Bayesian Inference for Longitudinal Data with Envelope Models

Abstract

Longitudinal studies play a crucial role in biomedical research, such as monitoring chronic diseases like osteoarthritis. Repeated measures over time enable the tracking of disease pro- gression and treatment effects. However, as the number of covariates increases, the resulting growth in model complexity and number of parameters can impede estimation efficiency and in- terpretability. These challenges are further amplified in the presence of multivariate responses, where analyzing each outcome separately may fail to capture their joint structure. Existing approaches often struggle with high-dimensional data, motivating the need for effective dimen- sion reduction techniques. The envelope model offers a promising solution by identifying the subspace of variation most relevant to the regression, enabling more parsimonious and efficient inference. In addition to dimensionality concerns, modern biomedical datasets often exhibit struc- tural complexities, such as longitudinal designs and binary outcomes. To address these chal- lenges, this dissertation introduces three Bayesian envelope models: the Robust Longitudinal Envelope Model (RoLEM), the Probit Scaled Envelope Model (ProSEM), and the Probit Lon- gitudinal Scaled Envelope Model (ProLSEM). These models extend the envelope model to ac- commodate high-dimensional covariates, repeated measures, and binary outcomes, providing an efficient framework for complex biomedical data analysis. RoLEM is designed for longitudinal data, incorporating a scale mixture of normal dis- tributions to model random errors, thereby enabling robustness to potential outliers. It also accommodates various correlation structures among repeated measures within subjects. Addi- tionally, we introduce a novel prior distribution and a proposal distribution on the Grassmann manifold to facilitate the Bayesian analysis of RoLEM. ii ProSEM extends the envelope framework to probit models with binary responses. It retains the scale-invariant property of the probit model while introducing a scaled-envelope structure for dimension reduction. To ensure model identifiability, a constraint is imposed on the error covariance matrix, and carefully constructed priors are introduced accordingly. This reparameterization reduces the number of free parameters and enhances estimation efficiency, especially when the envelope dimension is small. Building on ProSEM, ProLSEM further in- corporates longitudinal structures by modeling within-subject correlations, thus providing a unified framework for analyzing high-dimensional binary responses with repeated measures. Simulation studies and real data analysis using osteoarthritis data from the OAI study demonstrate the proposed models’ superior performance in terms of estimation accuracy and robustness. Overall, this dissertation makes novel methodological contributions to the analysis of longitudinal data with high-dimensional continuous and high-dimensional binary responses.