Time series and multilevel modeling for longitudinal item response theory data

Longitudinal Item Response Theory (IRT) data occurs when experimental units are submitted to measurement instruments (e.g., cognitive test, psychiatric questionaires, biological essays among others) along different assessment conditions, as different time points. Very often, in this kind of study, we are interested in the so-called latent variables (or latent traits) and their behavior along these conditions, including the modeling of their inter-dependency structure. In this work we use some stationay and nonstationary time series and multilevel models to represent longitudinal IRT data. More specifically, we consider first order auto-regressive (AR(1)), first order moving average (MA(1), first order auto-regressive- moving average (ARMA(1,1)) time series models as well as the Uniform and Hankel dependency structures, induced by appropriate multilevel models. These structures are studied under a time-homocedastic and time-heteroscedastic fashions. We developed a Bayesian inference framework, which includes parameter estimation, model fit assessment and model comparison, through MCMC algorithms. Simulation studies are conducted in order to measure the parameter recovery and model comparson tools. A real data analysis, concerning a longitudinal cognitive study for Mathmatics achievement, conducted by the Federal Brazilian government, is performed. All computational implementations are made through the WinBUGS program, using the R2WinBUGS package, from R program.