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.

A brief review of various information criteria is presented for the detection of differential item functioning (DIF) under item response theory (IRT). An illustration of using information criteria for model selection as well as results with simulated data are presented and contrasted with the IRT likelihood ratio (LR) DIF detection method. Use of information criteria for general IRT model selection is discussed.

The procedures required to obtain the approximate posterior standard deviations of the parameters in the three commonly used item response models for dichotomous items are described and used to generate values for some common situations. The results were compared with those obtained from maximum likelihood estimation. It is shown that the use of priors may reduce the instability of estimates of the item parameters, assuming the choice of priors is reasonable. The sample size required for acceptable accuracy for the purposes of practical applications of item response theory may be inferred from tables or computer programs. It is suggested that the careful selection of priors be exercised to obtain the required precision when three-parameter models are applied in these situations.

Covariates have been used in mixture IRT models to help explain why examinees are classed into different latent classes. Previous research has considered manifest variables as covariates in a mixture Rasch analysis for prediction of group membership. Latent covariates, however, are more likely to have higher correlations with the latent class variable. This study investigated effects of including latent variables as covariates in a mixture Rasch model, in presence of and in absence of interactions between the covariates. Results indicated the latent and manifest covariates influenced latent class membership but did not have much influence on class ability means or class proportions. The influence was relatively higher for latent covariates compared to manifest covariates. The effects of the covariates on class membership and on item parameters were class specific. Substantial effects of covariates on item parameters yielded smaller standard errors for item parameter estimates. A significant interaction term also had an effect on the coefficients for predicting and explaining latent class membership.