL&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NULL&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_width&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2017&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2019&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;country_average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;this_country&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;match.arg&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Could not find {the_country} in {paste(years, collapse = &apos;, &apos;)}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.null&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;check&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;check&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;warning&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Could not find {pastec(comparison_countries[!check])}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.null&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison: {pastec(comparison_countries[check])}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  

  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# some countries won&apos;t have data for all years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bad_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bad_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;warning&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Dropping {pastec(bad_countries)} as missing some years&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bad_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;This is fatal for drawing this graph...&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bad_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bad_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;one_wide&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;high_level_cod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the nervous system&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the nervous system (includes Alzheimer&apos;s and Parkinson&apos;s)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Mental and behavioural disorders&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Mental and behavioural disorders (includes Dementia)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;External causes of mortality&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;External causes of mortality (includes accidents, assaults and self-harm)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Neoplasms&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Neoplasms (eg cancer)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;country_average&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;one_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;country_average&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;30&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_diff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_average&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_average&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_average&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;one_wide&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distinct&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_average&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_viridis_c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Age standardised deaths per 100,000 in {the_country} compared to unweighted average of {length(unique(death_rates2$country))} countries&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: OECD. Analysis by freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Where do extra deaths in {the_country} come from in {pastec(years)}?&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;width&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_width&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;this_country&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Rate in {the_country}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country_average&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;All-country average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now that I’ve got a function to draw a chart for one country’s cause of death in a given subset of years, compared to the other countries that all also have data for that subset of years, I’m able to start producing graphs! Here’s the first one I actually made, even though I haven’t highlighted it in the “Overview” part of this post. USA 2017 to 2019, compared to all countries available:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-us-1719.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-us-1719.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The difference between this and the one I used in the overview is the comparison to all countries available, not just those with higher life expectancy than the USA which I’ll come to later. I think this makes it a bit less useful for the purpose of our fundamental question which is “what is wrong with the USA”, although the results aren’t radically different.&lt;/p&gt;

&lt;p&gt;I also looked at the USA in 2021, during the time of Covid-19:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-us-21.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-us-21.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;We see that for this period at least, the USA experienced a lower death rate from Covid-19 than other countries were.&lt;/p&gt;

&lt;p&gt;Those two USA-compared-to-all-countries charts were just made with these short snippets of code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;United States&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;country_average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison to all countries with available cause of death rates.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;United States&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;To draw the PDFs with one page per country, I made use of the fact that each time you print a new chart to an open PDF device it adds a page, making this a very quick and easy way to store all the charts available with some combination of parameters:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;Cairo&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CairoPDF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0285-all-countries-2017-2019.pdf&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;width&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;height&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cause of deaths 2017-2019&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sort&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;try&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;country_average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dev.off&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Cairo&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CairoPDF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0285-all-countries-2021.pdf&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;width&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;height&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cause of deaths 2021&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sort&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;try&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;country_average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dev.off&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;For comparsions with just those countries that had better life expectancy than the USA, I went back to my data for that life expectancy scatter plot and extracted the countries from there. One of them (South Korea) needed to have its name changed by hand to work with the OECD data. Once we have that vector of countries, the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;bar_one_country()&lt;/code&gt; function Just Works; here is the code to draw charts for the USA, Australia and Estonia at different times:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------comparisons with a smaller set of countries--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;better_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;h2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;United States&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;life_exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Korea, Rep.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fixed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Korea&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;United States&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;better_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2017&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2019&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;United States&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;better_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;better_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2022&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bar_one_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Estonia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comparison_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;better_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2022&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;And here are the three charts (the United States 2017 to 2019 chart has already been shown in the ‘Overview’ section of this blog):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-us-21-lim.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-us-21-lim.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-aus-22-lim.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-aus-22-lim.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-est-22-lim.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-est-22-lim.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Australia’s highest relative cause of death being ‘endocrine, nutritional and metabolic diseases’ (eg diabetes), and with relatively low death rate from circulatory diseases.&lt;/p&gt;

&lt;p&gt;Estonia in stark contrast having a very high death rate from circulatory disease, compared to these particular high-life-expectancy countries.&lt;/p&gt;

&lt;h3 id=&quot;which-causes-are-associated-with-which-countries&quot;&gt;Which causes are associated with which countries?&lt;/h3&gt;

&lt;p&gt;Finally, I also scratched my curiousity itch on whether principal components analysis and visualisations could help create and communicate a sense of which countries are associated with which cause of death. This blog post is already very long so I won’t go into much detail on this. I’ll just say that the changing country availability by year was again a fishhook, and the main visualisaiton problem was the long names of the causes of death. In the end, I had to hide most of the labels of causes of death other than those that really made a contribution to the variation in the first two principal components.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------biplot--------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_biplot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;min_loading&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.25&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;death_rates&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;high_level_cod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;spread&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cause_of_death&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# decided not to scale the countries, but it wouldn&apos;t be insane; here&apos;s how to do it:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#all_countries_m &amp;lt;- t(scale(t(all_countries_m)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;row.names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsub&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Diseases of the &quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsub&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Certain &quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#colnames(all_countries_m) &amp;lt;- gsub(&quot; system&quot;, &quot;&quot;, colnames(all_countries_m))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;princomp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;loadings&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;row.names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;loadings&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Comp.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;min_loading&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Comp.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;min_loading&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;princomp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggbiplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;row.names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;all_countries_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;varname.color&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cause of death by country: {pastec(years)}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_biplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2017&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2019&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;min_loading&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_biplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;min_loading&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cod_biplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2021&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Here are the three charts drawn by that code, each one representing a different time period.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-biplot-1719.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-biplot-1719.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-biplot-20.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-biplot-20.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0285-biplot-21.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0285-biplot-21.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;It never pays to overthink a biplot. Just go with the flow and interpret them intuitively.&lt;/p&gt;

&lt;p&gt;Do these plots help? I think they do a bit. For example, it helps to see in a single chart - say, the one for 2017 to 2019 - the eastern European countries associated with circulatory system diseases, and Latin America and others with diabetes; cancer in the Balkans.&lt;/p&gt;

&lt;p&gt;That’s all for now. Hopefully this helps at least someone understand the variation in cause of death across countries, and how the USA lowish life expectancy is a more complex phenomenon than just being the fault of its health insurance industry.&lt;/p&gt;

</description>
				<pubDate>Sat, 14 Dec 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/12/14/us-death-rates</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/12/14/us-death-rates</guid>
			</item>
		
			<item>
				<title>Simulating Ponzi schemes by @ellis2013nz</title>
					<description>&lt;p&gt;I recently read John Cox’s 2018 book &lt;a href=&quot;https://www.amazon.com/Fast-Money-Schemes-Deception-Framing/dp/0253025605&quot;&gt;Fast Money Schemes: Hope and Deception in Papua New Guinea&lt;/a&gt; and gave it a rare (for me) five star rating. It’s fascinating research into those involved in the &lt;a href=&quot;https://en.wikipedia.org/wiki/U-Vistract&quot;&gt;U-Vistract&lt;/a&gt; pyramid scheme.&lt;/p&gt;

&lt;p&gt;At the time of writing U-Vistract had been bankrupt and exposed for years yet many of those who had ‘invested’ in it were still hoping to get their money back. Cox conducted in-depth interviews not just with victims but with some of those involved in running the scam. He successfully puts it all in a wider context, not just of Melanesian history with cults and cons but its place in global “prosperity theology” (good grief), international financial cons such as the “Nigerian Prince scam” (not a Ponzi scheme) and the biggest Ponzi scheme of all, Bernie Madoff’s multi-billion dollar rip-off of some of the more privileged, educated and well-connected people on the planet.&lt;/p&gt;

&lt;p&gt;U-Vistract (which still has its true believers, apparently) targeted not pre-modern villagers in the Papua New Guinea highlands but the growing middle class that lives in the formal economy and is connected to or part of the governing elites. Part of the appeal of Cox’s book is that it is “post-village ethnology”, looking at society as it is now, not just interested in some pre-contact tribal life.&lt;/p&gt;

&lt;p&gt;Anyway, it got me thinking about Ponzi schemes. U-Vistract’s mode of operation was to get people to invest 100 or so kina and promise a 100% return per month. In fact, part of U-Vistract’s propaganda appeal was a criticism of traditional banks for giving you only 5% per year or so of the full 100% that must exist somewhere. This got me thinking, how fast does a Ponzi scheme need to grow new investor/victims to sustain a promised rate of return of doubling their money &lt;em&gt;every month&lt;/em&gt;? How long could such a scheme plausibly last?&lt;/p&gt;

&lt;p&gt;Of course, this depends not just on how many new investors come in, but other factors such as how often they seek to withdraw their money. In the U-Vistract world several early investors - often well known and apparently respectable - made early fortunes. Some subsequent investors sought more modestly just to withdraw 100 kina a month (still a stunning return on an initial investment of the same amount) or even to make a steady savings-like contribution of that amount, whereas others waited to see their on-paper fortune climb into the millions.&lt;/p&gt;

&lt;p&gt;To find out how it worked I wrote an R function that parameterises all this. Full details are to the end of the blog, but the key parameters are:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;number of new investors that come in each month&lt;/li&gt;
  &lt;li&gt;mean and variance of the amount invested by each investor&lt;/li&gt;
  &lt;li&gt;a four-way set of proportions indicating what happens to investors each month:
    &lt;ul&gt;
      &lt;li&gt;seek to withdraw it all&lt;/li&gt;
      &lt;li&gt;seek to withdraw a smaller amount about the same as initial contribution&lt;/li&gt;
      &lt;li&gt;invest a further amount roughly equal to the initial contribution&lt;/li&gt;
      &lt;li&gt;leave it alone and just roll over the existing investment&lt;/li&gt;
    &lt;/ul&gt;
  &lt;/li&gt;
  &lt;li&gt;proportion of the real money in the scheme that is extracted each month by the scammers running it&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Caveat - I’m not a financial fraud expert and haven’t bothered to do even a minimal literature review in this space. I’m just toying around, and am probably reinventing some wheels, and possibly falling into beginner traps (assuming that there are others who &lt;em&gt;are&lt;/em&gt; such experts).&lt;/p&gt;

&lt;p&gt;Here is a simulated Ponzi scheme with a plausible starting set of values for all of those&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0283-ponzi1.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0283-ponzi1.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;created with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ponzi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep_history&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ponzi_plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;In this version, all investors put in $100 on average (but random log-normal distribution with standard deviation of 200) and we start with 10 such people. 20% of them decide they are on a particularly good thing, and put in another ~$100 per month; 10% (the default value, so not shown explicitly) try to withdraw just ~$100, 1% try to withdraw everything at the end of any given month and the remainder (so 69%) just roll over whatever they have.&lt;/p&gt;

&lt;p&gt;Not shown here is the default growth path in new investor/victims, which I set to be 10 new investors in the first month, then 20, 30, 40, 50 etc. Which looks like a lot of new people but is actually a declining growth rate each month; by month 10 or so this is down to about 20% growth per month in investors and declining further, noticeably less than the promised growth rate of 100% per month in paper money. So it’s got to end in tears quickly!&lt;/p&gt;

&lt;p&gt;In fact, we see the total paper value of the scheme gets up to about $1m after about 10 months and then the scheme collapses - one investor too many tries to cash in on their paper value and the scheme goes bankrupt. The total paper value got nearly up to $1m; the real amount invested was about $64k, and the scammers got away with only $1,524.&lt;/p&gt;

