Flexible bayesian survival models with application in biometric studies

Resumen

Survival analysis groups a great variety of statistical methods for analysing data whose main response variable is the time until the occurrence of an event of interest. Its relevance in the field of statistics is very substantial due to its extensive application in many fields of science. Literature for survival analysis shows a use of both frequentist and Bayesian statistical approaches. However, in recent years Bayesian survival methods have proliferated considerably providing a suitable and robust methodology to perform analyses beyond the standard models. The general aim in this thesis focuses on providing an appropriate methodology to describe and illustrate the use and application of flexible survival models in many biometrical contexts from a Bayesian approach. Particularly, the specific objectives are: 1. To place on value the potentialities of Bayesian survival analysis in contexts in which that methodology has not been widely used. In that regard, we play special attention to some of the advantages that this approach offers compared to frequentist inference. 2. To propose and implement a general survival modelling framework in the context of Cox proportional hazards (CPH) models (Cox, 1972). There are many studies that need to go beyond the standard approach of CPH model (Cox,1972) in which the baseline hazard is usually unspecified or parametrically defined. Baseline hazard functions are a key component in the CPH model definition and its misspecification can imply a loss of valuable model information that can make impossible to fully report estimated outcomes of interest, such as posterior probabilities and survival curves for all relevant groups patterns. In that regard, different model scenarios are addressed and discussed based on: - Parametric and non-parametric specifications of the baseline hazard function. Weibull distribution is the default choice to illustrate the parametric specification while non-parametric specifications are defined by means of mixtures of piecewise constant functions (Sahu et al., 1997) and cubic B-spline functions (Hastie et al., 2009). - Different prior scenarios that introduce regularization procedures to avoid overfitting and instability (Breiman, 1996) in the estimation process of the models defined via non-parametric baseline hazard proposals. 3. To propose and implement a feasible extension to estimate standard mixture cure models by means of the integrated nested Laplace approximation (INLA, Rue et al., 2009). 4. To extend the methodological proposals described in objective 2 to the framework of Bayesian joint model of longitudinal and survival data. Furthermore, this Ph.D. project has a transversal objective based on comparing two of the most usual methods for accounting Bayesian inference in the context of survival analysis: Markov chain Monte Carlo (MCMC) simulation methods and the INLA methodology.