Bayesian hierarchical models for multivariate mixed responses with repeated measures: A case study in arterial occlusive disease

Modeling repeated measures of arterial occlusive diseases, such as peripheral artery disease (PAD), using data with mixed-type outcomes poses unique challenges due to complex dependency structures and diverse distributional assumptions. This study proposes a comprehensive Bayesian hierarchical modeling framework for the simultaneous analysis of binary and continuous outcomes observed repeatedly within individuals. We focus on the methodological comparison of three major Markov Chain Monte Carlo (MCMC) Bayesian computational methods-Metropolis-Hastings, Gibbs sampling, and Hamiltonian Monte Carlo (HMC) that are suitable for hierarchical models without random effects, as well as those with random intercepts and slopes, by utilizing arterial occlusive disease (AOD) data that includes repeated leg measurements on 16 patients with a total of 256 observations. We evaluate model performance across multiple criteria, including the widely applicable information criterion (WAIC), Leave-one-out information criteria (LOO-IC), K-fold cross-validation (K = 10), the Bayesian information criterion (DIC), and the BIC information complexity (ICOMP). Our results reveal that the full random effects model estimated via HMC performed better and achieved higher predictive accuracy across the considered information criteria for this small-sample, historical dataset used for modeling applications. This work emphasizes the importance of model selection strategies in hierarchical Bayesian analysis and highlights the advantages of employing modern MCMC techniques in medical applications. However, we realize that these findings may depend on the precise priors and parameterizations used and may not apply to all small-sample hierarchical datasets. Thus, expanding this model to larger, contemporary datasets will improve its generalizability and clinical relevance.