Date of Award

2026

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy in Statistics (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

Nominal outcomes frequently arise in health sciences, transportation, economics, market research, and related fields. These data often contain missing values, while longitudinal and panel studies generate multiple correlated nominal responses. Bayesian estimation of multinomial probit (MNP) and multivariate multinomial probit (MMNP) models provides a flexible framework for analyzing such data but remains computationally challenging due to high-dimensional likelihood integration, restrictive covariance identification constraints, and poor mixing of Markov chain Monte Carlo (MCMC) algorithms, particularly in the presence of missing data. This dissertation develops parameter-expanded data augmentation (PX-DA) methods for MNP and MMNP models with missing nominal outcomes by incorporating parameter expansion into the data augmentation framework. The proposed methods relax restrictive identification constraints and substantially improve the convergence and mixing of MCMC algorithms while preserving the target posterior distribution.

Chapter 2 develops a Bayesian PX-DA algorithm for univariate multinomial probit (MNP) models with missing nominal outcomes under ignorable missing-data mechanisms. Chapter 3 extends the proposed algorithm to MMNP models for correlated nominal outcomes, providing an efficient Bayesian estimation framework that accommodates both outcome dependence and missing data. The proposed methods are evaluated through extensive simulation studies under different missing-data mechanisms and are further illustrated through real-data analyses using the Mental Health Client-Level Data (MH-CLD) and the Health and Retirement Study (HRS). The performance of the proposed algorithms is compared with existing Bayesian approaches, including Metropolis–Hastings (MH) and standard Gibbs sampling (GS), using convergence diagnostics, mixing behavior, estimation accuracy, and computational efficiency.

Results from simulation and real-data analyses demonstrate that the proposed parameter-expanded algorithms improve convergence, mixing, and computational efficiency while maintaining accurate parameter estimation under different levels of missingness. By addressing key computational and methodological challenges in Bayesian estimation of multinomial probit models, this dissertation expands the practical applicability of both univariate and multivariate MNP models and provides an efficient computational framework for the Bayesian analysis of nominal data with missing values.

Download

Share

COinS