Score‐based measurement invariance checks for Bayesian maximum‐a‐posteriori estimates in item response theory

A family of score‐based tests has been proposed in recent years for assessing the invariance of model parameters in several models of item response theory (IRT). These tests were originally developed in a maximum likelihood framework. This study discusses analogous tests for Bayesian maximum‐a‐posteriori estimates and multiple‐group IRT models. We propose two families of statistical tests, which are based on an approximation using a pooled variance method, or on a simulation approach based on asymptotic results. The resulting tests were evaluated by a simulation study, which investigated their sensitivity against differential item functioning with respect to a categorical or continuous person covariate in the two‐ and three‐parametric logistic models. Whereas the method based on pooled variance was found to be useful in practice with maximum likelihood as well as maximum‐a‐posteriori estimates, the simulation‐based approach was found to require large sample sizes to lead to satisfactory results.