Topics on Survival Analysis with Long-term Survivors

Abstract

This thesis focuses on various topics in survival analysis when there are both "immunes" or "cured" individuals, those who will never experience the event of interest, and susceptibles, those who will eventually experience the event of interest, present in the dataset.

Traditional survival models are unsuitable for datasets where immunes are present. In those cases, cure models should be used instead. We introduce some of the commonly used cure models in practice in Chapter 2.

The issues that mixture cure models may face under insufficient follow-up are discussed in Chapter 3. Some existing tests and methods to detect insufficient follow-up are also reviewed.

The generalised F distribution embeds most of the commonly used parametric lifetime distributions as special cases. However, the connections between the generalised F distribution and its sub-models have never been clearly formulated in literature. We fill this gap by clearly outlining such connections in Chapter 4.

In Chapter 5, we provide a detailed methodology for researchers to select a parametric mixture cure model for the data given using a generalised F distribution. Using the results, we demonstrate the application of parametric mixture cured models in practice using real data examples, where we conduct tests for sufficient follow-up, parametric cure model selection and covariate analysis.

Cure proportion estimation is usually of interest to researchers. However, when the assumption that the follow-up period is sufficient does not hold, the parametric cure proportion estimator will be extremely unstable, and the nonparametric cure proportion estimator designed using the Kaplan-Meier estimator will overestimate the cure proportion. In Chapter 6, we develop a nonparametric cure proportion estimator under insufficient follow-up for models in the Gumbel maximum domain of attraction. We show our estimator is asymptotically consistent and normally distributed under reasonable assumptions and performs well in simulation studies with data under both sufficient and insufficient follow-up. We also demonstrate its use with an application to survival data where patients with different stages of breast cancer have varying degrees of follow-up.