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Konrath, Susanne and Kneib, Thomas and Fahrmeir, Ludwig (2008): Bayesian Regularisation in Structured Additive Regression Models for Survival Data. Department of Statistics: Technical Reports, No.35
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Abstract

During recent years, penalized likelihood approaches have attracted a lot of interest both in the area of semiparametric regression and for the regularization of high-dimensional regression models. In this paper, we introduce a Bayesian formulation that allows to combine both aspects into a joint regression model with a focus on hazard regression for survival times. While Bayesian penalized splines form the basis for estimating nonparametric and flexible time-varying effects, regularization of high-dimensional covariate vectors is based on scale mixture of normals priors. This class of priors allows to keep a (conditional) Gaussian prior for regression coefficients on the predictor stage of the model but introduces suitable mixture distributions for the Gaussian variance to achieve regularization. This scale mixture property allows to device general and adaptive Markov chain Monte Carlo simulation algorithms for fitting a variety of hazard regression models. In particular, unifying algorithms based on iteratively weighted least squares proposals can be employed both for regularization and penalized semiparametric function estimation. Since sampling based estimates do no longer have the variable selection property well-known for the Lasso in frequentist analyses, we additionally consider spike and slab priors that introduce a further mixing stage that allows to separate between influential and redundant parameters. We demonstrate the different shrinkage properties with three simulation settings and apply the methods to the PBC Liver dataset.