Bayesian Approach to Multivariate Component-Based Logistic Regression: Analyzing Correlated Multivariate Ordinal Data

Applications of component-based models have gained much attention as a means of accompanying dimension reduction in the regression setting and have been successfully implemented to model a univariate outcome in the behavioral and social sciences. Despite the prevalence of correlated ordinal outcome data in the fields, however, most of the extant component-based models have been extended to address the multivariate ordinal issue with a simplified but unrealistic assumption of independence, which may lead to biased statistical inferences. Thus, we propose a Bayesian methodology for a component-based model that accounts for unstructured residual covariances, while regressing multivariate ordinal outcomes on pre-defined sets of predictors. The proposed Bayesian multivariate ordinal logistic model re-expresses ordinal outcomes of interest with a set of latent continuous variables based on an approximate multivariate t-distribution. This contributes not only to developing an efficient Gibbs sampler, a Markov Chain Monte Carlo algorithm, but also to facilitating the interpretation of regression coefficients as log-transformed odds ratio. The empirical utility of the proposed method is demonstrated through analyzing a subset of data, extracted from the 2009 to 2010 Health Behavior in School-Aged Children study that investigates risk factors of four different forms of bullying perpetration and victimization: physical, social, racial, and cyber.