On Wald tests for differential item functioning detection

Wald-type tests are a common procedure for DIF detection among the IRT-based methods. However, the empirical type I error rate of these tests departs from the significance level. In this paper, two reasons that explain this discrepancy will be discussed and a new procedure will be proposed. The first reason is related to the equating coefficients used to convert the item parameters to a common scale, as they are treated as known constants whereas they are estimated. The second reason is related to the parameterization used to estimate the item parameters, which is different from the usual IRT parameterization. Since the item parameters in the usual IRT parameterization are obtained in a second step, the corresponding covariance matrix is approximated using the delta method. The proposal of this article is to account for the estimation of the equating coefficients treating them as random variables and to use the untransformed (i.e. not reparameterized) item parameters in the computation of the test statistic. A simulation study is presented to compare the performance of this new proposal with the currently used procedure. Results show that the new proposal gives type I error rates closer to the significance level.