Mathematical Sciences

Theses and Dissertations

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Suwash Silwal

Dissertation Defense -
Suwash Silwal

Date: Tuesday, July 28, 2026
Time: 9:00 - 12:00 p.m.
Location: Fisher Hall 325
Advisor: Dr. Xiao Zhang
Zoom: https://michigantech.zoom.us/j/7303724626?omn=87248027147 

Title: Bayesian Analysis of Nominal Outcomes with Missing Values Using Multinomial and Multivariate Multinomial Probit Models 

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 using the Mental Health Client-Level Data (MH-CLD) and the Health and Retirement Study (HRS). The performance of the 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.

Stephen Acheampong

Dissertation Defense -
Stephen Acheampong

Date: Friday, July 31, 2026
Time: 9:00 - 12:00 p.m.
Location: Fisher Hall 325
Advisor: Dr. Xiao Zhang
Zoom: https://michigantech.zoom.us/j/87865534307  

Title: Bayesian Analysis for Longitudinal Binary and Ordinal Data with Missing Values Using Multivariate Probit Model

Abstract: Longitudinal binary and ordinal outcomes are common in medicine, epidemiology, and the social sciences, where repeated measurements are often incomplete because of nonresponse, missed visits, or dropout. Bayesian multivariate probit models provide a flexible framework for correlated discrete outcomes but are computationally demanding under identifiable formulations. This dissertation develops efficient parameter-expanded Bayesian methods for incomplete longitudinal binary and ordinal data.
For both binary and ordinal outcomes, the proposed framework can estimate regression and correlation parameters; the ordinal model additionally estimates cut-points. Missing responses are assumed for missing completely at random and missing at random. Simulation studies are conducted to demonstrate the proposed methods. We also apply our methods to health insurance coverage and life satisfaction data from the Panel Study of Income Dynamics.
Parameter expansion improves computational efficiency for both binary and ordinal models, with the proposed Gibbs sampling methods providing the superior overall performance. These results demonstrate that parameter-expanded multivariate probit models are effective tools for analyzing incomplete longitudinal discrete data.

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