&lt;p&gt;That’s a little hard to read off the graphic with the log scales, so here’s the actual numbers I was quoting there:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; set.seed(123)
&amp;gt; ponzi(invest_more_rate = 0.2,
+             withdraw_all_rate = 0.01,
+             cv = 2, keep_history = TRUE) |&amp;gt;
+   ponzi_plot(return_numbers = TRUE)
                      [,1]
total_invested   64302.256
total_withdrawn 103190.149
total_extracted   1524.389
paper_value     888278.519
months               9.000
total_investors    450.000
leverage            14.000
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;Note that I’ve ducked the ugly question of exactly what happens when bankruptcy happens. The “total withdrawn” is $103k but only $64k was ever put in. Someone tried to withdraw $39k more than was available. That would have been ugly (as indeed would have been the case when the other people who thought they had nearly $900k of value missed out).&lt;/p&gt;

&lt;p&gt;The actual amount extracted is pretty modest compared to the high paper value of the scheme just before disaster. But this was a very unstable scheme. Here’s an alternative where I let the number of new investors grow steadily at 50% per month, plus have a generous 50% of existing investors put more in; and reduce the proportion of those seeking to withdraw their whole fortune to just 0.1%.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;ponzi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.001&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep_history&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ponzi_plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0283-ponzi2.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0283-ponzi2.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Now the scheme lasts a couple of years; the paper value got well into the billions; and the scammers were able to extract well over $1 million for themselves. Intuitively, the big difference here is that the number of new investor/victims was growing consistently at 50% per month. Nowhere near as much as the doubling that intuitively is needed if we are to keep the paper value of things doubling plausibly, but enough to keep things going for a couple of years.&lt;/p&gt;

&lt;p&gt;To explore more systematically the impact of these different parameters I ran a bunch of simulations with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#--------------systematic exploration-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;investor_growth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.001&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.001&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# give me three of each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# bit of randomness in one of the most important parameters&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;investor_growth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;investor_growth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runif&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# set up parallel processing cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;makeCluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;registerDoParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterEvalQ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterExport&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;params&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ponzi&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.combine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%dopar%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ni&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;investor_growth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_sim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ponzi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ni&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ceiling&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep_history&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_extracted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paper_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;months&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;leverage&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paper_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_sim&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This took half an hour or more to run (I didn’t time it). Most of the scenarios are quick to run, but those where the number of new investors increases by around 1.8 times per month (ie nearly doubling) can go for a long time. How long? Let’s see the impact of all those parameters on the time each scheme could run before it collapsed:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0283-months.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0283-months.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The interesting turning point in these charts comes when the number in new victim/investors hits the ceiling of 10 million people. Growth rates can’t conceiveably keep on forever like this. But what we see here is that, a few random cases aside, the best chance of prolonging a Ponzi scheme is to have one where the exit rate (ie the proportion of investors who try to pull all their paper value out of the scheme at once) is as low as possible. Which seems intuitive.&lt;/p&gt;

&lt;p&gt;We get a different angle when we focus not on how long the scheme continues, but on the total dollar value extracted by the scammers:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0283-extract.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0283-extract.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Again we see that things flattern out in those scams that hit the 10 million person ceiling (which by the way is absurdly too high to be realistic - I hope). But we can clearly see that the schemes that make the most money for the scammers are those with growth rates of 1.5 or higher, high extraction rates, and low withdrawal rates. Which is all intuitive.&lt;/p&gt;

&lt;p&gt;Those two charts were drawn with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;i&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Extrct: {extraction_rate}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Exit: {withdraw_all_rate}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;investor_growth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;months&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.ordered&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_jitter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variation in investor amounts&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Length of scam (in months)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Monthly growth in number of &apos;investors&apos;\n(2 = doubling per month; capped at 10 million)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;A low &apos;exit rate&apos; (by investors/victims) is key to prolonging a Ponzi scheme&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;How long can a Ponzi scheme last?&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;i&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&apos;Tax&apos; scammed:\n{extraction_rate}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Exit: {withdraw_all_rate}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;investor_growth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.ordered&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_jitter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variation in investor amounts&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Total extracted by scammers&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Monthly growth in number of &apos;investors&apos;\n(2 = doubling per month; capped at 10 million)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Key success factor for getting rich from a Ponzi scheme is to quickly and consistently grow your &apos;investors&apos; and extract as much &apos;tax&apos; you can.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;How much can the scammers extract from a Ponzi scheme?&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, I don’t know what set of parameters are realistic for different scams. In a different world, we could observe a bunch of these things - how long scams continue, the growth rate in victim/investors, what proportion of them withdraw their money or invest more, and so on, and create real-life calibrated models. In fact, it seems likely to me that someone has done this! I haven’t bothered to look, because my interest is basically whimsical and curiousity. And I’ve scratched that itch sufficiently to get an idea that maintaining a Ponzi scheme that promises to double investors’ money every month is going to be very, very hard to maintain for more than 10 months.&lt;/p&gt;

&lt;h2 id=&quot;supporting-code&quot;&gt;Supporting code&lt;/h2&gt;

&lt;p&gt;This section has the code defining functions used in the main part of the blog. You have to run these definitions before running the code above, if you are following along. Or you could just go to &lt;a href=&quot;https://github.com/ellisp/blog-source/blob/master/_working/0283-ponzi.R&quot;&gt;the source of the blog&lt;/a&gt; to see it directly as run by me.&lt;/p&gt;

&lt;p&gt;I’ve put this stuff at the end of the blog because it’s quite long and I thought likely to put off people who wanted to just read about Ponzi schemes.&lt;/p&gt;

&lt;h3 id=&quot;log-normal-distrbution&quot;&gt;Log-normal distrbution&lt;/h3&gt;

&lt;p&gt;First there’s a little function to generate samples from a log normal distribution. Now of course there is a function in R to do this already, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;rlognorm()&lt;/code&gt;, but it is parameterised by the mean and standard deviation of the underlying normal distribution that you get by taking the logarithm of the distribution we actually observe. I wanted, for easy interpretability, to be able to specify the mean and standard deviation (or in the end I chose the coefficient of variation, which is just the standard deviation except, well, standardised so it is described relative to the mean). I was surprised to find there wasn’t already a function doing this; probably if I looked I harder I’d find one. But this is what I ended up with from rolling my own:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;doParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; Generate samples from a log normal distribvution given E(X) and coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; of variation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param n number of samples to generate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param ex expected value of the distribution, on its observed (not log) scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param cv coefficient of variation (ie standard deviation as a proportion of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   the mean) of the distribution, on its observed (not log) scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @details Only needed because rlnorm() has the parameters of the Normal&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   distribution that log(X) follows and I wanted a version that used the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   actual mean of X and its coefficient of variation.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;output&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sdlog&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;meanlog&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;log&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sdlog&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;output&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;meanlog&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;meanlog&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sdlog&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sdlog&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;output&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check that this function works&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# If cv is zero, should just return a whole bunch of ex (i.e. 100):&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# If cv is not zero, check that the mean and coefficient of variation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# are as expectd:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;321&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Those tests are all passed ok. Here are a couple of demo distributions:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0283-log-normal.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0283-log-normal.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;produced with:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mfrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Log-normal distribution,\nmean 100 and cv 0.5&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Log-normal distribution\nmean 100 and cv 2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h3 id=&quot;ponzi-scheme&quot;&gt;Ponzi scheme&lt;/h3&gt;

&lt;p&gt;Here’s the code for the main workhorse function simulating the Ponzi scheme&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; Simulate ponzi scheme that doubles in paper value each time period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param number_investors vector of number of new investors joining the scheme&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   each time period. If this is less than 2000 time periods long it will be&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   filled out with repeats of the last element.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param mu average investment of each investor when they first join or invest&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   further&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param cv coefficient of variation of the amount that investors each invest&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param invest_more_rate the proportion of investors in each time period that&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   invest additional funds&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param withdraw_small_rate the proportion of investors in each time period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   that withdraw a small amount equivalent to their original investment&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param withdraw_all_rate the proportion of investors in each time period that&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   withdraw the entire paper value of their investment&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param roll_over_rate the proportion of investors who simply leave their&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   investment as it is at the end of each investment, neither withdrawing or&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   investing further&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param ceiling the maximum number of investors that can ever be involved in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   the scheme, including those that have already withdrawn all funds. If the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   cumulative sum of number_investors exceeds ceiling then no more investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   are added ie \code{ceiling} overrides \code{number_investors}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param extraction_rate The proportion of the total real value of the scheme&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   that the owners extract for their own use, each time period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param keep_history whether to return the state of the scheme at each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   time_period (if TRUE), or only the final state (if FALSe)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param time_multiplier how much the investors are promised their investment&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   increases by each time period; defaults to 2 ie doubling&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ponzi&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_small_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;roll_over_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_small_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ceiling&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep_history&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_multiplier&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------checks on number of investors------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;number_investors should be a vector of numbers of 0 or greater&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;First element of number_investors should be greater than 0&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# make sure we have enough for 2000 time periods&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nni&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nni&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                          &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nni&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------Other checks--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_small_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;roll_over_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;invest_more_rate, withdraw_small_rate, withdraw_all_rate and roll_over_rate should add to one&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;extraction_rate should be between zero and one&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------------------set up month 1------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cash_on_hand&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;with&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#---------------------months 2 and onwards---------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;while&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cash_on_hand&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ceiling&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we make a new state for our existing investors, which is going to later be&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# appended to the state so far:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;update&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# limit to the state from the last time_period:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Double the &apos;value&apos; of existing investments&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_multiplier&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Decide for each investor what they are going to do this time period:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;withdraw_all&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;withdraw_small&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rollover&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;invest&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_all_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdraw_small_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                                      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;roll_over_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invest_more_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# An incremental amount which might be used for furhter investments or for withdrawals:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;incr_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# some people try to withdraw their paper value. This increases the amount&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# they have ever withdrawn:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;withdraw_all&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;withdraw_small&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;incr_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# ... and decreases their paper value by the same amount&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;withdraw_all&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;withdraw_small&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;incr_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rollover&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;invest&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;incr_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Some other people choose to invest more:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;invest&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;incr_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the people running the Ponzi scheme extract a percentage of the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# real money available, calculated per investor (so fully withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# investors not included):&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extraction_rate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;action&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;incr_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract_tmp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# time period goes up one:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# We are also going to get a number of new investors:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;update&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rlognormal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_new_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value_tmp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;update&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# depending on whether the user wants all the history, we either append our&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# new update for the previous investors and our new investors to the old state,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# or we just keep the latest update on previous investors plus our new investors:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep_history&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;update&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;update&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_investors&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# We need to determine if the scheme has gone bust, by calculating its&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# actual cash available (the amount invested so far, minus the total&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# withdrawn and total extracted)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cash_on_hand&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;h3 id=&quot;ponzi-summary-table-and-plot&quot;&gt;Ponzi summary table and plot&lt;/h3&gt;

&lt;p&gt;Finally, here’s a function that takes the output of a single Ponzi simulation and draws the chart showing extraction, paper value, and real value invested over time.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;ponzi_plot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;return_numbers&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numbers&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_invested&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_withdrawn&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_extracted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paper_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;months&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_investors&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;leverage&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paper_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;t&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;status&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Paper value`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Real value`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invested&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;withdrawn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Extracted by scammers`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extracted&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of months&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;return_numbers&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numbers&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all folks. Take care out there, and if the rate of return on a proposed investment is too good to be true, it probably isn’t true.&lt;/p&gt;

