Bayesian General Cholesky Decomposition Based Modeling of Longitudinal Multiple-Group IRT Data with Skewed Latent Distributions and Growth Curves

José Roberto Silva dos Santos
Caio Lucidius Naberezny Azevedo

In this work we introduce a multiple-group 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. In addition, we consider growth curve models for the mean of the latent traits. 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. Using an appropriate augmented data structure, 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) and Metropolis-Hastings algorithms are used. Simulation studies indicate that the parameters are well recovered. Furthermore, a longitudinal study extracted from the Amsterdam Growth and Health Longitudinal Study (AGHLS), that monitor health and life-style of Dutch teenagers, was analyzed to illustrate our model.