Thesis submitted in total fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Statistics at Strathmore UniversityIn this study, we apply the methodology of smooth tests of goodness-of-fit to hazard functions. The smooth test formulation applied here is an extension of Neyman’s smooth test and is obtained by nesting the null hypothesis in a larger class of probability and hazard rate functions. The study revisits Neyman’s smooth tests and its data-driven versions in the context of classical probability and survival analysis. Though several authors have theoretically looked at the development of Neyman’s smooth tests, the main contribution of this study is modelling loss to follow-up in HIV retention. To the best of our knowledge, this issue has not been given its due share of coverage in the literature. We extend methods proposed by Rayner et al. (2009); Pena (1998a,b) and Kraus (2007a), to an HIV retention setting. The applications dealt with in this thesis also covers performance of other goodness-of-fit (GOF) approaches and compares them with that of smooth tests. Three main methodological approaches are covered under the research methodology. Part I revisits smooth tests for various probability distributions and applies the test when assessing the fit for the two-parameter Weibull distribution to an HIV retention data under the complete and uncensored data scenario. Part II looks at the application of smooth test to Cox proportional hazards models. We assess the proportionality assumption in the two sample problem in cancer survival studies. Part III covers recurrent event situation. We fit Block, Borges and Savits (BBS) minimal repair model to loss to follow-up (LFTU) data and assesses the performance of the smooth test in terms of power.More specifically, Chapter 1 deals with background of GOF in classical probability and survival distributions. The motivation for the study, overview of the smooth test of GOF and comparison with other GOF tests is also covered in this chapter. In Chapter 2, we provide a review of the literature. Chapter 3 details research methodology. We present analysis and results in Chapter 4. Chapter 5 discusses important findings using simulated and real data in the context of HIV retention and overall survival in cancer studies. Chapter 6 covers summary of the thesis, the limitations of the study and possible extensions of the smooth GOF to discrete probability cases. All computations have been implemented in R and the scripts are briefly described in Appendix A. The chapters are self-contained in order to achieve our objective of covering the applications smooth tests of goodness-of-fit approach from distributions with noncensored data to extensions in recurrent events.A major limitation of this study, is that, in clinical studies, particularly involving LTFU data, incomplete data is frequently encountered. Analysis of severity of data incompleteness is a subject of future research
