Computes the Item Information Function (IIF) for the Graded Response Model. The information function indicates how precisely an item measures ability at different theta levels.
Details
The information is Samejima's (1969) item information for the GRM,
computed identically to exametrika::grm_iif (>= 1.15.0.9000)
so that ggExametrika plots match the base plots of the parent
package:
$$I(\theta) = \sum_{k=1}^{K} \frac{[P_k'(\theta)]^2}{P_k(\theta)}$$
where \(P_k(\theta) = P_{k-1}^*(\theta) - P_k^*(\theta)\) is the category response probability, \(P_k^*(\theta)\) is the cumulative (boundary) probability with \(P_0^* = 1\) and \(P_K^* = 0\), and \(P_k^{*\prime}(\theta) = a P_k^*(\theta) [1 - P_k^*(\theta)]\). The logistic metric of the parent package's estimation is used as is (no 1.702 scaling constant).
Examples
# Information at ability = 0 for a 5-category item
ItemInformationFunc_GRM(theta = 0, a = 1.5, b = c(-1, -0.5, 0.5, 1))
#> [1] 0.7024022