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Hennerfeind, Andrea and Brezger, Andreas and Fahrmeir, Ludwig (2005): Geoadditive survival models. Collaborative Research Center 386, Discussion Paper 414
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Abstract

Survival data often contain small-area geographical or spatial information, such as the residence of individuals. In many cases the impact of such spatial effects on hazard rates is of considerable substantive interest. Therefore, extensions of known survival or hazard rate models to spatial models have been suggested recently. Mostly, a spatial component is added to the usual linear predictor of the Cox model. We propose flexible continuous-time geoadditive models, extending the Cox model with respect to several aspects often needed in applications: The common linear predictor is generalized to an additive predictor, including nonparametric components for the log-baseline hazard, time-varying effects and possibly nonlinear effects of continuous covariates or further time scales, and a spatial component for geographical effects. In addition, uncorrelated frailty effects or nonlinear two-way interactions can be incorporated. Inference is developed within a unified fully Bayesian framework. We prefer to use penalized regression splines and Markov random fields as basic building blocks, but geostatistical (kriging) models are also considered. Posterior analysis uses computationally efficient MCMC sampling schemes. Smoothing parameters are an integral part of the model and are estimated automatically. Propriety of posteriors is shown under fairly general conditions, and practical performance is investigated through simulation studies. We apply our approach to data from a case study in London and Essex that aims to estimate the effect of area of residence and further covariates on waiting times to coronary artery bypass graft (CABG). Results provide clear evidence of nonlinear time-varying effects, and considerable spatial variability of waiting times to bypass graft.