Model Selection for Multilevel Mixture Rasch Models

Mixture item response theory (MixIRT) models can sometimes be used to model the heterogeneity among the individuals from different subpopulations, but these models do not account for the multilevel structure that is common in educational and psychological data. Multilevel extensions of the MixIRT models have been proposed to address this shortcoming. Successful applications of multilevel MixIRT models depend in part on detection of the best fitting model. In this study, performance of information indices, Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and sample-size adjusted Bayesian information criterion (SABIC), were compared for use in model selection with a two-level mixture Rasch model in the context of a real data example and a simulation study. Level 1 consisted of students and Level 2 consisted of schools. The performances of the model selection criteria under different sample sizes were investigated in a simulation study. Total sample size (number of students) and Level 2 sample size (number of schools) were studied for calculation of information criterion indices to examine the performance of these fit indices. Simulation study results indicated that CAIC and BIC performed better than the other indices at detection of the true (i.e., generating) model. Furthermore, information indices based on total sample size yielded more accurate detections than indices at Level 2.