Bayesian modeling of multivariate mixed longitudinal responses with scale mixtures of multivariate normal distributions

Program Summary (Abstract) Health-related studies generally involve more than one longitudinal response composed of multiple types of data, such as binary, ordinal, nominal or continuous variables. Since these responses are collected from the same individual or unit, it is desirable to analyze them jointly instead of separately to understand the data as

a whole. The multivariate probit models have been widely utilized for analyzing multivariate longitudinal binary and ordinal data and especially for mixed binary/ordinal and continuous data due to the assumption of the latent multivariate normal variables. However, this only option of the underlying multivariate normal variables

makes limited model comparisons and diagnostics. Furthermore, the identifiable multivariate probit models constrain the covariance matrix of the latent multivariate normal variables to be a correlation matrix, which brings a rigorous task for both likelihood-based estimation and Markov chain Monte Carlo (MCMC) sampling.

Similar issues also exist in multinomial probit models for analyzing nominal data. In this proposal we focus on developing MCMC methods to analyze multivariate mixed longitudinal data with three main purposes. The first purpose is to use scale mixtures of multivariate normal (SMMVN) distributions, which provide flexible multivariate distributions for latent variables, such as multivariate normal, multivariate-

t and multivariate logistic distributions. The second purpose is to propose identifiable models using SMMVN distributions and develop the MCMC sampling methods. The third purpose is to tackle the model identification issue by proposing non-identifiable models and develop MCMC methods to circumvent a Metropolis-Hastings

algorithm to sample restricted covariance matrices by a Gibbs sampling covariance matrix without restrictions. The Specific Aims are to: (1) Construct both identifiable and non-identifiable multivariate models for multivariate longitudinal binary/ordinal data with SMMVN distributions and develop the MCMC sampling

methods; (2) Construct both identifiable and non-identifiable multivariate models for multivariate longitudinal nominal data with SMMVN distributions and develop the MCMC sampling methods; (3) Extend the multivariate models proposed in (1) and (2) to multivariate mixed longitudinal data and develop the MCMC

sampling methods for data with missing values and perform model assessment; (4) Implement, distribute, support and maintain user friendly software packages for the methods proposed in this application. This proposal is consistent with the objectives of NIH AREA Program (R15) by enhancing the infrastructure

of research and education at Michigan Technological University (MTU). This application will offer a unique opportunity to expose a diverse group of undergraduates and graduates to health-related research involving statistical theories, statistical applications, computational methods and data applications at the cutting-edge

of modern research and strengthen the health-related research and research environment at MTU.