A General Cholesky Decomposition Based Modeling of Longitudinal IRT Data: Handling Skewed Latent Traits Distributions

José Roberto Silva dos Santos
Caio Lucidius Naberezny Azevedo

In this work we develop a longitudinal IRT model considering skewed latent traits distribution, based on the work of Pourahmadi (1999), which uses the Cholesky decomposition of the matrix of variance and covariance (dependence) of interest related to the latent traits. A kind of multivariate skew-normal distribution for the latent traits is induced by an antedependence model with centered skew-normal erros. We focus on dichotomous responses considering skewed latent traits distributions and a single group of individuals followed over several evaluation conditions (time-points). In each of these evaluation conditions the subjects are submitted to a (possibly different along these time-points) measuring instruments which have some common items structure. Using an appropriate augmented data framework, a longitudinal IRT model is developed through the Pourahmadi’s approach. The parameter estimation, model fit assessment and model comparison were implemented through a hybrid MCMC algorithm, such that when the full conditionals are not known, the SVE (Single Variable Exchange) algorithm is used. Simulation studies indicate that the parameters are well recovered. In addition, a longitudinal study in education, promoted by the Brazilian federal government, is analyzed to illustrate the methodology developed.