Bayesian general multivariate latent variable modeling of longitudinal item response data

Longitudinal item response data are characterized by examinees that are assessed at different time points or measurement occasions such that time-specific measurements are nested within examinees. Besides the usual nesting of response observations within examinees, the time-specific latent traits are also nested within examinees. In the well-known hierarchical modeling approach, the complex dependencies due to the nested structure of the data are commonly modeled by introducing random effects such that observations and latent traits are conditionally independently distributed. However, the implied compound symmetry structure is often not sufficient to model the complex time-heterogenous dependencies.Therefore, a Bayesian general multivariate item response modeling framework is proposed that accounts for the complex within-examinee latent trait dependencies. Flexible parametric covariance structures are considered to modelspecific within-examinee dependencies. Furthermore, it can handle many measurement occasions, different item response functions, and different latent trait population distributions, and generalizes some works of current literature. Due to identification rules and restricted parametric covariance structures, a conditional modeling approach is pursued to specify proper priors for the unrestricted parameters and to implement an efficient MCMC algorithm, by conditioning on baseline population parameters. The study is motivated by a large-scale longitudinal research program of the Brazilian Federal government to improve the teaching quality and general structure of schools for primary education. It is shown that the growth in math achievement can be accurately measured when accounting for complex dependencies over grades using time-heterogenous covariances structures.