A Bayesian method for analyzing combinations of continuous, ordinal, and nominal categorical data with missing values

From a Bayesian perspective, we propose a general method for analyzing a combination of continuous, ordinal (including binary), and categorical/nominal multivariate measures with missing values. We assume multivariate normal linear regression models for multivariate continuous measures, multivariate probit models for correlated ordinal measures, and multivariate multinomial probit models for multivariate categorical/nominal measures. Then we assume a multivariate normal linear model on the continuous vector comprised of continuous variables and those underlying normal variables for ordinal variables from multivariate probit models and for categorical variables from multinomial probit models. We develop a Markov chain Monte Carlo (MCMC) algorithm to estimate unknown parameters including regression parameters, cut-points for ordinal data from the multivariate probit models, and the covariance matrix encompassing both continuous variables and the underlying normal latent variables. Combining the continuous variables and the normal latent variables allows us to model combinations of continuous, ordinal, and categorical multivariate data simultaneously. The framework incorporates flexible priors for the covariance matrix, provides a foundation for inference about the underlying covariance structure, and imputes missing data where needed. The method is illustrated through simulated examples and two real data applications.