Bayesian Non-Parametric Estimation in Proportional Hazard Models : Survival Data Analysis with WinBUGS

In survival analysis, the proportional hazard model (PHM) was introduced by Cox (1972) in order to estimate the effects of different covariates influencing the times-to-event data. The PHM has been used extensively in biomedicine, reliability engineering and, recently, interest in its application in different areas of knowledge (e.g., social, economic, actuarial sciences) has increased. However, PHM makes a number of assumptions, which may be violated. In contrast to PHM, the additive hazard models (AHM) assume that the hazard function is the sum of, rather than the product of, the baseline hazard function and a non-negative function of covariates. The survival data may include studies having another source of heterogeneous items; that is, the items under study are dissimilar with respect to some factors other than covariates. Such factors have been referred to as frailty. In general, the frailty models in survival data deal with the unobserved heterogeneity among items under study. In this thesis, both Bayesian and non Bayesian approaches to the estimation of the parameters in the PHM and AHM are reviewed. Development of estimation of PHM and AHM parameters using hierarchical construction are introduced in chapter IV. The methodologies developed are exemplified with two illustrated well-known dataset. Frailty concept is introduced using probabilistic model. The implication in introducing such concept in the estimation process and model validity are given in chapter V. Non-parametric Bayesian is also discussed and implemented in the analysis of PHM. An illustrated example is provided. Since the implementation of Bayesian approaches that obtaining the posterior distribution often requires the integration of high-dimensional functions, a Marcov Chain Monte Carlo (MCMC) is used to compute the features of such posterior distributions for the parameters of interest. Hence, this thesis demonstrates how both AHM and PHM can be implemented by means of MCMC methods using the software WinBUGS. Finally, in chapter VI, a collected dataset from "Insurance Industry" studding the suitable premiums should have paid from an insured to an insurer has been investigated used the survival data analysis