Modelling survival data

Abstract

This dissertation investigates several modelling techniques used in survival analysis. One of the chapters provides a comprehensive theoretical review of traditional methods. These include non-parametric methods, including the Kaplan Meier and Nelson Aalen estimators; semi-parametric methods including the Cox proportional hazards model; and parametric methods based on the assumption that the survival distribution has a functional form. The Exponential, Weibull and Gompertz distributions are the most widely used for proportional hazard survival analysis. These traditional methods assume that the population is fairly homogenous and that the variation in survival durations can be explained by a small number of observed explanatory variables. In the presence of heterogeneity, frailty models are more appropriate to model survival data by introducing random effects that account for the variability generated from unobserved covariates. This dissertation presents three chapters on frailty models. The unshared frailty model assumes that different individuals have distinct frailties; and shared frailty models assume that the population comprises clusters, where individuals in the same cluster share the same frailty. Moreover, semi-parametric frailty models extend proportional hazards Cox models by introducing random effects to account for unobserved heterogeneity in the data. The Gamma and the Inverse Gaussian distributions are the most popular choices for the frailty distribution because of their nice mathematical properties. These modelling techniques will be investigated both from a frequentist and a Bayesian approach. One of the chapters in this thesis describes the Bayesian paradigm and sampling methods from the posterior distribution, including the Metropolis-Hasting Algorithm with the Gibbs sampler. One of the benefits of the Bayesian approach is that it allows prior information to be incorporated in a survival model. Another advantage is that MCMC sampling methods enable exact inference for any sample size without relying on any asymptotic properties. All these survival modelling techniques are applied to a data set using the facilities of R and STATA. The participants are patients who underwent an aortic valve replacement procedure at Mater Dei Hospital between 2003 and 2019. The dependent variable is the duration till death or censoring and the eleven explanatory variables provide information about the patients’ health condition; surgery operative procedures; and duration of convalesce period. Moreover, in shared frailty models the patients were clustered by their diabetic condition since it is known that diabetic patients are more at risk of dying following aortic surgery.