[docs] add example on se

src/information/information.jl

@@ -220,6 +220,8 @@ item_observed_information(
 It computes the sum of the item observed informations across the responses.
 Items (latents) must be of the same type.
 
+# Example 1
+
 ```@example
 examinees = [Examinee(n) for n = 1 : 100]; #default examinee factory (1-D latent)
 items_2PL = [Item(i, Parameters2PL()) for i = 1 : 30]; #2PL item factory
@@ -233,6 +235,28 @@ responses = answer(examinees, items) # generate responses
 obs_items_info_2PL = item_observed_information(items_2PL, examinees, responses)
 obs_items_info_3PL = item_observed_information(items_3PL, examinees, responses)
 ```
+
+# Example 2. Compute the standard errors of item parameter estimates using the observed responses and estimated abilities.
+
+```@example
+# The package LinearAlgebra is required to compute the inverse of the matrix.
+using LinearAlgebra
+
+# Randomly generate examinees, items and responses.
+examinees = [Examinee(n) for n = 1 : 10]
+items = [Item(i) for i = 1:40];
+responses = answer(examinees, items);
+responses_per_item = map(i -> get_responses_by_item_id(i.id, responses), items);
+
+items_obs_info = [sum([item_observed_information(items[i], examinees[r.examinee_idx], r) for r in responses_per_item[i]] for i in 1:40];
+
+#Using the inverse of the matrix:
+inv_items_obs_info = inv.(items_obs_info);
+item_parameters_standard_errors_inv = [[sqrt(i[1,1]), sqrt(i[2,2])] for i in inv_items_obs_info]
+
+#Using the inverse of the diagonal:
+item_parameters_standard_errors_diag = [[sqrt(1/i[1,1]), sqrt(1/i[2,2])] for i in items_obs_info]
+```
 """
 function item_observed_information(
     items::Vector{<:AbstractItem},

src/information/item_expected_information.jl

@@ -50,9 +50,10 @@ A ``2 \times 2`` matrix of the expected informations.
 """
 function _item_expected_information(parameters::Parameters2PL, latent::Latent1D)
     p = _probability(latent, parameters)
-    i_aa = (1 - p) * p * (latent.val - parameters.b)^2
-    i_ab = -parameters.a * (1 - p) * p * (latent.val - parameters.b)
-    i_bb = parameters.a^2 * (1 - p) * p
+    p_1_p = (1 - p) * p
+    i_aa = p_1_p * (latent.val - parameters.b)^2
+    i_ab = -parameters.a * p_1_p * (latent.val - parameters.b)
+    i_bb = parameters.a^2 * p_1_p
     return [i_aa i_ab; i_ab i_bb]::Matrix{Float64}
 end
 

src/information/item_observed_information.jl

@@ -28,7 +28,10 @@ function _item_observed_information(
     latent::Latent1D,
     response_val::Float64
 )
-    return _item_expected_information(parameters, latent)::Matrix{Float64}
+    p = _probability(latent, parameters)
+    i_bb = - (- p^2) * (1 - p) * p / (p^2) # v p.49 Kim, Baker: - L_22
+    return i_bb::Float64
+   # return _item_expected_information(parameters, latent)::Matrix{Float64}
 end
 
 ########################################################################
@@ -45,7 +48,9 @@ _item_observed_information(
 
 # Description
 
-It is equal to the item expected information.
+Theoretically, it should be equal to its expected counterpart.
+However, it is used to compute the standard errors of item parameter estimates which are based on the log-likelihood (which is based on the responses).
+Look at the example in the documentation of the exported method.
 
 # Arguments
 
@@ -54,7 +59,7 @@ It is equal to the item expected information.
 - **`response_val::Float64`** : Required. A scalar response. 
 
 # Output
-A ``2 \times 2`` matrix of the observed (expected) informations. 
+A ``2 \times 2`` matrix of the observed informations. 
 """
 function _item_observed_information(
     parameters::Parameters2PL,
@@ -63,9 +68,7 @@ function _item_observed_information(
 )
     #return _item_expected_information(parameters, latent)::Matrix{Float64}
     p = _probability(latent, parameters)
-    i = (1 - p) * p
-    h = (- p^2) * i
-    j = response_val * i
+    h = (- p^2) * (1 - p) * p
     den = p^2
     i_aa = - h * (latent.val - parameters.b)^2 / den # v p.49 Kim, Baker: - L_11
     i_ab =