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.