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Konrath, Susanne (2013): Bayesian regularization in regression models for survival data. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics

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This thesis is concerned with the development of flexible continuous-time survival models based on the accelerated failure time (AFT) model for the survival time and the Cox relative risk (CRR) model for the hazard rate. The flexibility concerns on the one hand the extension of the predictor to take into account simultaneously for a variety of different forms of covariate effects. On the other hand, the often too restrictive parametric assumptions about the survival distribution are replaced by semiparametric approaches that allow very flexible shapes of survival distribution. We use the Bayesian methodology for inference. The arising problems, like e. g. the penalization of high-dimensional linear covariate effects, the smoothing of nonlinear effects as well as the smoothing of the baseline survival distribution, are solved with the application of regularization priors tailored for the respective demand. The considered expansion of the two survival model classes enables to deal with various challenges arising in practical analysis of survival data. For example the models can deal with high-dimensional feature spaces (e. g. gene expression data), they facilitate feature selection from the whole set or a subset of the available covariates and enable the simultaneous modeling of any type of nonlinear covariate effects for covariates that should always be included in the model. The option of the nonlinear modeling of covariate effects as well as the semiparametric modeling of the survival time distribution enables furthermore also a visual inspection of the linearity assumptions about the covariate effects or accordingly parametric assumptions about the survival time distribution. In this thesis it is shown, how the p>n paradigm, feature relevance, semiparametric inference for functional effect forms and the semiparametric inference for the survival distribution can be treated within a unified Bayesian framework. Due the option to control the amount of regularization of the considered priors for the linear regression coefficients, there is no need to distinguish conceptionally between the cases p<=n and p>n. To accomplish the desired regularization, the regression coefficients are associated with shrinkage, selection or smoothing priors. Since the utilized regularization priors all facilitate a hierarchical representation, the resulting modular prior structure, in combination with adequate independence assumptions for the prior parameters, enables to establish a unified framework and the possibility to construct efficient MCMC sampling schemes for joint shrinkage, selection and smoothing in flexible classes of survival models. The Bayesian formulation enables therefore the simultaneous estimation of all parameters involved in the models as well as prediction and uncertainty statements about model specification. The presented methods are inspired from the flexible and general approach for structured additive regression (STAR) for responses from an exponential family and CRR-type survival models. Such systematic and flexible extensions are in general not available for AFT models. An aim of this work is to extend the class of AFT models in order to provide such a rich class of models as resulting from the STAR approach, where the main focus relies on the shrinkage of linear effects, the selection of covariates with linear effects together with the smoothing of nonlinear effects of continuous covariates as representative of a nonlinear modeling. Combined are in particular the Bayesian lasso, the Bayesian ridge and the Bayesian NMIG (a kind of spike-and-slab prior) approach to regularize the linear effects and the P-spline approach to regularize the smoothness of the nonlinear effects and the baseline survival time distribution. To model a flexible error distribution for the AFT model, the parametric assumption for the baseline error distribution is replaced by the assumption of a finite Gaussian mixture distribution. For the special case of specifying one basis mixture component the estimation problem essentially boils down to estimation of log-normal AFT model with STAR predictor. In addition, the existing class of CRR survival models with STAR predictor, where also baseline hazard rate is approximated by a P-spline, is expanded to enable the regularization of the linear effects with the mentioned priors, which broadens further the area of application of this rich class of CRR models. Finally, the combined shrinkage, selection and smoothing approach is also introduced to the semiparametric version of the CRR model, where the baseline hazard is unspecified and inference is based on the partial likelihood. Besides the extension of the two survival model classes the different regularization properties of the considered shrinkage and selection priors are examined. The developed methods and algorithms are implemented in the public available software BayesX and in R-functions and the performance of the methods and algorithms is extensively tested by simulation studies and illustrated through three real world data sets.