</description>
				<pubDate>Sat, 30 Nov 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/11/30/ponzi-schemes</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/11/30/ponzi-schemes</guid>
			</item>
		
			<item>
				<title>Design effects for stratified sub-populations by @ellis2013nz</title>
					<description>&lt;p&gt;So, some months ago, I spent a few hours over a few days puzzled by something that turned out to be straighforwardly written up in the Stata manual, but not easily findable anywhere else. So I want to write it up, if only to have somewhere for future me to find it easily.&lt;/p&gt;

&lt;p&gt;It’s all about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Design_effect&quot;&gt;design effect in a complex survey&lt;/a&gt;. The definition of a design effect given by Wikipedia is fine:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“In survey methodology, the design effect … is a measure of the expected impact of a sampling design on the variance of an estimator for some parameter of a population. It is calculated as the ratio of the variance of an estimator based on a sample from an (often) complex sampling design, to the variance of an alternative estimator based on a simple random sample (SRS) of the same number of elements.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;So a design effect of 1 means the variance of your estimator (and hence its standard error) is the same in your complex survey design as if you’d just sampled simply from the population at random. Values higher than 1 are the norm, and mean that you have a higher variance in your estimate as a price of something else in your complex design (for example, maybe you are sampling clusters that live closer together to make it easier for interviewers to interview several households at once, which reduces cost but increases variance).&lt;/p&gt;

&lt;p&gt;My puzzle related to when you are estimating some measure of interest over different levels of a categorical variable that features in the sample design (say, the unemployment rate in different states of Australia, in a survey where the sample was stratified by state), and in particular when some levels been over-sampled and some under-sampled. By which I mean, the fraction of the total population that has been sampled is different for different levels of this variable. Perhaps Tasmania (a small state) was over-sampled from to make sure there were enough people to report on unemployment separately for Tasmania.&lt;/p&gt;

&lt;p&gt;A concrete example I’m going to use is a survey that was stratified by province, and some provinces have had a high proportion of the population sampled for some reason that we don’t need to worry about, but let’s presume it’s a good one (like, wanting a minimum number of people from each province to report the estimate of interest with a reasonably small confidence interval). The variable of interest in this survey is &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;likes_cats&lt;/code&gt; which is either 1 or 0 and indicates whether the survey respondent likes cats. (Warning - no attempt has been made to make the proportion who likes cats in this made up data resemble reality).&lt;/p&gt;

&lt;p&gt;The thing that started me on this as a problem was noticing differences between Stata and R output, so I’ll be structuring the blog around that. I guess this also serves the useful case of an example of the same survey being analysed in both environments.&lt;/p&gt;

&lt;p&gt;The data I’m using is made up, but based on real life data that I’ve mangled, re-simulated, and generally fiddled with to avoid any privacy questions. Let’s start by downloading it, importing it to R, and specifying the survey design using Thomas Lumley’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;survey&lt;/code&gt; package.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;survey&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;url&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://raw.githubusercontent.com/ellisp/blog-source/refs/heads/master/_working/simulated-survey.csv&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;simulated-survey.csv&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;url&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Specify survey design - stratified by province, and 2 stages, both with finite populations&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv_des&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svydesign&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;neighborhood&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ind_id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fweight&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;strata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fpc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nb_in_province&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_in_neighborhood&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nest&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;And here’s doing the same in Stata, other than the downloading part given we’ve already done that once:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-stata&quot; data-lang=&quot;stata&quot;&gt;&lt;span class=&quot;c1&quot;&gt;** Simple Stata demo of survey design and calcualtion, assumes data has been&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;** previously downloaded in the accompanying R script&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;clear&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;import&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;delimited&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;simulated&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-survey.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;

&lt;span class=&quot;c1&quot;&gt;** make a numeric version of province, as over() can&apos;t work with strings:&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;encode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gen&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c1&quot;&gt;** Set up survey design = complex 2 stages design stratified sample&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;** first step of sampling design is selection of neighborhoods from each province with finite neighborhoods,&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;** second step is individuals select from the finite population in each neighborhood&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;svyset&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;neighborhood&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pweight&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fweight&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;singleunit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;centered&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;/// &lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;strata&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fpc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nb_in_province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;||&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ind_id&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fpc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_in_neighborhood&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;		&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;These are pretty similar workflows. Stata has the extra step - for Reasons - of encoding a string variable we’re going to want to use later as numeric. R’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;svydesign&lt;/code&gt; function and the Stata &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;svyset&lt;/code&gt; function do essentially identical jobs in a pretty similar way, telling the software the stratified, two stage design we’re going to be using and giving it the columns to look at for the finite population corrections. Our sample design is to pick a number of neighbourhoods in each province, and then a number of individuals in each neighbourhood, so we’re interested in both the total number of neighbourhoods in each province and the total number of individuals in each neighbourhood.&lt;/p&gt;

&lt;p&gt;(Note - in real life I would be declaring that the sun of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;fweight&lt;/code&gt; in each province has been calibrated to match the population total in that province, but doing this introduced some further limitations with Stata that I decided were distractions, to look at some other time.)&lt;/p&gt;

&lt;p&gt;To check that we’re in agreement, let’s start by asking both environments the proportion of people that like cats, the standard error of that proportion, and the design effect.&lt;/p&gt;

&lt;p&gt;In R:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svymean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv_des&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;SE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5320009&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.0065751&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.8673&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;And in Stata:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-stata&quot; data-lang=&quot;stata&quot;&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;**&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;level&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;effects&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svy&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;estat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;effect&lt;/span&gt;

&lt;span class=&quot;o&quot;&gt;----------------------------------------------------------&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Linearized&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;Std.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Err&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEFF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEFT&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;-------------+--------------------------------------------&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.5320009&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0065751&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.86726&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.31766&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;----------------------------------------------------------&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;Note&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;must&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;represent&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;totals&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kr&quot;&gt;to&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;be&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;correct&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;when&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;using&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;an&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FPC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;however&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deft&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;is&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;invariant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kr&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kr&quot;&gt;of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weights&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;(Some of Stata’s output has been edited out)&lt;/p&gt;

&lt;p&gt;OK, so we see that both systems agree that 0.5320009 of people like cats, that the standard error for this is 0.0065751, and the design effect is 1.86726.&lt;/p&gt;

&lt;p&gt;So far so good, but what we are interested in is what proportion of people &lt;em&gt;in each province&lt;/em&gt; like cats. Here is what R has to say about this:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svyby&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv_des&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svymean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEff.likes_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6657223&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01910098&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.676309&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6345405&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01423610&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.257485&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6096537&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.02125743&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.050192&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;D&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;D&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5706099&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.02251630&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.852858&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;E&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;E&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.4392845&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01633982&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.748840&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6783198&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01831859&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.103528&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;G&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;G&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.4153480&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01535588&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.217967&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.4066567&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.01329190&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.258354&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;…and here is Stata:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-stata&quot; data-lang=&quot;stata&quot;&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;**&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Strata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;level&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;effects&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svy&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;over&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;estat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;effect&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;                          &lt;/span&gt;

&lt;span class=&quot;o&quot;&gt;----------------------------------------------------------&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Linearized&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Over&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;Std.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Err&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEFF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEFT&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;-------------+--------------------------------------------&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6657223&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.019101&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.637551&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.769941&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6345405&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0142361&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;      &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.3455&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.11852&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6096537&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0212574&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.31479&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.46709&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;D&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.5706099&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0225163&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3.13267&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.7067&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;E&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.4392845&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0163398&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.01139&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.36757&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6783198&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0183186&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.895&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.32741&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;G&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.415348&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0153559&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.597809&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.745558&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.4066567&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0132919&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.74908&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.27528&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;----------------------------------------------------------&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This time the point estimates and the standard errors are different, but the design effects are radically different. For example, in province A, R gives a design effect of 1.68 (meaning that the variance of the estimate is 68% bigger as a result of the complex survey sampling design), but Stata gives 0.64 (meaning the variance of the estimate is 36% &lt;em&gt;lower&lt;/em&gt; because of the design).&lt;/p&gt;

&lt;p&gt;Why is this so? In particular, how can the design effects be different, when the estimated standard errors are identical?&lt;/p&gt;

&lt;p&gt;Well, the answer turns out to be pretty straightforward. In &lt;a href=&quot;https://www.stata.com/manuals/svyestat.pdf&quot;&gt;the Stata manual for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;estat effect&lt;/code&gt;&lt;/a&gt; it says we have an option &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;srssubpop&lt;/code&gt; which will “report design effects, assuming simple random sampling &lt;strong&gt;within subpopulation&lt;/strong&gt;” (emphasis added) and:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;…by default, DEFF and DEFT are computed using an estimate of the SRS variance for sampling &lt;strong&gt;from the entire population&lt;/strong&gt;.&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;As per Wikipedia, the design effect is a comparison of the actual estimated standard error of the proportion of people who like cats in each province, with the standard error “you would have got with a simple random sampling design” (it is actually the ratio of the variances, which is the squared standard error, but as we are typically thinking in terms of standard error and that is what both R and Stata return, I’m emphasising that part of the concet).&lt;/p&gt;

&lt;p&gt;But there are two ways of looking at this variance that “you would have got with a simple random sampling design”.&lt;/p&gt;

&lt;p&gt;One way is to say that you have the sample numbers in each province that you have, but let’s pretend the design that gave it to you was just simple random sampling, and calculate the standard error accordingly; square both standard errors and divide one by the other. This is what is happening in R.&lt;/p&gt;

&lt;p&gt;The other ways is to reflect on one more step, and say “hang on, if we’d really had a simple random sample, we wouldn’t have these sample sizes in the first place. For example, we’ve only got 864 samples in province A because we deliberately oversampled there. If we’d used simple random sampling, it would have been more like 362. Let’s compare our standard errors from the actual sample to &lt;em&gt;the standard errors we would have likely got from simple random sampling, including less people in province A and more in province G than we actually got&lt;/em&gt;.” This is what is happening in Stata. That’s why province A has that design effect of 0.64 - the stratified sample design gave us extra samples in province A, so the design is actually giving us &lt;em&gt;reduced&lt;/em&gt; variance there, compared to a genuine simple random sample.&lt;/p&gt;

&lt;p&gt;So the reason this second method has a design effect of less than 1 for province A is that it says “the only reason you have so many people in your sample from province A is the stratification part of the sampling design. So when we want to say ‘what impact has the design had on our estimates’, it’s only fair to say that the estimate you’ve got is a lot more precise than you would have under simple random sampling.”&lt;/p&gt;

&lt;p&gt;I actually find this a pretty intutitive interpretation, if we want to use the design effect as a short hand for “what impact has our design had on this particular estimate’s variance”. Of course a lot of this is largely academic; design effects are only really worried about by specialists, and they are often focused on as a single number for the survey as a whole (focusing on the impact of the design on the key variable of interest, estimated for the whole population), which doesn’t vary by either method. And the two methods of getting design effect will only vary much when, like in our case, the categorical variable you are splicing your sample up into for estimation purposes happens to be the very variable that you used as a basis for over- and under-sampling in the first place.&lt;/p&gt;

