A Copula Based Modeling for Longitudinal IRT Data with Skewed Latent Distributions

In this work we introduce longitudinal IRT model considering skewed latent traits distribution, based on a Gaussian copula function. Differently of the antedependence approach proposed by Santos et al. (2017a) and Santos et al. (2017b), the copula modeling allows the entire control of the respective marginal latent trait distributions, but as the first one, it accommodates several dependence structures. A three parameters probit model for dichotomous items is considered. We assume tests administered to subjects clustered into independent groups, which are followed along several time-points (not necessarily equally spaced). Test have common items and may differ among groups and or time-points. Estimation algorithms, model fit assessment and model comparison tools were developed under the Bayesian paradigm through hybrid MCMC algorithms, such that when the full conditionals are not known, the SVE (Single Variable Exchange) and Metropolis-Hastings algorithms are used. Simulation studies indicate that the parameters are well recovered. Furthermore, a longitudinal study in education, promoted by the Brazilian federal government, is analyzed to illustrate our methodology.