information: Fisher and Observed Information for an Item Response Model

In ltbayes: Simulation-Based Bayesian Inference for Latent Traits of Item Response Models

Description

information computes the Fisher (test, item, and category) and observed (test only) information for an item response model. Fisher information can be computed at any specified value of zeta but observed information is computed only at the MLE.

Usage

information(fmodel, y, zeta, observed = FALSE, ...)

Arguments

fmodel	Function with first argument zeta which returns a list of the (unnormalized) natural logarithm of the posterior distribution evaluted at zeta and a m by r matrix of item category response probabilities. These must be named post and prob, respectively. The posterior should assume an (improper) uniform prior for zeta, and will impose this proper if fmodel has a prior argument.
y	Vector of length m for a single response pattern, or matrix of size s by m of a set of s item response patterns. In the latter case the posterior is computed by conditioning on the event that the response pattern is one of the s response patterns. Elements of y should be integers from 0 to r-1 where r is the number of response categories.
zeta	The value of the latent trait at which to compute Fisher information. Observed information is always computed at the MLE regardless of zeta. The default is the MLE.
observed	Logical to determine if the observed information is computed. The default is FALSE, but only the observed information can be computed if y has two or more rows.
...	Additional arguments to be passed to fmodel such as item parameters, or to the numerical routines for calculating first- and second-order (parital) derivatives.

Details

The Fisher information is defined here as the negative of the expected value of the second-order derivative of the log-likelihood function for ζ_i. This is the test information function. The item and category Fisher information functions are defined by decomposing this quantity by item and category, respectively (see Baker & Kim, 2004). The observed information is the second-order derivative of the log-likelihood evaluated evaluated at the MLE of ζ_i which is computed using postmode. The observed information function is only computed here for the test. The Fisher information cannot be computed by information if y has more than one row (i.e., more than one response pattern).

Value

test	Test information at zeta.
item	Item information at zeta (NA if observed = FALSE).
category	Category informtation at zeta (NA if observed = FALSE).

Note

For generality information computes Fisher and observed information using numerical (partial) differentiation even when closed-form solutions exist. Thus even though it does not depend on y that argument must still be provided for computational purposes. General and some model-specific closed-form formulas for test/item/category Fisher information are given by Baker and Kim (2004).

Author(s)

Timothy R. Johnson

References

Baker, F. B. & Kim, S. H. (2004). Item response theory: Parameter estimation techniques (2nd ed.). New York, NY: Marcel-Dekker.

Examples

alph <- c(1.27,1.34,1.14,1,0.67)   # discrimination parameters
beta <- c(1.19,0.59,0.15,-0.59,-2) # difficulty parameters
gamm <- c(0.1,0.15,0.15,0.2,0.01)   # lower asymptote parameters

# Fisher information of a three-parameter logistic binary model
information(fmodel3pl, y = c(0,1,1,1,1), apar = alph, bpar = beta, cpar = gamm)

# plot of Fisher information functions for each item
zeta <- seq(-5, 5, length = 100)
info <- matrix(NA, 100, 5)
for (j in 1:100) {
	info[j,] <- information(fmodel3pl, c(0,1,1,1,1), zeta = zeta[j], 
		apar = alph, bpar = beta, cpar = gamm)$item
}
matplot(zeta, info, type = "l", ylab = "Information", bty = "n", xlab = expression(zeta))
legend(-3, 0.3, paste("Item", 1:5), lty = 1:5, col = 1:5)

# observed information given a sum score of 4
information(fmodel3pl, patterns(5, 2, 4), apar = alph, bpar = beta, cpar = gamm,
	observed = TRUE)

ltbayes documentation built on May 2, 2019, 12:40 p.m.