Title: On Wald Tests for Differential Item Functioning Detection

Authors: Michela Battauz

Affiliation: University of Udine

Abstract:
Lord's chi-square test is a Wald-type test commonly used for Differential 
Item Functioning detection using Item Response Theory (IRT) methods. 
However, simulation studies reported in the literature showed that empirical 
type I error rates are largely greater than the significance level for the 
3PL model, while they are smaller for the 2PL model.
Two improvements of the test are proposed.
1. Item parameter estimates and their covariance matrix should be transformed 
to a common scale before calculating the test statistics. This transformation 
is performed using the equating coefficients, which are estimated by applying 
the methods proposed in the literature. However, these coefficients are treated 
as known constants in the computation of the covariance matrix. The proposal 
is to account for the estimation of the equating coefficients in the 
computation of the covariance matrix of the transformed item parameters.
2. The parameterization of the IRT model used to estimate the item parameters 
is usually different from the usual IRT parameterization. Item parameters 
estimates (and their covariance matrix) are then transformed to the usual IRT 
parameterization. The proposal is to apply the Wald test to the untransformed 
item parameters.
Simulation studies show that these two proposals lead to empirical type I error
rates closer to the significance level for both the 2PL and the 3PL models.
The procedure applies also to multiple groups.