A General Cholesky Decomposition Based Modeling of Longitudinal IRT Data.

In this work we proposed an approach for modeling longitudinal Item Response
Theory (IRT) data based on the work of Pourahmadi (1999), which uses the Cholesky
decomposition of the matrix of variance and covariance (dependence) of interest
which, in our case, is related to the latent traits. One of the most important features
of this approach is that it handle unbalanced data (inclusion and dropouts of
subjects) easily. Also, our modeling can accommodate various covariance (dependence)
structures relatively easily, facilitates the choice of prior distributions for the
parameters of the dependence matrix, facilitates the implementation of estimation
algorithms, allows to consider dierent distributions for latent traits, makes it easier
the inclusion of regression (growth curves) and multilevel structures for the latent
traits, among other advantages. We focus on dichotomous responses, symmetric
normal latent trait 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 measuring instrument, that can be dierent along
the time-points but have some structure of common items. For inference purposes,
an appropriate augmented data structure is considered. The parameter estimation,
model t 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 proposed
methodologies.