Bayesian analysis of multivariate mixed longitudinal ordinal and continuous data

Multivariate longitudinal ordinal and continuous data exist in many scientific fields. However, it is a rigorous task to jointly analyse them due to the complicated correlated structures of those mixed data and the lack of a multivariate distribution. The multivariate probit model, assuming there is a multivariate normal latent variable for each multivariate ordinal data, becomes a natural modeling choice for longitudinal ordinal data especially for jointly analysing with longitudinal continuous data. However, the identifiable multivariate probit model requires the variances of the latent normal variables to be fixed at 1, thus the joint covariance matrix of the latent variables and the continuous multivariate normal variables is restricted at some of the diagonal elements. This constrains to develop both the classical and Bayesian methods to analyse mixed ordinal and continuous data. In this investigation, we proposed three Markov chain Monte Carlo (MCMC) methods: Metropolis--Hastings within Gibbs algorithm based on the identifiable model, and a Gibbs sampling algorithm and parameter-expanded data augmentation based on the constructed non-identifiable model. Through simulation studies and a real data application, we illustrated the performance of these three methods and provided an observation of using non-identifiable model to develop MCMC sampling methods.