Bayesian estimation of a multilevel multidimensional item response model using auxiliary variables method: an exploration of the correlation between multiple latent variables and covariates in hierarchical data

Within the framework of Bayesian analysis, we present a multilevel multidimensional item response modeling and estimation method to study the relations among multiple abilities and covariates in a hierarchical data structure. The proposed method is well suited to examining a scenario in which a test measures multidimensional latent traits (e.g., reading ability, cognitive ability, and computing ability) and in which students are nested within classes or schools. The developed Gibbs sampling algorithm based on auxiliary variables can accurately estimate the correlations among multidimensional latent traits, along with the correlation between person- and school-level covariates and latent traits. Three information criteria and the pseudo-Bayes factor approach are used to evaluate model fit and make model comparison. Simulation studies show that the proposed method works well in estimating all model parameters across a broad spectrum of scenarios. A case study on an educational assessment data is investigated to demonstrate the practical application of the proposed procedure.