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Date of Award

2024

Document Type

Campus Access Dissertation

Degree Name

Doctor of Philosophy in Computational Science and Engineering (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Xiao Zhang

Committee Member 1

Kui Zhang

Committee Member 2

Qiuying Sha

Committee Member 3

Hairong Wei

Abstract

This dissertation explores advanced Bayesian inference methods for analyzing longitudinal binary and ordinal data, with a focus on multivariate probit models. In medical and survey research, responses often follow an ordinal structure, presenting unique challenges due to their discrete, ordered nature. Traditional statistical methods, including Generalized Linear Mixed Models (GLMMs), Generalized Estimating Equations (GEE), and joint maximum likelihood estimation model, partially address these challenges but face limitations in computational complexity and efficiency, particularly when analyzing high-dimensional longitudinal data. Extensive simulation studies demonstrate the superiority of PX-GS and PX-GSM models over traditional methods in terms of estimation accuracy and convergence speed while real data applications further validate the practical effectiveness of these models in handling complex data structure and dependencies. Part 1 focuses on the analysis of multivariate longitudinal ordinal data. It reviews GLMMs for capturing individual variability through random effects in repeated measures, as well as GEE, which estimate population-averaged effect without assuming independence between observations. This section compares two frequentist approaches (MIXOR and mvord) with three Bayesian sampling algorithms: one for identifiable model (PX-MH) and two for non-identifiable model (PX-GS and PX-GSM). Part 2 extended the Bayesian framework to encompass multivariate binary and ordinal data, concluding with the comparative assessment of estimation performance across the three Bayesian methods. This research provides integrated and comprehensive insights by comparing different approaches to multivariate longitudinal data analysis.

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