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* new formula * h_getBetaMixPost reasoning * clean * clean * added my name+email * clean * notations cleaned * removed unused term in wordlist * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * clean * clean * clean * clean * checking formulars first * clean * clean * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * clean * Update vignettes/introduction.Rmd Co-authored-by: Daniel Sabanes Bove <[email protected]> * Auto stash before merge of "45_h_getbetaMixPost_simplify" and "origin/45_h_getbetaMixPost_simplify" * clean * clean --------- Co-authored-by: Daniel Sabanes Bove <[email protected]>
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@@ -3,6 +3,8 @@ title: "`phase1b`: Tools for Decision Making in Phase 1b Studies" | |
| package: phase1b | ||
| bibliography: '`r system.file("REFERENCES.bib", package = "phase1b")`' | ||
| author: | ||
| - name: Audrey Yeo | ||
| email: [email protected] | ||
| - name: Tony Pourmohamad | ||
| email: [email protected] | ||
| - name: Jiawen Zhu | ||
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@@ -242,7 +244,7 @@ Under the Bayesian approach, the parameter $P_E$ is treated as a random unknown | |
| quantity in the model. The prior distribution for $P_E$ can be elicited from | ||
| external knowledge or expert opinion, however, the use of a conjugate prior is | ||
| employed here. The prior used in `postprob()` is of the following form: | ||
| $P_E\sim \sum_{i=1}^kw_i\text{Beta}(a_i,b_i)$, so a mixture of Beta | ||
| $P_E\sim \sum_i=1}^kw_i\text{Beta}(a_i,b_i)$, so a mixture of Beta | ||
| distributions, but for simplicity, the default prior in `postprob()` is | ||
| set to be the single conjugate Beta distribution, $\text{Beta}(\alpha,\beta)$. | ||
| Here we make the assumption that a (mixture) Beta prior is a reasonable choice | ||
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@@ -266,17 +268,76 @@ following Beta distribution | |
| P_E\given x\sim\text{Beta}(a+x,b+n-x) | ||
| \end{equation} | ||
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| Of course we could have carried out this calculation using the more general prior, | ||
| $P_E\sim \sum_{i=1}^kw_i\text{Beta}(a_i,b_i)$, however, for sake of illustration | ||
| (and to avoid tedious bookkeeping) we chose to use the simpler non-mixture of | ||
| Beta distributions for ease of calculation. | ||
| Here we illustrate a posterior of a simple prior as an illustration | ||
| $P_E\sim \sum_{j=1}^kw_i\text{Beta}(a_j,b_j)$. In section \@ref(updateposterior), we illustrate a mixture Beta distribution and how weights and parameters are updated with data. | ||
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| All further probabilistic calculations follow naturally under the Bayesian approach | ||
| once the posterior distribution has been identified. For example, as seen in | ||
| the function `postprob()`, one could calculate the posterior probability that the | ||
| rate $P_E$ is greater than (or less than) some threshold $p$, i.e., $\Pr(P_E>p\given x)$, | ||
| by simply calculating the area under the corresponding Beta distribution. | ||
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| ## Updating Posterior Distribution parameters and weights {#updateposterior} | ||
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| In order to incorporate prior information and evaluate posterior probabilities, updated weights and parameters are required especially where mixed Beta-Binomial distribution is involved. To demonstrate the calculation of how parameters and weights are updated, we | ||
| illustrate the reasoning as follows. | ||
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| We present the Binomial likelihood and the mixture of Beta distributions prior to begin: | ||
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| \begin{align} | ||
| f(x|\pi) &= \binom{n}{k} \pi^x (1-\pi)^{n-x} \\ | ||
| f(\pi) &= \sum_{j = 1}^{k} w_j \frac {1} {B(\alpha_j, \beta_j)} \pi^{\alpha_j-1} (1-\pi)^{\beta_j-1} | ||
| (\#eq:binomial) | ||
| \end{align} | ||
| <!-- (\#eq:weightedBetaPrior) --> | ||
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| Where the sum $w_j$ has to equal 1: $\sum_{j = 1}^{k} w_j = 1$. | ||
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| Each component $j$ has the Beta density: | ||
