A multilevel mixture IRT model for DIF analysis
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The usual methodology for detection of di_erential item functioning (DIF) is to examine di_erences among manifest groups formed by such characteristics as gender, ethnicity, age, etc. Unfortunately, membership in a manifest group is often only modestly related to the actual cause(s) of DIF. Mixture item response theory (IRT) models have been suggested as an alternative methodology to identifying groups formed along the nuisance dimension(s) assumed to be the actual cause(s) of DIF. A multilevel mixture IRT model (MMixIRTM) is described that enables simultaneous detection of DIF at both examinee- and school-levels. The MMixIRTM can be viewed as a combination of an IRT model, an unrestricted latent class model, and a multilevel model. Three perspectives on this model were presented: First, the MMixIRTM can be formed by incorporating mixtures into a multilevel IRT model; second, the MMixIRTM can be formed by incorporating a multilevel structure into a mixture IRT model; and third, the model can be formed by including an IRT model in a multilevel unrestricted latent class model. A fully Bayesian estimation of the MMixIRTM was described including analysis of label switching, use of priors, and model selection strategies along with a discussion of scale linkage. A simulation study and a real data example were presented.