&lt;p&gt;Stata conveniently gives you an easy option to use the alternative method of calculating a design effect, just adding the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;, srssubpop&lt;/code&gt; option to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;estat effect&lt;/code&gt; command:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-stata&quot; data-lang=&quot;stata&quot;&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;estat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;effect&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;srssubpop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;                          &lt;/span&gt;

&lt;span class=&quot;o&quot;&gt;----------------------------------------------------------&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Linearized&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Over&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;Std.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Err&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEFF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEFT&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;-------------+--------------------------------------------&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6657223&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.019101&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.67631&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.18258&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6345405&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0142361&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.25748&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.08378&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6096537&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0212574&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.05019&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.38623&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;D&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.5706099&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0225163&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.85286&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.63379&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;E&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.4392845&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0163398&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.74884&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.28105&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.6783198&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0183186&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.10353&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.39313&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;G&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.415348&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0153559&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.21797&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.02739&lt;/span&gt;
&lt;span class=&quot;w&quot;&gt;           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.4066567&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;.0132919&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;     &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.25835&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.09248&lt;/span&gt;
&lt;span class=&quot;o&quot;&gt;----------------------------------------------------------&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;We see these design effects exactly match those from R, eg the design effect for province A is now 1.676.&lt;/p&gt;

&lt;p&gt;Here’s what the Stata manual says about which method to choose:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“Typically, srssubpop is used when computing subpopulation estimates by strata or by groups of strata.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;This is a little terse but I presume the logic here is that if you stratified your sample, perhaps you want to compare to simple random sampling within those strata - not to from the population as a whole. In otherwords, you’re not really comparing your complex survey design to the simples of all simple random sampling, but to stratified random sampling.&lt;/p&gt;

&lt;p&gt;The &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;survey&lt;/code&gt; package in R defaults to the equivalent of Stata’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;srssubpop&lt;/code&gt; option. As far as I could tell, R doesn’t give an equivalently easy way to get the design effects that Stata defaults to. But they aren’t hard to calculate yourself. The table below is meant to be a step-by-step guide to how this was done:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;avg_weight&lt;/code&gt; is the average individual weight of sample units in each province. Lower weights mean that each sampled person represents less of the total population than is the case for heigher weights.&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;p&lt;/code&gt; is the estimated proportion of people in that province that like cats, calculated straight from the data&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;n&lt;/code&gt; is the actual sample size&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;N&lt;/code&gt; is the population size in that province&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;even_n&lt;/code&gt; is the hypothetical approximate sample size you would have drawn from that population under simple random sampling&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;se_with_actual_n&lt;/code&gt; is the standard error, pretending there is no complex survey design to worry about, with the actual number of samples in that province. In other words &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sqrt(p * (1 - p) / n * (N - n) / (N - 1))&lt;/code&gt; which is the usual standard error for a proportion, with finite population correction.&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;se_with_even_n&lt;/code&gt; is the same but calculated with the hypothetical more even sample size if you’d hard real simple random sampling. So same as the above, but using &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;even_n&lt;/code&gt; instead of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;n&lt;/code&gt;.&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;se_complex&lt;/code&gt; is the actual standard error, taking into account the complex survey design.&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;DEff_1&lt;/code&gt; is &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;(se_complex / se_with_even_n) ^ 2&lt;/code&gt; ie the Stata default method&lt;/li&gt;
  &lt;li&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;DEff_2&lt;/code&gt; is &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;(se_complex / se_with_actual_n) ^ 2&lt;/code&gt; ie the R default method&lt;/li&gt;
&lt;/ul&gt;

&lt;hr /&gt;

&lt;table&gt;
  &lt;thead&gt;
    &lt;tr&gt;
      &lt;th style=&quot;text-align: left&quot;&gt;province&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;avg_weight&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;p&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;n&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;N&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;even_n&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;se_with_actual_n&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;se_with_even_n&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;se_complex&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;DEff_1&lt;/th&gt;
      &lt;th style=&quot;text-align: right&quot;&gt;DEff_2&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;A&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;6.033866&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.6657223&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;854&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;5152.921&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;362&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0147457&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0239095&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0191010&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.6382190&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.677948&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;B&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;15.167991&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.6345405&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1345&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;20400.948&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1432&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0126908&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0122711&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0142361&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.3458988&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.258358&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;C&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;15.946476&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.6096537&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1013&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;16153.780&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1134&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0148393&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0139691&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0212574&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;2.3157141&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;2.052091&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;D&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;15.539426&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.5706099&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1291&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;20061.399&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1408&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0133260&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0127205&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0225163&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;3.1331916&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;2.854927&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;E&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;16.230320&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.4392845&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1515&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;24588.934&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1726&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0123520&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0115194&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0163398&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;2.0120421&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.749924&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;F&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;12.927955&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.6783198&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1263&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;16328.008&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1146&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0126258&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0133060&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0183186&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.8953402&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;2.105066&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;G&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;7.497966&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.4153480&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1088&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;8157.787&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;572&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0139086&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0198700&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0153559&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.5972474&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.218938&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;td style=&quot;text-align: left&quot;&gt;H&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;19.407325&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.4066567&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1631&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;31653.346&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;2221&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0118457&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0100508&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;0.0132919&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.7489189&lt;/td&gt;
      &lt;td style=&quot;text-align: right&quot;&gt;1.259086&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;

&lt;hr /&gt;

&lt;p&gt;For variables not highly correlated with the over/under-sampling structure, the design effects will not vary by much the two methods. For variables that &lt;em&gt;are&lt;/em&gt;, the Stata manual suggests using the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;srssubpop&lt;/code&gt;. What this suggests to me is that it is safe to use the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;srssubpop&lt;/code&gt; option more generally - when it makes much difference is precisely the time that it is more likely to be used.&lt;/p&gt;

&lt;p&gt;I (literally) can’t have a blog post without a plot, so here is an attempt to show some of the key information from that table visually. I tried a few different ways to show this visually, and ultimately opted for just comparing two ratios - that of the design effect from one method compared to the other, and the actual sample size compared to the expected sample size under unstratified simple random sampling.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0263-scatter.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0263-scatter.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Here’s the code that calculated that table and drew the plot:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;avg_weight&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fweight&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;weighted.mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fweight&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fweight&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;even_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_with_actual_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_with_even_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;even_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;even_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_complex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svyby&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;likes_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;design&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_surv_des&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;svymean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEff_1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_complex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_with_even_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEff_2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_complex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se_with_actual_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio_n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;even_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio_deff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEff_2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DEff_1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio_deff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;province&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_abline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slope&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;intercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fontface&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;bold&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_log10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Oversampling ratio for actual sample\n(&amp;gt;1 is oversampled, &amp;lt;1 is undersampled)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Design effect ratio&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ratio of &apos;srssubpop&apos; design effect to Stata default design effect
Grey line shows equality of the two ratios.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Two types of design effect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Cheerio.&lt;/p&gt;
</description>
				<pubDate>Sat, 16 Nov 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/11/16/design-effects</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/11/16/design-effects</guid>
			</item>
		
			<item>
				<title>Regressions where the coefficients are a simplex. by @ellis2013nz</title>
					<description>&lt;p&gt;I have been thinking in my spare time a bit about synthetic control methods as an approach to evaluation, and am working on a blog post. But a side issue that popped up was sufficiently interesting to treat separately.&lt;/p&gt;

&lt;p&gt;Technically, it is a question of fitting a regression where the coefficients are constrained to be non-negative and add up to one; in other words, the coefficients are a &lt;em&gt;simplex&lt;/em&gt;. The reason this comes up is in that in synthetic control methods, such a regression is used to determine the weights to use in constructing a weighted average of multiple units (countries, firms, people) that is as comparable as possible to the unit that received the intervention being evaluated.&lt;/p&gt;

&lt;p&gt;The argument for the weights adding up to one is to avoid extrapolating from the range of the real data. I don’t find this a convincing argument, but I’ll leave that for a later post if I get round to it. For now, I’m interested in the technical question of how to get such a set of non-zero weights adding up to one; or abstracting it away from the application, how to fit a regression where the coefficients are constrained to be a simplex.&lt;/p&gt;

&lt;p&gt;Naturally, there are multiple ways you might do this. There was &lt;a href=&quot;https://stats.stackexchange.com/questions/21565/how-do-i-fit-a-constrained-regression-in-r-so-that-coefficients-total-1&quot;&gt;a good question and answer on this on Cross-Validated&lt;/a&gt; way back in 2012. Two of the four methods I’m going to go through below come pretty much straight from answers or comments there. The other two are my own invention (I’m not claiming to be the first, just that I haven’t seen others doing it this way).&lt;/p&gt;

&lt;p&gt;First, let’s get started by simulating some data. I’m going to make a matrix called &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;X&lt;/code&gt; with five columns, all from different distributions. Then I create a vector of true coefficients that will relate the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;y&lt;/code&gt; I’m about to make to that &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;X&lt;/code&gt;. To make the exercise interesting and life-like, the true coefficients are going to involve a negative number, and aren’t going to add to exactly 1. One of them is going to be zero.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glmnet&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;MASS&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for eqscplot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pracma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for lsqlincon&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rstan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;options&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mc.cores&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;parallel&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;detectCores&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Roboto&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runif&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# note true data generating process coefficients add up&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# to more than 1 and have a negative, to make it interesting&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coefs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coefs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, I’m going to fit various regressions (with no intercept) to this data and extract the coefficients and compare them to the true data generating process. First let’s start with ordinary least squares with no constraints&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-------------method 1 - unconstrained regression------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# This is the best way to recover the true DGP&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# plotting function we will use for each model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;newx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;eqscplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coefs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;newx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;l&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xlab&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficients from original data generating process&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ylab&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficients from model&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;abline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;OLS&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This approach is very good at getting close to the true data generating process, as we’d expect. But of course, like the true process, it has a negative coefficient for x3, and it sums to more than 1:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; round(coefs1, 2)
  Xx1   Xx2   Xx3   Xx4   Xx5 
 0.20  0.61 -0.11 -0.02  0.45 
&amp;gt; sum(coefs1)
[1] 1.122484
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;This is the plot we get, comparing the model’s coefficients to those from the real data generating process:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0281b-mod1.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0281b-mod1.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;OK, all very good but it makes no effort (other than just hoping it works out) to constrain the coefficients to add up to one. So here’s my first real attempt at that, which is based on a suggestion that whuber made in the comments on the Cross-Validated post. He points out with some basic algebra that if you subtract one of the X columns from the others and from y, and then fit a regression with OLS to that transformed data, you get the correct coefficients on all the columns of X that you left in, and can calculate the remaining one by just subtracting them all from 1.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------method 2 - transform-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# This way will get you coefficients that add up to 1,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# but might not be in (0, 1). This is suggested by whuber in &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# his comment on Elvis&apos; solution at:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://stats.stackexchange.com/questions/21565/how-do-i-fit-a-constrained-regression-in-r-so-that-coefficients-total-1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Transformation&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw plot&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This time the coefficients add up exactly to one as required, but unfortunately there is still a negative in there&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; round(coefs2, 2)
 X2x1  X2x2  X2x3  X2x4       
 0.24  0.40 -0.04  0.00  0.40 
&amp;gt; sum(coefs2)
[1] 1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0281b-mod2.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0281b-mod2.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;This isn’t news - it was identified as a problem in the discussion on Cross-Validated, but with the simulated data used there it didn’t manifest. In fact, this is one reason why my simulated data is a bit dirtier, with original coefficients that don’t add to one and which have a genuine negative value among them.&lt;/p&gt;

