Item Information Function for GRM — grm

Calculates the value of the Item Information Function for the Graded Response Model.

Usage

grm_iif(theta, a, b)

Arguments

theta: Latent trait value of the subject
a: Discrimination parameter of IRF
b: Vector of difficulty parameters (thresholds) of IRF

Value

Value of the Item Information Function

Details

The information is Samejima's (1969) item information for the GRM, $$I(\theta) = \sum_{k=1}^{K} \frac{[P_k'(\theta)]^2}{P_k(\theta)},$$ 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 estimation routine is used as is (no 1.702 scaling constant), so the information is consistent with the parameters returned by GRM and with the posterior standard deviations (PSD).

Examples

if (FALSE) { # \dontrun{
# Example for an item with 3 categories
a <- 1.5
b <- c(-1.0, 1.0)
thetas <- seq(-3, 3, by = 0.1)
info <- sapply(thetas, function(t) grm_iif(t, a, b))
plot(thetas, info, type = "l", xlab = "Theta", ylab = "Information")
} # }