Bayesian semiparametric correlation models for longitudinal data with applications to an HIV/AIDS biomarker study

In longitudinal data analysis, parametric covariance models rely on strong assumptions, while the unstructured covariance model has too many parameters and can not often be fit to high dimensional unbalanced data. As an alternative, I start with the variance-correlation decomposition and propose two rich families of Bayesian semi-parametric stationary correlation models. One approach is a mixture of simple structure correlation matrices; the second approach models correlations as a convex monotone B-spline function of time lags. Both correlation models satisfy the positive definite constraint. They are capable of handling large dimensional, highly unbalanced data and are feasible in a Bayesian framework. Further I develop a matrix mixture correlation model which extends my stationary correlation models to the nonstationary correlation situation. The nonstationary model inherits strengths from the stationary mixture correlation model and allows the correlation to change in both value and structure simultaneously. For correlations with a known change point or correlations continuously changing over time, there is an appropriate version of the model. I present proper but uninformative priors for correlation parameters and illustrate how substantive prior information can be incorporated in the prior specification. I compare my models to each other and to standard models using DIC, cross-validation and log marginal likelihood. Simulations show that our model works well and DIC, cross-validation and log marginal likelihood choose similar models. I illustrate my methods with an unbalanced dataset of CD4 cell counts and mice growth data.