&lt;p&gt;So much for model 2.&lt;/p&gt;

&lt;p&gt;Model 3 is one I invented myself and makes use of the fact that &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;glmnet&lt;/code&gt; lets you not only regularise your coefficients (squash them towards zero, to help handle multicollinearity, overfitting, and related issues) it lets you specify upper and lower bounds for coefficients. So we could get our first fully compliant set of regression coefficients (non-negative, adding to one) by using the data transformation approach in model2 but fitting the model with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;glmnet&lt;/code&gt; instead. Like this:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------method 3 - transform, regularise and constrain-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# This method is guaranteed to get you coefficients that add up to 1,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# and they will also be in the (0, 1) range, plus they will be regularised&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# alpha = 0 is ridge regression, but could use 1 for lasso (or anything in between)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cv.glmnet&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;intercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower.limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper.limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Coefficients from glmnet include the intercept even though we forced it to be zero&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# which is why we need the [-1] in the below&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.vector&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Transform and ridge&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw plot&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;The results are all non-negative, and add correctly to one:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; round(coefs3, 2)
[1] 0.11 0.24 0.00 0.00 0.64
&amp;gt; sum(coefs3)
[1] 1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;They do differ markedly from the true coefficients:&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0281b-mod3.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0281b-mod3.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;We’ll come back to that.&lt;/p&gt;

&lt;p&gt;Next is what we might call the “proper” solution, which is to treat the problem as the classic case for &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratic_programming&quot;&gt;quadratic programming&lt;/a&gt; that it is. Quadratic programming was developed exactly to deal with this situation - optimise a multivariate quadratic function (like, least squares in a regression) subject to linear constraints (like, coefficients must be non-negative and add up to one). Hans Borchers’  R package (“Practical Numerical Math Functions”) has an easy to understand interface to do this without having to invert your own matrices, so the only thing you need to do is to express the constraints in the right way. Here’s how we do this with our problem:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------method 4 - quadratic programming-----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# This is based on dariober&apos;s solution, which itself is Elvis&apos; solution&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# but using the easier API from pracma::lsqlincon&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# lsqlincon &quot;solves linearly constrained linear least-squares problems&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Equality constraint: We want the sum of the coefficients to be 1.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# I.e. Aeq x == beq  &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Aeq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;beq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Lower and upper bounds of the parameters, i.e [0, 1]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ub&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# And solve:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lsqlincon&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Aeq&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Aeq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;beq&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;beq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lb&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lb&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ub&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ub&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Quadratic programming&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Just like the previous method, our result is all non-negative and sums to zero:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; round(coefs4, 2)
[1] 0.17 0.44 0.00 0.00 0.38
&amp;gt; sum(coefs4)
[1] 1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;This time the coefficients look a bit closer to the original ones:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0281b-mod4.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0281b-mod4.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Finally, I the other method I invented was to go all-in Bayesian and simply specify in Stan that the coefficients have to be a simplex and that they have a &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirichlet_distribution&quot;&gt;Dirichlet distribution&lt;/a&gt;. This actually seems the most intuitive solution to me - what better way is there of making something a simplex than by just deeming it to have a Dirichlet distribution?&lt;/p&gt;

&lt;p&gt;So I wrote this program in Stan:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-stan&quot; data-lang=&quot;stan&quot;&gt;&lt;span class=&quot;c1&quot;&gt;// Stan program for a regression where the coefficients are a simplex&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;// Probable use case is when y is some kind of weighted average of the X&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;// and you want to guarantee the weights add up to 1 (note - note sure I think&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;// this is a good idea but it is out there)&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;//&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;// Peter Ellis November 2024&lt;/span&gt;

&lt;span class=&quot;nn&quot;&gt;data&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;na&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;   &lt;span class=&quot;c1&quot;&gt;// number of data items must match number rows of X&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;na&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;   &lt;span class=&quot;c1&quot;&gt;// number of predictors must match number columns of X&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;nv&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;   &lt;span class=&quot;c1&quot;&gt;// predictor matrix. Might include intercept (column of 1s)&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;vector&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;nv&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;      &lt;span class=&quot;c1&quot;&gt;// outcome vector&lt;/span&gt;
&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;
&lt;span class=&quot;nn&quot;&gt;parameters&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;simplex&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;nv&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;       &lt;span class=&quot;c1&quot;&gt;// coefficients for predictors&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;real&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;na&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;  &lt;span class=&quot;c1&quot;&gt;// error scale&lt;/span&gt;
  &lt;span class=&quot;kt&quot;&gt;real&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;na&quot;&gt;lower&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;na&quot;&gt;upper&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;c1&quot;&gt;// parameter for dirichlet distribution&lt;/span&gt;
&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;
&lt;span class=&quot;nn&quot;&gt;model&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;
  &lt;span class=&quot;nv&quot;&gt;alpha&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;~&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;beta&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;
  &lt;span class=&quot;nv&quot;&gt;b&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;~&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;dirichlet&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;rep_vector&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nv&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;));&lt;/span&gt;
  &lt;span class=&quot;nv&quot;&gt;y&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;~&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;normal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nv&quot;&gt;X&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nv&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;  &lt;span class=&quot;c1&quot;&gt;// likelihood&lt;/span&gt;
&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;… and called it from R with this:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------method 5 - Bayesian model with a Dirichlet distribution------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stan_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.vector&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0281b-dirichlet-coefs.stan&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stan_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extract&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;b&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This is now the third method for which our result is correctly all non-negative and sums to zero:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; round(coefs5, 2)
[1] 0.17 0.45 0.00 0.00 0.38
&amp;gt; sum(coefs5)
[1] 1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;
&lt;p&gt;… and this is what it looks like, basically very similar to the quadratic programming solution:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0281b-mod5.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0281b-mod5.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;OK, which method is best? We need some sort of benchmark to test them against, better than the original coefficients from the data generating process in complying with the fundamental restrictions. Given the ultimate purpose is to construct a weighted average, we want something that has similar proportions to the original coefficients. So, just wanting to be pragmatic about this, I created a “normalised” set of coefficients by removing the negative one and scaling the remainder to add up to one.&lt;/p&gt;

&lt;p&gt;This lets us define a success criteria - the winning method is the one that meets the constraints (ruling out models 1 and 2) and most closely matches those proportions. Here’s a nice pairs plot comparing everything at a glance:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0281b-pairs.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0281b-pairs.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;What we see is that models 4 and 5 are clear, effectively equal, winners. Not only are they nearly identical, they also match the normalised true coefficients (correlations of 0.98 or above). Sadly, the glmnet method doesn’t perform very well at recovering the original proportions. Perhaps it would have done better if I’d let the columns of X be correlated with eachother rather than independent (dealing with multi-collinearity being elastic net regression’s home turf). Perhaps also it would get closer to the quadratic programming solution if I forced the amount of regularisation down to nearly zero. But with two good solutions already I’m not inclined right now to experiment further on that.&lt;/p&gt;

&lt;p&gt;The quadratic programming approach was easier to write and debug and quicker to run than the explicit modelling of the Dirichlet distribution in Stan, so pragmatically it becomes the winner; with the sole caveat being that question mark in the last paragraph about what happens if the columns of X are multi-collinear.&lt;/p&gt;

&lt;p&gt;Oh, let’s finish with the code to draw that pairs plot. For one reason or another, I used good old base R &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pairs()&lt;/code&gt; to do this, rather the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;GGally&lt;/code&gt; package I more normally use for scatterplot matrices. I like the easy control &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pairs()&lt;/code&gt; gives you over the upper and lower triangles of the scatterplot matrix, by just defining your own function on the fly. So for the old-timers, here’s that base R code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------comparisons------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;norm_coefs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coefs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# remove negative values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;norm_coefs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;norm_coefs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;norm_coefs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# force to ad dup to one&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pairs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Actual data\ngenerating process&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coefs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Normalised to\nadd to one and &amp;gt; 0&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;norm_coefs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ordinary least squares,\nwon&apos;t add to 1, not\nconstrained to (0,1)&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Transform and OLS, add to 1\nbut can be negative&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Transform and ridge regression,\nconstrain to within (0,1)&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Quadratic programming&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;Dirichlet &amp;amp; Stan&apos;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefs5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lower.panel&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;abline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;points&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pch&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;19&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;},&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;upper.panel&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;range&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;range&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;font&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;},&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;main&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of different methods for creating a weighted average with weights adding up to one&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

</description>
				<pubDate>Wed, 06 Nov 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/11/06/simplex-regression-coefficients</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/11/06/simplex-regression-coefficients</guid>
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			<item>
				<title>Git, peer review, tests and toil by @ellis2013nz</title>
					<description>&lt;p&gt;This week I was in Auckland New Zealand to deliver the third and final of the 2024 series of &lt;a href=&quot;https://www.auckland.ac.nz/en/science/about-the-faculty/department-of-statistics/ihaka-lecture-series.html&quot;&gt;the Ihaka Lectures&lt;/a&gt;, named after legendary denizen of University of Auckland’s statistics department Ross Ihaka, one of the two co-founders of the statistical computing language R.&lt;/p&gt;

&lt;p&gt;I have added links to the video of the talk (it was live-streamed), my slides, and the ‘storyline’ summary I used to help me structure the talk to my mostly-neglected &lt;a href=&quot;/presentations/index.html&quot;&gt;presentations page on this blog&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;Here is perhaps the key image from the talk, a slide showing an all-purpose workflow for an analytical project, drawing on a large and persistent data warehouse, plus project specific data, and having a deliberate processing stage to combine the two into an analysis-ready “project-specific database”. I’ve been using variants of this diagram for more than 10 years now, and it will be familiar to anyone from my days with New Zealand’s Ministry of Business, Innovation and Employment, or international management consultancy Nous Group.&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;https://freerangestats.info/img/git-project-diagram-spc-themed.png&quot; title=&quot;Diagram of workflows showing an enterprise data warehouse, project specific data, R combining the two into a project-specific database, R used for analysis, and output in Word, PowerPoint, Quarto, Shiny, Tableau or Power BI&quot; width=&quot;95%&quot; /&gt;&lt;/p&gt;

&lt;p&gt;Overall, I emphasised the importance of R being part of a broader toolkit and a broader transformation - with Git and SQL the two non-negotiable must-have partners to successfully make R work in government.&lt;/p&gt;

&lt;p&gt;I also talked a bit about how errors in analysis are universal, invisible, and catastrophic. If that doesn’t motivate people to start doing some decent quality control, I don’t know what will!&lt;/p&gt;

&lt;p&gt;The ‘storyline’ is a great technique I was trained on in a course on writing for the New Zealand public sector. I always find it helps to structure reports and presentations if I take the time to plan them first. In case people are interested in making their own summaries of this sort, you could use the &lt;a href=&quot;https://github.com/ellisp/ihaka-lecture-2024/blob/main/ellis-ihaka-storyline.Rmd&quot;&gt;RMarkdown source code of that storyline&lt;/a&gt;, which of course is available in GitHub (or I’d be a bit of a hypocrite wouldn’t I). It uses the flexdashboard template. Of course a storyline doesn’t need to be written in RMarkdown, but I find it a simple and disciplined way to write them without having to worry about formatting.&lt;/p&gt;

