«Bayesian Inference for the Defective distributions used for Cure Rate modelling»

Survival analysis consists of a set of statistical methods in the field of biostatistics, whose main aim is to study the time until the occurrence of a specified event, such as death. For the majority of these methods it is assumed that all the individuals taking part in the study are subject to the event of interest. However, there are situations where this assumption is unrealistic, since there are observations not susceptible to the event of interest or cured. For this reason, there have been developed some survival models which allow for patients that may never experience the event, usually called long-term survivors. These models, called Cure Rate Models, assume that, as time increases, the survival function tends to a value p ∈ (0,1), representing the cure rate, instead of tending to zero as in standard survival analysis.
Recently, Rocha (2016) proposed a new approach to modelling the situations in which there are long-term survivors in survival studies. His methodology was based on the use of defective distributions to model cure rates. In contrast to the standard distributions, the defective ones are characterized by having probability density functions which integrate to values less than one for certain choices of the domain of some of their parameters. The aim of the present thesis is to provide new Bayesian estimates for the parameters of the defective distributions used for cure rate modelling under the assumption of right censoring. We will develop Markov chain Monte Carlo (MCMC) algorithms for inferring the parameters of a broad class of defective models, both for the baseline distributions (Gompertz & Inverse Gaussian), as well as, for their extension under the Marshall-Olkin family of distributions. The Bayesian estimates of the distributions’ parameters, as well as their associated credible intervals, will be obtained from the samples drawn from their joint posterior distribution. In addition, Bayesian estimates’ behavior will be evaluated and compared with the maximum likelihood estimates obtained by Rocha (2016) through simulation experiments. Finally, we will apply the competing models and approaches to real datasets and we will compare them through various statistical measures. This work will be the first attempt to explore the advantages of the
Bayesian approach to inference for defective cure rate models under the assumption of right censoring mechanism, as well as the first presentation of new Bayesian estimates for several defective distributions, but without incorporating covariate information.
Keywords: Defective distributions, Cure fraction, Bayesian Inference, Maximum likelihood, Right censoring, Survival Analysis,
Gompertz distribution, Inverse Gaussian distribution, Marshall Olkin family.
References
[1] Rocha, R. F. D. (2016). Defective models for cure rate modeling.

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