Flexible Bayesian Modeling for Longitudinal Binary and Ordinal Responses

TL;DR

This paper introduces a flexible Bayesian functional data analysis approach for modeling longitudinal binary and ordinal responses, overcoming limitations of traditional mixed effects models by nonparametrically modeling subject-specific processes.

Contribution

It develops a hierarchical Bayesian framework using Gaussian and Inverse-Wishart processes to model the mean and covariance of subject-specific stochastic processes for binary and ordinal data.

Findings

Flexible inference for response evolution and correlation

Effective borrowing of strength across subjects

Handles unbalanced longitudinal data efficiently

Abstract

Longitudinal studies with binary or ordinal responses are widely encountered in various disciplines, where the primary focus is on the temporal evolution of the probability of each response category. Traditional approaches build from the generalized mixed effects modeling framework. Even amplified with nonparametric priors placed on the fixed or random effects, such models are restrictive due to the implied assumptions on the marginal expectation and covariance structure of the responses. We tackle the problem from a functional data analysis perspective, treating the observations for each subject as realizations from subject-specific stochastic processes at the measured times. We develop the methodology focusing initially on binary responses, for which we assume the stochastic processes have Binomial marginal distributions. Leveraging the logits representation, we model the discrete space processes through sequences of continuous space processes. We utilize a hierarchical framework to model the mean and covariance kernel of the continuous space processes nonparametrically and simultaneously through a Gaussian process prior and an Inverse-Wishart process prior, respectively. The prior structure results in flexible inference for the evolution and correlation of binary responses, while allowing for borrowing of strength across all subjects. The modeling approach can be naturally extended to ordinal responses. Here, the continuation-ratio logits factorization of the multinomial distribution is key for efficient modeling and inference, including a practical way of dealing with unbalanced longitudinal data. The methodology is illustrated with synthetic data examples and an analysis of college students' mental health status data.