Methods for Analyzing Survival Data with Non-Proportional Hazards and Complex Covariate Effects

Studies of time-to-event outcomes are among the most common in many areas of scientific research, particularly in medicine. Ubiquitous in subject-area literature for studies of this type, is the Cox model; a model which assumes that the instantaneous risks of failure are proportional for different groups of people with similar covariate values. The Cox model has become so pervasive in communicating results that the verification of this assumption is rarely mentioned in subject-area literature and alternative methods are are even more rarely attempted. Unfortunately, the violation of this assumption occurs in situations where there are some specific characteristics which could be incorporated into models while still yielding similar interpretations. Specifically,situations where the proportionality of two hazards varies over time.

Motivated by a dataset capturing survival following coronary artery bypass graft surgery and another containing longitudinal tree growth and mortality, this thesis will describe, compare/contrast and provide interpretations of several models to address certain pathologies leading to a violation of the proportional hazards assumption of the Cox model.

Three different pathologies will be investigated, including complex (ie: multi-phase) hazard functions, latent mixtures of individuals subject to distinct hazards, and effects of covariates which change over time either through a direct erosion of the effect or more indirectly through more complex mechanisms. All models investigated are based on standard relative-hazard interpretations similar to a Cox models but include additional information defining phases of risk, group membership, or time-dependent effects. Connections to the rapidly evolving field of joint modeling of longitudinal and time-to-event data will be made to demonstrate the additional utility of modeling longitudinal data to fully characterize the mechanisms underlying an overall treatment effect. Novel methods for modeling covariates' influence on transitions from one phase of risk to another and for imputing failure-times for interval-censored data by exploiting the relationship between longitudinal data and the failure mechanism will be described.