Bayesian Transformation Models for Multivariate Survival Analysis with Applications in Large Data

Abstract

This dissertation broadly focuses on the advancement and extension of Bayesian methods for multivariate survival analysis with applications in larger data settings. Increased sample information allows researchers to develop more flexible Bayesian multivariate time-to-event models, but at the cost of increased computational time associated with Markov Chain Monte Carlo (MCMC) sampling. Therefore, an additional challenge addressed in this dissertation is the testing and implementation of scalable algorithms that allow models to be applied to large data. In the first portion of this dissertation, we focus on generalizing various bivariate copula survival models that allow for separate specification of marginal and association components. We extend the Bayesian formulation of these models to include generalized forms of the marginal survival functions as well as to include flexible B-spline formulations of the baseline hazard. To demonstrate the approach, we apply the techniques to a myocardial infarction dataset. In the second portion of this thesis, we develop a scalable Bayesian framework to accommodate time to first event of multivariate survival outcomes with ordinal severity. This is done using a flexible Bayesian multivariate frailty model that de-restricts the form of the survival function in order to simultaneously study the correlated covariate effects on differing severity levels of the outcome and to provide a mechanism for combining these profiles into an overall effect. Using an additional data source correlating the multivariate survival outcomes with ordinal severity scores, we provide a systematic and flexible way to determine the overall direction of the smoking effect size on infant bronchiolitis over the multivariate survival events. Using Bayesian methods at the scale of an insurance claims database is challenging due to the computational bottlenecks of typical MCMC routines, which motivated the use of approximate Bayesian inference in this paper. Variational Bayesian inference is one such approximate approach, wherein the posterior density of a model parameter is approximated by a member of a distributional family closest to the true posterior in terms of some statistical distance. It is unclear how valid these approximations are outside of relatively simple statistical models. Therefore, as a secondary focus of the paper, we study the efficacy of variational Bayesian methods on multivariate survival models through a variety of simulations. As a final component to this dissertation, we provide software for the previously described time-to-event methods using the python Bayesian module `pymc3' which is built upon the deep learning library `theano' This software includes flexible implementations of the methods developed in this dissertation with user-friendly syntax to reduce the barrier-of-entry to researchers.