Bayesian Analysis of Longitudinal Ordinal Data Using Non-Identifiable Multivariate Probit Models

Abstract

Multivariate probit models have been explored for analyzing longitudinal ordinal data. However, the inherent identification issue in multivariate probit models requires the covariance matrix of the underlying latent multivariate normal variables to be a correlation matrix and thus hinders the development of efficient Bayesian sampling methods. It is known that non-identifiable models may produce Markov Chain Monte Carlo (MCMC) samplers with better convergence and mixing than identifiable models. Therefore, we were motivated to construct a non-identifiable multivariate probit model and to develop efficient MCMC sampling algorithms. In comparison with the MCMC sampling algorithm based on the identifiable multivariate probit model, which requires a Metropolis-Hastings (MH) algorithm for sampling a correlation matrix, our proposed MCMC sampling algorithms based on the non-identifiable model circumvent an MH algorithm by a Gibbs sampler for sampling a covariance matrix and thus accelerate the MCMC convergence. We illustrate our proposed methods using simulation studies and two real data applications. Both the simulation studies and the real data applications show that constructing nonidentifiable models may improve the convergence of the MCMC algorithms compared with the identifiable models. The marginalization of the redundant parameters in the non-identifiable models should be considered in developing efficient MCMC sampling algorithms. This investigation shows that construction of non-identifiable models is valuable in developing MCMC sampling methods and illustrates advantages and disadvantages of construction of non-identifiable models to improve the convergence of the MCMC sampling components.