&lt;p&gt;Big thanks to the University of Auckland Department of Statistics and all the good folks there for inviting me to give this talk and looking after me so nicely while in New Zealand.&lt;/p&gt;
</description>
				<pubDate>Sat, 28 Sep 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/09/28/ihaka-lecture</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/28/ihaka-lecture</guid>
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			<item>
				<title>Prime numbers as sums of three squares. by @ellis2013nz</title>
					<description>&lt;p&gt;I was interested by a &lt;a href=&quot;https://www.linkedin.com/posts/fermatslibrary_397-is-conjectured-to-be-the-largest-prime-activity-7242947116719915008-BRz7?utm_source=share&amp;amp;utm_medium=member_desktop&quot;&gt;LinkedIn post about the number 397&lt;/a&gt;:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“397 is conjectured to be the largest prime that can be represented uniquely as the sum of three positive squares”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;That is, 3^2 + 8^2 + 18^2 = 397&lt;/p&gt;

&lt;p&gt;This led to some confusion in the comments as people found other prime numbers that can be created as the sum of three squares. But the wording is sloppy; better wording would be:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“397 is conjectured to be the largest prime that can be represented as the sum of three positive squares of integers in exactly one way”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Let’s confirm that. The only method I know for something like this is brute force. I can make a data frame with three columns, each with all the squares of positive integers up to some maximum point - so the data frame has every combination of those, discarding duplicate combinations by making the order of the columns strictly non-decreasing. Then we sum those three squares, and store the results:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;primes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;30&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;primes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;generate_primes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# This is the part that gets slower with larger k as you make k^3 combinations&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum_3_sq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# example:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum_3_sq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;397&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This gets us our one combination that adds up to 397:&lt;/p&gt;
&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;     s1    s2    s3 sum_3_sq
  &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt;    &amp;lt;dbl&amp;gt;
1     9    64   324      397
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Next step is to count the number of times each result appears.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;s3s_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum_3_sq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# example:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sum_3_sq&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;So for example we see that 54 (which of course is not a prime - we haven’t yet filtered to primes) can be made 3 ways: as the sum of the squares of 1, 2, 7; of 2, 5, 5; and 3, 3, 6:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;  sum_3_sq number_3_square_sums    s1    s2    s3
     &amp;lt;dbl&amp;gt;                &amp;lt;int&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt;
1       54                    3     1     4    49
2       54                    3     4    25    25
3       54                    3     9     9    36
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Then it’s a simple matter of joining that summary (of counts of the number of ways to get a given total of three squares) to a data frame of the prime numbers, and drawing a plot of the results:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;primes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s_sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;p&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sum_3_sq&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace_na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;point&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;397&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;397&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;500&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Prime number&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of ways&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of ways to make a prime number as sum of three positive squares of integers&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;397 (circled) is the largest with exactly one way, of primes up to {comma(k^2)}.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;panel.grid.minor.y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_blank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;There’s a bit of fiddling there to make the charts look nice for low values of k (where k is the maximum number I square). In my real life code the above is surrounded by a loop of different values of k, with the results saved as SVG and PNG images for use in this blog. All of which gets me this result:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0280-primes-squares-k30.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0280-primes-squares-k30.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here we see for some larger results of k:&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0280-primes-squares-k100.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0280-primes-squares-k100.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;As k gets bigger of course the program gets slower to run. This method doesn’t scale well; for primes above about 250,000 the step of making a data frame of all the combinations of three squares starts taking too long. If I wanted to extend this further I’d have to find some ways to do this more efficiently, or put that step into a database that can handle bigger-than-memory data objects.&lt;/p&gt;

&lt;p&gt;I’m sure there’s some interesting maths behind why this is just a “conjecture” and no-one has been able to prove it!&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0280-primes-squares-k300.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0280-primes-squares-k300.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That’s all for today.&lt;/p&gt;
</description>
				<pubDate>Sat, 21 Sep 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/09/21/primes-squares</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/21/primes-squares</guid>
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			<item>
				<title>Stepwise selection of variables in regression is Evil. by @ellis2013nz</title>
					<description>&lt;p&gt;I’ve recently noticed that stepwise regression is still fairly popular, despite being well and truly frowned upon by well-informed statisticians. By stepwise regression, I mean any modelling strategy that involves adding or subtracting variables from a regression model on the basis that they are “significant”, reduce the Akaike Information Criterion, or increase adjusted R-squared, or in fact any other data-driven statistics.&lt;/p&gt;

&lt;p&gt;This might be an automated variable procedure or it might be a matter of eyeballing the results of the first model you fit and saying (for example) “Let’s take literacy out, it’s p-value is not significant, and it will be a more parsimonious model once we do that.”.&lt;/p&gt;

&lt;p&gt;And then people produce and report t tests, F tests, and so on, as though the end model was the one they always intended to run.&lt;/p&gt;

&lt;p&gt;Let me clear about this. This is wrong. It’s not as disastrously wrong as, say, sorting the data separately one column at a time before you fit your model, but it’s still objectively bad. As my professor once told our class:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“If you choose the variables in your model based on the data and then run tests on them, you are Evil; and you will go to Hell.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Why is it wrong? Here are the seven reasons given by Frank Harrell in his must-read classic, &lt;em&gt;Regression Modeling Strategies&lt;/em&gt;:&lt;/p&gt;

&lt;ol&gt;
  &lt;li&gt;The R-squared or even adjusted R-squared values of the end model are biased high.&lt;/li&gt;
  &lt;li&gt;The F and Chi-square test statistics of the final model do not have the claimed distribution.&lt;/li&gt;
  &lt;li&gt;The standard errors of coefficient estimates are biased low and confidence intervals for effects and predictions are falsely narrow.&lt;/li&gt;
  &lt;li&gt;The p values are too small (there are severe multiple comparison problems in addition to problems 2. and 3.) and do not have the proper meaning, and it is difficult to correct for this.&lt;/li&gt;
  &lt;li&gt;The regression coefficients are biased high in absolute value and need shrinkage but this is rarely done.&lt;/li&gt;
  &lt;li&gt;Variable selection is made arbitrary by collinearity.&lt;/li&gt;
  &lt;li&gt;It allows us to not think about the problem.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;At the core of the problem is using statistical inference methods like p values, confidence intervals and ANOVA F tests that were designed and valid for a pre-specified model, but applying them instead to a model we have structured based on the data. The variables are selected partly based on chance, and we are giving ourselves a sneaky headstart in making a variable being significant.&lt;/p&gt;

&lt;p&gt;Basically, this is the sort of thing that leads to the reproducibility crisis in science.&lt;/p&gt;

&lt;p&gt;Some of the problems don’t matter as much if your goal for the model is just prediction, not interpretation of the model and its coefficients. But most of the time that I see the method used (including recent examples being distributed by so-called experts as part of their online teaching), the end model is indeed used for interpretation, and I have no doubt this is also the case with much published science. Further, even when the goal is only prediction, there are better methods like the Lasso, of dealing with a problem of a high number of variables.&lt;/p&gt;

&lt;p&gt;Let’s look at a couple of simulations to show how this is a problem.&lt;/p&gt;

&lt;h2 id=&quot;increases-the-false-positive-rate-even-with-white-noise&quot;&gt;Increases the false positive rate even with white noise&lt;/h2&gt;

&lt;p&gt;First, let’s take a case where we simulate data that is known to have no relation at all to the response variable. In the code below I simulate 1,000 observations with 100 explanatory X variables and 1 response variable y. All of these variables are unrelated to eachother and are just normally distributed with a mean of zero and standard deviation of 1.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;MASS&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;doParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------------X not related to y--------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;42&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvrnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we get 6 variables that look &apos;significant&apos; - about what&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we&apos;d expect, about 5% false positives:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;When I fit a regression model of y ~ X, I should expect about five of the columns to appear ‘significant’ by conventional p value of 0.05 or less - because that’s more or less the definition of that critical cut-off value. That is, we tolerate a 1 in 20 false positive rate. In this case we have six variables below the cut-off, about what we’d expect:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;       Estimate Std. Error   t value   Pr(&amp;gt;|t|)
V2  -0.06606672 0.03213918 -2.055644 0.04010514
V38  0.06881778 0.03406473  2.020206 0.04365814
V62  0.06263414 0.03137298  1.996436 0.04618768
V91  0.06826250 0.03302463  2.067018 0.03901824
V94 -0.07923079 0.03423568 -2.314275 0.02087724
V96 -0.07962012 0.03290373 -2.419790 0.01572689
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Now let’s use stepwise selection, “both” directions (so we can remove variables from the model or add them), using the Akaike Information Criterion to choose a ‘better’ model at each step. This is better than just using p values, and much better than using p values and a low cut-off like 0.05, so I’m giving the stepwise method a fair go here.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gets us these variables showing up as ‘significant’:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;       Estimate Std. Error   t value    Pr(&amp;gt;|t|)
V2  -0.06312445 0.03046917 -2.071748 0.038550346
V4  -0.07353379 0.03222313 -2.282019 0.022702106
V32 -0.06120508 0.03094750 -1.977707 0.048241917
V38  0.06383031 0.03227612  1.977633 0.048250238
V90  0.06288076 0.03094938  2.031729 0.042450732
V91  0.07450724 0.03105172  2.399456 0.016605492
V94 -0.06617689 0.03208892 -2.062297 0.039442821
V96 -0.08052606 0.03073565 -2.619957 0.008930218
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;So the net impact of this fancy-looking automated procedure is to worsen our false positive rate from 6% to 8%.&lt;/p&gt;

&lt;p&gt;OK, that’s just one dataset. Let’s try it with a range of others, of different sample sizes, and to make things more interesting let’s let the X variables sometimes be correlated with eachother. The stepwise selection process can be a bit slow so I spread the 700 runs of the simulation below over seven parallel processes:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# set up parallel processing cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;makeCluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# only any good if you have at least 7 processors :)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;registerDoParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterEvalQ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;MASS&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise_results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;700&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.combine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%dopar%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;400&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;800&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1600&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runif&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvrnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepAIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;trace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# count the false positives&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise_results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise_results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`All variables included`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Stepwise selection of variables`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sample size: {n}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_hline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;gam&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;formula&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of false positives -\nvariables returned as &apos;significant&apos;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation of the X predictor variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;False positive rates when using stepwise variable selection&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Models with 100 X explanatory variables that are in truth unrelated to Y; expecting 5 falsely &apos;significant&apos; variables. 
Small sample sizes make the false positive problem for stepwise selection of variables; multicollinearity in the X when no relation to the Y doesn&apos;t matter.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;The average false positive rate of the full model is 5.1%; for the stepwise variable selection it is 9.5%. In the chart below we can see that sample size relative to the number of variables in X matters a lot here:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-noisy.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-noisy.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;For example, the case with the sample size of only 200 observations gets a false positive rate above 15% from the stepwise method. But even with larger samples, we take a hit in false positives from the stepwise approach. The degree of multicollinearity in the X doesn’t seem to make much difference.&lt;/p&gt;

&lt;p&gt;It might seem unfair to have a model with 100 explanatory variables and only 200 observations, but out there on the internet (I’m not going to link) there are guides telling you it is ok to do this procedure even when you have more variables than observations. In fact I have a horrible fear that this practice might be common in some parts of science. You can imagine how doing &lt;em&gt;that&lt;/em&gt; is basically a machine for generating false, non-reproducible findings.&lt;/p&gt;

&lt;h2 id=&quot;even-the-correctly-retained-variables-coefficients-are-biased-big&quot;&gt;Even the correctly-retained variables’ coefficients are biased big&lt;/h2&gt;