| \begin{align} | ||
| f(\pi | \alpha_j, \beta_j) = \frac {1} {B(\alpha_j, \beta_j)} \pi^{\alpha_j-1} (1-\pi)^{\beta_j-1} | ||
| (\#eq:unweightedBetaPrior) | ||
| \end{align} | ||
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| Here $B(\alpha, \beta)$ is the beta function evaluated with parameters $\alpha$ and $\beta$. | ||
| Then, the product of likelihood and prior $f(\pi | x) \propto f(\pi) f(x | \pi)$ gives the kernel for the posterior where we exclude the normalising constants $\frac{1}{f(x)}$ as well as any other multiplicative constants not depending on $\pi$: | ||
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| \begin{align} | ||
| f(\pi | x) &\propto \pi^x (1-\pi)^{n-x}\sum_{j = 1}^{k} \ w_j \frac {1}{B(\alpha_j, \beta_j)} \pi^{\alpha_j-1}(1-\pi)^{\beta_j-1} | ||
| \\ | ||
| &\propto \sum_{j = 1}^{k} \frac {w_j}{B(\alpha_j, \beta_j)} \pi^{\alpha_j + x -1} (1-\pi)^{\beta_j + n - x -1} \frac{B(\alpha_j + x, \beta_j + n - x)}{B(\alpha_j + x, \beta_j + n - x)} | ||
| (\#eq:rearrangingWeightedBayes) | ||
| \end{align} | ||
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| From \@ref(eq:rearrangingWeightedBayes), we know that the updated Beta prior $f(\pi | \alpha + x_j, \beta_j + n -x) = \frac {1} {Beta(\alpha_j + x, \beta_j + n - x)}\pi^{\alpha_j + x -1} (1-\pi)^{\beta_j + n - x -1}$ is embedded and can be simplified to the following: | ||
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| \begin{align} | ||
| f(\pi | x) \propto \sum_{j = 1}^{k} {w_j} \frac {B(\alpha_j + x, \beta_j +n -x )}{B(\alpha_j, \beta_j)} f(\pi | \alpha_j + x, \beta_j + n - x) | ||
| (\#eq:simplifiedWeightedBayesone) | ||
| \end{align} | ||
| \ | ||
| We can identify the shape of the density as that of a mixture distribution with weights being proportional to $w_j$ : | ||
| \frac {B(\alpha_j + x, \beta_j +n -x )}{B(\alpha_j, \beta_j)} | ||
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| \begin{align} | ||
| f(\pi | x) \propto \sum_{j = 1}^{k} w_j \frac {B(\alpha_j + x, \beta_j + n -x )}{B(\alpha_j, \beta_j)} | ||
| (\#eq:simplifiedWeightedBayestwo) | ||
| \end{align} | ||
| \ | ||
| The normalizing constant is thus: | ||
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| \begin{align} | ||
| \frac {1}{\sum_{j=1}^{k}w_j\frac{B(\alpha_j + x, \beta_j + n - x)}{B(\alpha_j, \beta_j)}} | ||
| (\#eq:normalizingconstant) | ||
| \end{align} | ||
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| such that the updated weights are given as : | ||
| \ | ||
| \begin{align} | ||
| \tilde{w}_j = w_j \frac {\frac {B(\alpha_j + x, \beta_j +n -x )}{B(\alpha_j, \beta_j)}} {\sum_{j} w_j \frac {Beta(\alpha_j + x, \beta_j +n -x )}{B(\alpha_j, \beta_j)}} | ||
| (\#eq:calcweights) | ||
| \end{align} | ||
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| ## Difference in beta random variables {#betadiff} | ||
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| One may wish to know the distribution of the difference in response rates between groups. | ||
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@@ -1050,7 +1111,7 @@ while the advanced predictive probability design should be used. | |
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| An advantage of such a design is that it allows an option of further evaluation | ||
| on additional clinical endpoints or PD markers when the decision falls into the | ||
| "gray zone". Moreover, a seperate delta can be specified for the futility decision | ||
| "gray zone". Moreover, a separate delta can be specified for the futility decision | ||
| to represent a "pool clinical improvement", which makes it a more flexible | ||
| decision rule. | ||
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@@ -1091,7 +1152,7 @@ calculate the *PPs*, if there are 18 PET-CR at the first interim analysis: | |
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| ```{r ex3:predictive, echo=TRUE} | ||
| # PP(event A) | ||
| predprobDist( | ||
| result1 <- predprobDist( | ||
| x = 18, n = 25, Nmax = 80, | ||
| delta = 0.15, | ||
| thetaT = 0.6, | ||
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@@ -1101,7 +1162,7 @@ predprobDist( | |
| ## PP(event B) | ||
| 1 - predprobDist( | ||
| result2 <- 1 - predprobDist( | ||
| x = 18, n = 25, Nmax = 80, | ||
| delta = 0.05, | ||
| thetaT = 1 - 0.6, | ||
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