Multidimensional Multiple Group IRT Models with Skew Normal Latent Trait Distributions

Item response theory (IRT) models are one of the most important statistical tools for psychometric data analysis. Their applicability goes from educational
assessment to biological essays. The IRT models combine, at least, two sets of unknown quantities: the latent traits (person parameters) and item parameters (related to measurement instruments of interest). The multidimensional item response theory (MIRT) models are quite useful to analyze data sets involving multiple skills or latent traits, which occurs in many of the applications. However, most of the works in the literature consider the usual assumption of multivariate (symmetric) normal distribution to the latent traits and do not deal with the multiple group framework (few groups with many of subjects in each one). They, in general, consider a limited
number of model t assessment tools, and do not investigate the measurement instrument dimensionality in a detailed way, while also dealing with the model nonidentiability in a dierent way than that we presented here and only for one group model. In this work, we propose a MIRT multiple group model with multivariate skew normal distributions for modeling the latent traits of each group under the centered parameterization, presenting simple and feasible conditions for model identication. A full Bayesian approach for parameter estimation, structural selection (model comparison and determination of the dimensionality of the measurement instrument) and model
t assessment are developed through Markov Chain Monte Carlo (MCMC) algorithms. The developed tools are illustrated through the analysis of a real data set related to the rst stage of the University of Campinas 2013 admission exam.