&lt;p&gt;The above simulation was pure noise so everything was a false positive. What does stepwise variable selection do in a more realistic case where some of the variables are correctly in the model and are related to y?&lt;/p&gt;

&lt;p&gt;To explore this I wrote a function (code a little way further down the blog) to simulate data with 15 X correlated variables and 1 y variable. The true model is:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;y = V1 + V2 + V3 + V4 + V5 + V6 + V7 + 0.1 (V8 + V9 + V10) + e
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;That is, the true regression coefficients for variables V1 to V7 are 1; for V8, V9 and V10 they are 0.1; for the the remaining 5 variables there is no structural relationship to y.&lt;/p&gt;

&lt;p&gt;When we simulate 50 data sets of this sort and use stepwise variable selection to regress y on X, here are the coefficients we get. Each point represents the coefficient for one variable from one of those runs.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-main-sim.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-main-sim.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The large dots on zero indicate the multiple runs in which that particular variable was not included in the final model. We see:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;Many occasions, variables V11 to V15 were rightly excluded, but a smattering of occasions they do get included in the model.&lt;/li&gt;
  &lt;li&gt;A lot of false negatives - variables V1 to V10 that should be found in the model and aren’t&lt;/li&gt;
  &lt;li&gt;Worse, when one of variables V1 to V10 is correctly included in the final model, the coefficient estimated for it is &lt;em&gt;always&lt;/em&gt; (in this dataset) larger than the true coefficient (which remember should be 1 or 0.1 - the correct values shown by the red crosses).&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;For comparison, here is an equivalent chart for when we fit the full model to these data. There’s a lot of variation in the coefficient estimates, but at least they’re not biased (that is, on average they are correct, their expected value is the true value):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-main-sim-full.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-main-sim-full.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Here’s the code for the function simulating that data and drawing the plots:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------------when X is related to y--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param xcm correlations of the X variables with eachother, as a multiplier of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   their standard deviation (all X variables have the same variance / sd of 1)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param ysdm standard deviation of the y variable, expressed as a multiplier of variance&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   of the X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param n sample size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param k number of columns in X. Only currently works if this is 15 (because of the hard-coded true_coef)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param runs number of simulations to run&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param seed random seed for reproducibility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_steps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;321&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Sigma, not sigma squared&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvrnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step_mod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepAIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;trace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step_mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;run&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;run&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Full model&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;levels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;complete&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;run&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;levels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient.x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient.y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r.squared&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step_mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r.squared&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mclabel&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;negligible&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.19&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;mild&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.39&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;            &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;strong&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;multicollinearity&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficient value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of observations:\n(usually only one, except when coefficient dropped altogether)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise regression returns coefficient estimates biased away from zero&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Black dots show coefficient estimates from one run of a stepwise (AIC-based) model fitting. Red squares show correct values.
Coefficients of variables left in the model are on average {biases[&apos;Stepwise&apos;]} too large (compared to real value of 0, 0.1 or 1).
Also, real explanatory variables are often dropped. Fake ones are often included.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simulated data with a model that explains about {percent(summary(step_mod)$r.squared)} of variation in response variable, with {mclabel}, by https://freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficient value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Using the full model returns unbiased coefficient estimates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Black dots show coefficient estimates from one run of a all-variables-in model fitting. Red squares show correct values.
Coefficients of variables left in the model are on average {biases[&apos;Full model&apos;]} too large (compared to real value of 0, 0.1 or 1).&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simulated data with a model that explains about {percent(summary(mod)$r.squared)} of variation in response variable, with mild multicollinearity, by https://freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_steps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sim&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# plot results for stepwise selection&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sim&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# plot results for full model&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, you can see that there are a few arbitrary aspects to that simulation - in particular the sample size, the variance of the y relative to the X, and the multicollinearity of the X. The idea of having it as a function is that you can play around with these and see the impacts; for example, if you make the variance of y smaller compared to that of the X, the model gets better at explaining the variance and the stepwise algorithm is less prone to getting things wrong. Rather than include a whole bunch of individual cases, I ran some more simulations covering a range of such values so we can see the relationship to those parameters of the average bias in the estimated regression coefficients remaining in the model.&lt;/p&gt;

&lt;p&gt;So here is the relationship of that bias to the R-squared of the model, at various levels of correlation between the X variables.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-rsquared-bias.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-rsquared-bias.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here is the relationship of the bias to sample size, standard deviation of Y, and correlation between the X all in one chart:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-corr-sd-bias.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-corr-sd-bias.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Of the two visualisations I probably prefer that last one, the heat map. First, it dramatically shows (all that white) that the regression estimates from the true model aren’t biased at all. Secondly, it nicely shows that the bias in the estimates returned by stepwise regression are worse&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;for smaller samples&lt;/li&gt;
  &lt;li&gt;with higher correlation between the X variables&lt;/li&gt;
  &lt;li&gt;and with high variance of the y variable&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;So, finally, here’s the code for those simulations:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# parameters to run this for:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# export onto the cluster some objects we need to use:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterExport&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sim_steps&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;var_params&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# run all the simulations&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;many_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.combine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%dopar%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_steps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# linechart plot:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;many_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;n = {n}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.ordered&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;loess&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;formula&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;span&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;R-squared (proportion of Y&apos;s variance explained by model)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Average bias of estimated variable coefficients&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias in regression coefficients after stepwise selection of variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias is worst with small samples, models with low R-squared, and correlation in the explanatory variables (shown from 0.1 to 0.9).
Model with 15 explanatory &apos;X&apos; variables. Correct values of coefficients are 0, 0.1 or 1;  so a bias of +1 is very serious.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model fitting procedure&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# heatmap plot:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;many_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;n = {n}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_gradientn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colours&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;darkred&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation between the X variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Standard deviation of the Y variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Average bias of estimated variable coefficients:&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias in regression coefficients after stepwise selection of variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias is worst with small samples, high variance response, and correlation in the explanatory variables.
Model with 15 explanatory &apos;X&apos; variables. Correct values of coefficients are 0, 0.1 or 1;  so a bias of +1 is very serious.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all folks. Just don’t use stepwise selection of variables. Don’t use automated addition and deletion of variables, and don’t take them out yourself by hand “because it’s not significant”. Use theory-driven model selection if it’s explanation you’re after, Bayesian methods are going to be good too as a complement to that and forcing you to think about the problem; and for regression-based prediction use a lasso or elastic net regularization.&lt;/p&gt;
</description>
				<pubDate>Sat, 14 Sep 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/09/14/stepwise</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/14/stepwise</guid>
			</item>
		
			<item>
				<title>Gender and sexuality in Australian surveys and census by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;the-2021-abs-standard-for-sex-gender-variations-of-sex-characteristics-and-sexual-orientation-variables&quot;&gt;The 2021 ABS Standard for ‘Sex, Gender, Variations of Sex Characteristics and Sexual Orientation Variables’&lt;/h2&gt;

&lt;p&gt;Over the past two weeks there has been &lt;a href=&quot;https://www.abc.net.au/news/2024-09-01/anthony-albanese-census-gender-identity-intersex/104296702&quot;&gt;quite a controversy&lt;/a&gt; relating to questions about sexuality and gender in the next Australian Census of Population and Housing, in 2026. Things hit the news when the testing of “new questions” was withdrawn, then there was an announcement that “the question” will be included; then apparent clarification that one of “the questions” will be but not all of them.&lt;/p&gt;

&lt;p&gt;My work includes understanding risks for these sorts of controversies (in other countries, not Australia) so I spent a bit of time getting my head around it. The Australian Broadcasting Corporation reports that the &lt;a href=&quot;https://www.abc.net.au/news/2024-09-06/2026-census-questions-revealed/104321662&quot;&gt;three originally proposed questions were&lt;/a&gt;:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;What is the person’s gender?&lt;/li&gt;
  &lt;li&gt;How does the person describe their sexual orientation?”&lt;/li&gt;
  &lt;li&gt;Has the person been told they were born with a variation of sex characteristics?&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The are essentially three of &lt;a href=&quot;https://www.abs.gov.au/statistics/standards/standard-sex-gender-variations-sex-characteristics-and-sexual-orientation-variables/latest-release&quot;&gt;the standard questions on sex, gender, sexual orientation&lt;/a&gt; in the relevant Australian Bureau of Statistics (ABS) Standard that was released in early 2021 and has had some minor tweaks and corrections since. A fourth question, apparently not up for debate, is:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;What was [your/the person’s name/their] sex recorded at birth?&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The standard options for this are “Male”, “Female”, and “Another term (please specify)”.&lt;/p&gt;

&lt;p&gt;At the time of writing, it looks like the sexual orientation and sex at birth questions &lt;em&gt;will&lt;/em&gt; be included in the Census but not those on gender and variation of sex characteristics. I guess this reflects a perception that Australian society / politics has a definite widespread acceptance of variation in sexual orientation - it’s generally seen that people can sleep with and indeed marry whomever they choose. What hasn’t yet become as common apparently is acceptance of a modest degree of complexity in sexual characteristics and gender identity.&lt;/p&gt;

&lt;p&gt;The Australian Standard is well worth a read in full for anyone who has to ask or analyse survey questions relating to sex and gender. It’s a good piece of work, was very thoroughly researched and consulted on. The concepts are explained clearly, most critically the difference between sex relating to physical characteristics and gender as a social and cultural concept.&lt;/p&gt;

&lt;p&gt;The standard isn’t just a list of questions and the wording to use for the answer options (although it does give this, and the carefully defined acceptable variants in limited circumstances, in explicit detail of course). Some of the elements that may not be obvious to people new to this field include, for example, strict rules that the questions have to be asked exactly as written. Interviewers are forbidden from inferring either sex or gender from appearance, names, etc. Or that the preferred word to use for statistics relating to the whole population is “Persons”, not “Total” or “People”, when presenting total population counts for sex.&lt;/p&gt;

&lt;p&gt;As per the Standard, the key variable in the “Cisgender and Trans and Gender Diverse Classification” has to be derived by comparing the answer to “What was this person’s sex recorded at birth” and “What is this person’s gender”.&lt;/p&gt;

&lt;p&gt;There’s a decision table in the Standard on how to do this; it gets complicated when one or both of the questions had “Another Term” selected for sex at birth, or “Prefer not to answer” to gender now. In either of these cases the correct value for Cisgender and Trans and Gender Divsere variable becomes “Inadequately described”.&lt;/p&gt;

&lt;p&gt;If sex at birth differs from gender now, or gender now is “non-binary” or “different term”, the person is classified as “trans”. If sex at birth matches gender now, the person is classified as “cis”. All this of course looks straightforward and simple when set out clearly. It makes sense, and flows naturally from the realisation that cis/trans status is quite different from gender now .&lt;/p&gt;

&lt;p&gt;It is both offensive and leads to poor quality data for example to give people a choice of male, female, trans-male, trans-female in response to a question on gender. This is precisely because the whole point of “transitioning” for many people is that now you are just whatever the final gender is, not an amalgam of previous and current genders - “trans-women &lt;em&gt;are women&lt;/em&gt;”, and so on. So the correct statistical procedure is to ask separate questions on sex at birth and gender now, and combine the two to determine cis / trans status.&lt;/p&gt;

&lt;h2 id=&quot;whats-the-information-for-and-what-we-know&quot;&gt;What’s the information for, and what we know&lt;/h2&gt;

