The Advancements on the Interface of Statistical Computing, Survival Analysis, and Degradation Analysis

Abstract

In survival analysis and degradation analysis, analyzing complex data often requires advanced computing methods to carry out model estimation and inference. Motivated by computational challenges in survival analysis and a specific dataset in degradation analysis, this dissertation focuses on the applications of advanced computational methods. Furthermore, this dissertation develops novel computational methods to meet the demands of these applications. In competing risk analysis, computing the exact partial likelihood becomes challenging due to the presence of tied events. The primary challenge comes from computing the denominator of the partial likelihood, which is the risk score for the risk set at a given failure time. Historically, no method has efficiently addressed this issue. Approximation methods have been the predominate approach for decades even though their estimates for coefficients are biased. This dissertation presents a novel computational method for the partial likelihood. The method re-represents the denominator as a part of the probability mass function (pmf) of a Poisson multinomial distribution (PMD). However, efficient methods for computing the pmf of the PMD have yet to be developed. To bridge this gap, this dissertation introduces three distinct methods to calculate the pmf of the PMD under different circumstances. Additionally, this dissertation explores the potential applications of the PMD in voting theory, ecological inference, and machine learning. In degradation analysis, a dataset from a field study presents challenges in degradation modeling with the existence of multiple degradation characteristics (DCs) and dynamic covariates. A nonlinear general path model with random effects is employed to capture the data's complexity. A Bayesian framework helps solve the estimation difficulties in the model. Subsequently, Markov Chain Monte Carlo (MCMC) is employed to draw posterior inference conclusions. This dissertation also introduces a spline-based method to describe the effects of covariates. Unlike traditional spline regression methods, this approach does not suffer from overfitting and can model covariate effects under shape constraints.