A Power Formula for the Mantel-Haenszel Test for Differential Item Functioning.

The asymptotic power of the Mantel-Haenszel (MH) test for the differential item function (DIF) is derived. The formula describes the behavior of the power when the number of items is large, so that the measured latent trait can be considered as the matching variable in the MH test. As shown in the derived formula, the power is related to the sample size, effect size of DIF, the item response function (IRF), and the distribution of the latent trait in the reference and the focal groups. The formula provides an approximation of the power of the MH test in practice and thus provides a guideline for DIF detection in practice. It also suggests analytical explanations of the behavior of the MH test as observed in many previous simulation studies. Based on the formula, this study shows how to conduct the sample size calculation. The power of MH test under some practical models such as the two-parameter logistic (2PL) and three-parameter logistic (3PL) item response theory (IRT) models is discussed.