&lt;p&gt;The outcry from the LGBTIQ+ community in response to some of these questions being reported as scrapped from the next Census had it seemed to me two substantive points:&lt;/p&gt;

&lt;ol&gt;
  &lt;li&gt;More detailed information on these issues is needed for government policy and program decision-making, particularly in health and social policy.&lt;/li&gt;
  &lt;li&gt;There is a “right to be counted” and recognised for who people are, and the Census is an important part of the state validating this aspect of people’s existence.&lt;/li&gt;
&lt;/ol&gt;

&lt;blockquote&gt;
  &lt;p&gt;“Without comprehensive and inclusive Australian census data, the full diversity of LGBTQ+ communities remain invisible and marginalised. This unnecessary lack of data hinders proper coordination of public health measures and social policy. We need political leadership.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://aus.social/@Lukas/113035241786325182&quot;&gt;Lukas Parker on Mastodon&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;I think the second of these two arguments - about the rights and symbolism - is actually the stronger.&lt;/p&gt;

&lt;p&gt;Important background on why this is such a sore point is that in the 2021 Census, there was a question on sex that was designed before the 2021 Standard came fully into force (Census questionnaires take years to develop). The very small proportion of people who picked “non-binary” in response to the question on sex was &lt;a href=&quot;https://www.abc.net.au/news/2022-10-07/latham-one-nation-transgender-abs-census-population-problematic/101507074&quot;&gt;unhelpfully picked up by a politician as evidence that there were hardly any transgender people in Australia&lt;/a&gt;. This is clearly an invalid interpretation of a question about sex (not about gender, and certainly not about cis/trans gender status), but one that is unfortunately likely to be repeated in the lack of clearly more appropriate estimates.&lt;/p&gt;

&lt;p&gt;Note that, as described above, the Standard now would require “Another term (please specify)” as the valid third choice for a question on sex at birth. But this still gets one nowhere in terms of counting the number of trans people; a second question on “gender now” is required.&lt;/p&gt;

&lt;p&gt;For health or social policy planning, any question in the Census needs to be justified in terms of the extra benefits gained from making &lt;em&gt;everyone&lt;/em&gt; in the country (not just a survey sample) fill it in. This might be justified, for example, because very fine level of spatial detail is needed that could never be provided by surveys; or a definitive population total is needed for survey design purposes; or we need to know this information on all individuals for later analysis with integrated data.&lt;/p&gt;

&lt;p&gt;I have in fact seen the argument that these population totals are needed for survey calibration and design made in this recent debate but unfortunately have lost the link. But I don’t think it’s particularly compelling, and have yet to see a good technical argument to the contrary. I believe that survey data could give us good enough estimates of the totals for decision-making, down to number of persons per state, and understanding of characteristics, and creating purposive samples for any given relationship to population proportions.&lt;/p&gt;

&lt;p&gt;This is certainly the case for sexual orientation, where we can go (for example) to the latest (2020) &lt;a href=&quot;https://www.abs.gov.au/statistics/people/people-and-communities/general-social-survey-summary-results-australia/2020#data-downloads&quot;&gt;General Social Survey Table 5&lt;/a&gt; and see that 773,000 (our of 20.3m) persons aged 15 or over are estimated to be Gay, Lesbian, Bisexual or ‘Other’.&lt;/p&gt;

&lt;p&gt;The relative margin of error for that estimate is 9.0%, so perhaps a confidence interval would be around 700 and 850 thousand persons - between 3.5% and 4.2% of the total persons aged 15 and over. 9% isn’t great, but I’m confident the measurement error is a bigger risk than sampling error here. That is, people who either don’t understand, don’t know, or don’t want to tell an interviewer (or a government form online) their sexual orientation. A census would remove the sampling error but doesn’t help measurement error.&lt;/p&gt;

&lt;p&gt;Relating to the number of trans persons in Australia, an estimate could be derived from the &lt;a href=&quot;https://www.abs.gov.au/methodologies/national-health-survey-methodology/2022#overview&quot;&gt;2022 National Health Survey&lt;/a&gt;, which included all four necessary questions in the sex, gender, etc Standard mentioned above. Surprisingly given the public interest in the issue, I haven’t been able to find such an estimate published anywhere. From eyeballing and back-of-enveloping the size of confidence intervals in one of the reports based on it, the &lt;a href=&quot;https://www.abs.gov.au/statistics/health/mental-health/national-study-mental-health-and-wellbeing/2020-2022&quot;&gt;National Study of Mental Health and Wellbeing&lt;/a&gt;, I infer that trans people were between 1% and 4% of the &lt;em&gt;sample&lt;/em&gt; of the National Health Survey. However, as young people were deliberately oversampled in the survey and we definitely think age is correlated to trans status, this is surely an overestimate of the population compared to what you could do with the original microdata.&lt;/p&gt;

&lt;p&gt;Someone should do this! Or if it exists, let me know in the comments. One hour’s work (for someone with access to the microdata), compared to the very large cost of including a question in the census. Alternatively, if the data isn’t good enough for estimating the number of trans persons in Australia, we should have a discussion of why and what to do about it. Many of the candidate reasons why - to do with measurement or validity error of some kind - would also mean that using the Census for this purpose wouldn’t help.&lt;/p&gt;

&lt;p&gt;Anyway, my blog is about playing around with data and charts and not politics, so let’s go to a visualisation of some of the published results from the 2020 General Social Survey. This survey included the question on sexual orientation and a good range of results are reported from it. After a lot of iteration I came up with this chart:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0278-lollipops.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0278-lollipops.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;You may need to zoom in on this, but it should be readable on most screens. In this chart I have selected the survey questions that had the largest estimated ratio of difference between heterosexuals and gay, lesbian or bisexual choosing that answer, having first filtered out any questions with less than 5 percentage points of absolute difference. There’s a lot packed in but there’s some interesting things here:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;Why are more young people reporting as gay, lesbian or bisexual? Obviously there are several candidate explanations: people become straighter as they get older (unlikely); more people are gay, lesbian or bisexual than before; or more people are &lt;em&gt;admitting to themselves and interviewers&lt;/em&gt; that they are gay, lesbian or bisexual than before. Most likely the answer is some version of this latter point. Cultural norms have changed astonishingly fast, in historical terms, in this space so it’s not surprising that there is a generational difference in reported prevalence.&lt;/li&gt;
  &lt;li&gt;Gay, lesbian and bisexual people apparently less likely to be unemployed and more likely to be in the highest income quintile&lt;/li&gt;
  &lt;li&gt;Gay, lesbian and bisexual people much more likely to be not married or in a de facto marriage than heterosexual people.&lt;/li&gt;
  &lt;li&gt;Sadly but unsurprisingly, gay, lesbian and bisexual people much more likely to have experienced discrimination or disagree with the statement that the police and justice system can be trusted&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Note that I am very deliberately using the expression “LGB+” here in these charts, not the familiar “LGBTIQ+” expression. LGBTIQ of course does not aspire to be a statistical classification and is more like a social or political coalition; it combines sexual orientation (LGB), sexual characteristics (the “I” for Intersex ) and trans/cis gender status (the “T” for Trans and possibly the “Q” for Queer, although that could really refer to some combination of the three concepts).&lt;/p&gt;

&lt;p&gt;Here’s the code that downloads the data, tidies it up and presents that chart:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readxl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://www.abs.gov.au/statistics/people/people-and-communities/general-social-survey-summary-results-australia/2020/GSS_Table5.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;gss_table5.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;demog_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sex&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Whether currently smokes&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Age group&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Employed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Level of highest non-school qualification&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Engagement in employment or study&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Family composition of household&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Marital status&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Main Source of Household Income&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Current weekly household equivalised gross income quintiles&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_excel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;gss_table5.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;skip&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sheet&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Table 5.1_Estimate&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;down&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_wrap&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;demog_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Characteristics&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Experiences and attitudes&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gay, Lesbian or Bisexual&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gay, Lesbian, Bisexual or Other&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check no typos in the demography categories&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;demog_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# some categories have long answers and are difficult to present on a chart&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difficult_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Community involvement&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cultural tolerance and discrimination&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Family and community support&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crime and safety&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stressors&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw basic faceted barchart&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difficult_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Total persons&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Persons aged 15 years and over&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;dodge&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;free&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of LGB+ and heterosexual attitudes&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Selected questions from Australia&apos;s General Social Survey, 2020&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;axis.text.x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;angle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;45&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# better visualisation that focuses on results.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# number of questions to pick to highlight:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;22&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;spread&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rabs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Age group&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Marital status&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Employed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Engagement in employment or study&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
      &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{category}:\n&apos;{variable}&apos;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lab_seq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;employ&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ignore.case&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-150&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;income&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ignore.case&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Characteristics&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lab_seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rabs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# don&apos;t show any where the absolute difference is less than 5 percentage points:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.026&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.026&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# variable for using to draw colour of segments and arrows:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gay, Lesbian, Bisexual or Other&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Heterosexual&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw lollipop / arrow plot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;free_y&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_segment&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stage&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after_scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prismatic&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clr_lighten&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;space&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;combined&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;angle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;inches&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Prevalence of response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Key differences between LGB+ and heterosexual Australians&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Responses from the General Social Survey 2020&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;In case anyone’s interested, I did over 100 iterations of this chart, looking for something that really presents some analytical findings in an interpretable and aesthetically pleasing way. It’s much harder to do presentation charts than the straightforward ones that just show all the numbers at once. One interesting point is that the great majority of the work is in reshaping and tidying data - in this case, different (but obviously derivative) data frames for drawing the points and for drawing the arrow segments. A nice example of how data management and plot polishing become an iterative process. You’d never publish the data in the format for a plot like this, it needs to be up to the end analyst to identify this as the plot they want, and reshape the data accordingly.&lt;/p&gt;

&lt;p&gt;There was also a lot of effort going into decisions like “Do I need the ‘category’ (e.g. Age group) for each label on the charts, or are some self-explanatory, and how do I code this in without retyping all the data?”.&lt;/p&gt;

&lt;p&gt;That code also produced a more straightforward chart which I show here for comparison. It’s the same info (although question categories with long names are excluded for readability), but puts a lot more work on the reader to pick out the key messages. The final plot I used tries to help the reader with this by doing a bunch of analysis ourselves first and presenting in a way that highlights the most important aspects of it.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0278-many-facets.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0278-many-facets.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That’s all for today. Do read (and follow, in your next survey questionnaire with questions on sex or gender) that standard on sex, gender, etc. - it’s good stuff.&lt;/p&gt;

&lt;h3 id=&quot;late-edits--update&quot;&gt;Late edits / update&lt;/h3&gt;

&lt;ul&gt;
  &lt;li&gt;thanks to @baptnz@social.nz who pointed out on Mastodon a better method of making the arrows appear light rather than the transparency hack I originally used. His method, using the ‘prismatic’ package, I have edited my code above and is now in use.&lt;/li&gt;
  &lt;li&gt;immediately after I wrote all the above I discovered that while I was publishing the original post, the Government has now announced that the questions on sexual orientation, sex at birth, and gender now will all be included in the Census; but not the question on sexual characteristics. So this is a good outcome for those wanting to estimate gender issues and trans/cis status, although those with interest in sexual characteristics are still disappointed.&lt;/li&gt;
&lt;/ul&gt;
</description>
				<pubDate>Sun, 08 Sep 2024 00:00:00 +0800</pubDate>
				<link>https://freerangestats.info/blog/2024/09/08/sex-gender</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/08/sex-gender</guid>
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