UC Santa Cruz

This dissertation presents the development of innovative Bayesian nonparametric models tailored for the complex demands of analyzing time-to-event data, specifically for problems in survival analysis. These models offer flexibility and computational efficiency in estimating various functionals of the survival distribution. The first thesis component introduces a flexible Erlang mixture model for survival analysis, structured on a weighted combination of Erlang densities with integer shape parameters, and a common scale parameter. The mixture weights are constructed through increments of a distribution function on the positive real line, which is assigned a Dirichlet process prior. The model balances general inference for survival functionals with efficient posterior simulation. The modeling approach is extended to accommodate multiple experimental groups through a dependent Dirichlet process prior. Moving to the second part of the dissertation, a Dirichlet process mixture model with a log-logistic kernel is proposed. The model incorporates covariates through a density regression framework, allowing variations in mixture weights and mixing parameters as functions of covariates. The model yields flexible inference for density, survival, and hazard functions across the covariate space. The final dissertation component explores a joint modeling approach for recurrent events and survival time, relevant for medical studies where the recurrent events process and the risk of death are related. Here, the density functions for the survival times and the gap times of recurrent events are modeled by dependent Dirichlet process mixtures with a log-logistic kernel. This modeling approach builds dependence between survival times and recurrent events through bivariate random effects. The joint modeling framework aims to provide flexibility in inferring marginal and conditional functionals of survival and gap times. For all proposed models, we discuss model properties, prior specification, and posterior simulation techniques, illustrating their effectiveness through synthetic and real data examples.