Survival models for data arising from multiphase hazards, latent subgroups or subject to time-dependent treatment effects

Studies of time-to-event outcomes are among the most common in many areas of scientific research, particularly medicine. Ubiquitous in subject-area literature for this research, is the Cox model; a model which assumes that the instantaneous risk of failure is 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 even more rarely attempted. Unfortunately, the mechanisms leading to violation of this assumption can often be accounted for with alternative models yielding only slightly more complex interpretation than the standard Cox model. 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 addressing specific pathologies which lead to violation of the proportional hazards assumption. In these models, the proportional hazards structure will be maintained in part, but be augmented to accommodate specific situations. Interpretations of these augmented proportional-hazards models will be a key element of the comparisons. 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 indirectly through complex longitudinal mechanisms. Additional scientific questions related to inference on duration of different risk phases or latent group membership will be enabled by these models. Related to these questions, a novel procedure incorporating covariate information into risk-phase durations will be presented. Further, through connections to the rapidly evolving field of joint modeling of longitudinal and time-to-event data, the utility of joint models to fully characterize the mechanisms underlying an overall, possibly time-dependent treatment effect will be explored. An application of joint models to interval-censored survival data including a novel, recursive, event-time imputation method exploiting the relationship between longitudinal data and the failure mechanism will be described.