Survival Analysis with Internal Categorical Time-varying Covariates
In the study of time-to-event data, an important feature is the ability to include time-varying covariates. Concerns have been raised over the use of Cox models in the presence of internal time-varying covariates. Multi-state models have been proposed as an acceptable alternative approach for internal time-varying covariates. A motivating example throughout this thesis is the development of bipolar disorder. It has been proposed that bipolar disorder progresses in a predictable sequence of clinical stages. In Chapter 2 the objective is to compare Cox models and non-parametric multi-state models in the presence of other psychiatric diagnoses. These diagnoses are coded as binary time-varying covariates in a Cox model or as states in a non-parametric multi-state model. A common assumption in Cox models with time-varying covariates is that the effect of a covariate on the event of interest is constant and permanent after it has changed. Chapter 3 presents a modification to the usual Cox model for binary time-varying covariates that allows the influence of a covariate to exponentially decay over time. Methods for generating data using the inverse cumulative density function for the proposed model are developed. Likelihood ratio tests and AIC are investigated as methods for comparing the proposed model with the commonly used permanent exposure model. A simulation study is performed and three different example data analyses are presented. One advantage to parametric multi-state models is the inclusion of misclassification. Until now, this approach has largely been confined to exponential waiting times within each state. In Chapter 4 we introduce Bayesian parametric multi-state models with unknown misclassification of states and Weibull distributed waiting times between states. Weibull waiting times allow transitions between states to depend on the time spent in the current state, a feature lacking in exponential waiting times. To fit the proposed Bayesian model, a Markov chain Monte Carlo (MCMC) Metropolis-Hastings algorithm is employed. The motivating example on the progression of bipolar disorder is presented along with simulation results. From the example analysis, there is evidence that assuming Weibull waiting times is an improvement over assuming exponential waiting times in the study of bipolar disorder.