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In studies of recurrent events, there can be a lot of information about a cohort over a period of time, but it may not be possible to extract as much information from the data as would be liked. This thesis considers data on individuals experiencing recurrent events, before and after they are randomised to treatment. The prerandomisation outcome is a period count, while the post-randomisation outcome is a survival time. Standard survival analysis may treat the pre-randomisation period count as a covariate, but it is proposed that point process models will give a more precise estimate of the treatment effect. A joint model is presented, based on a Poisson process with individual frailty. The pre-randomisation seizure counts are distributed as Poisson variables with rate depending on explanatory variables as well as a random frailty. The model for the post-randomisation survival times is the exponential distribution with the same individual seizure rate, modified by a multiplicative treatment effect. A conjugate mixing distribution (frailty) is used, and alternative mixing distributions are also discussed. The model is motivated by and illustrated on individual patient data from five randomised trials of two treatments for epilepsy. The data are presented, and the standard analyses are contrasted with the results of the joint model. This thesis also considers the relative efficiency of the joint model compared to other survival models. Finally, some extensions to the model are considered, including a more general non-conjugate mixing distribution, and alternative ways of including explanatory variables in the joint model.

This thesis is focused on the topic of competing risks survival analysis. The first chapter provides an introduction and motivation with a brief literature review. Chapter 2 considers the fundamental functional of all competing risks data: the crude incidence function. This function is considered in the light of the counting process framework which provides powerful mathematics to calculate confidence bands in an analytical form, rather than bootstrapping or simulation. Chapter 3 takes the Peterson bounds and considers what happens in the event of covariate information. Fortunately, these bounds do become tighter in some cases. Chapter 4 considers what can be inferred about the effect of covariates in the case of competing risks. The conclusion is that there exist bounds on any covariate-time transformation. These two preceding chapters are illustrated with a data set in chapter 5. Chapter 6 considers the result of Heckman and Honore (1989) and investigates the question of their generalisation. It reaches the conclusion that the simple assumption of a univariate covariate-time transformation is not enough to provide identifiability. More practical questions of modeling dependent competing risks data through the use of frailty models to induce dependence is considered in chapter 7. A practical and implementable model is illustrated. A diversion is taken into more abstract probability theory in chapter 8 which considers the Bayesian non-parametric tool: P61ya trees. The novel framework of this tool is explained and some results are obtained concerning the limiting random density function and the issues which arise when trying to integrate with a realised P61ya distribution as the integrating measure. Chapter 9 applies the theory of chapters 7 and 8 to a competing risks data set of a prostate cancer clinical trial. This has several continuous baseline covariates and gives the opportunity to use a frailty model discussed in chapter 7 where the unknown frailty distribution is modeled using a P61ya tree which is considered in chapter 8. An overview of the thesis is provided in chapter 10 and directions for future research are considered here.

MSc (Statistics) / Department of Statistics / Selection of variables is vital in high dimensional statistical modelling as it aims to identify the right subset model. However, variable selection for discrete survival analysis poses many challenges due to a complicated data structure. Survival data might have unobserved heterogeneity leading to biased estimates when not taken into account. Conventional variable selection methods have stability problems. A simulation approach was used to assess and compare the performance of Least Absolute Shrinkage and Selection Operator (Lasso) and gradient boosting on discrete survival data. Parameter related mean squared errors (MSEs) and false positive rates suggest Lasso performs better than gradient boosting. Frailty models outperform discrete survival models that do not account for unobserved heterogeneity. The two methods were also applied on Zimbabwe Demographic Health Survey (ZDHS) 2016 data on age at first marriage and did not select exactly the same variables. Gradient boosting retained more variables into the model. Place of residence, highest educational level attained and age cohort are the major influential factors of age at first marriage in Zimbabwe based on Lasso. / NRF