Bayesian Model Selection and Averaging in Additive and Proportional Hazards Models

Lifetime Data Analysis, 11, 213232, 2005

2005 Springer Science+Business Media, Inc. Printed in The Netherlands.DAVID B. DUNSON [e-mail] Branch, National Institute of Environmental Health Sciences, MD A3-03, P.O. Box 12233,Research Triangle Park, NC 27709, USAAMY H. HERRINGDepartment of Biostatistics, The University of North Carolina, Chapel Hill, NC, USAReceived September 11, 2003; Revised May 5, 2004; Accepted September 24, 2004Abstract. Although Cox proportional hazards regression is the default analysis for time to event data,there is typically uncertainty about whether the eects of a predictor are more appropriately characterizedby a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prioron the coecients in an additive-multiplicative hazards model. This prior assigns positive probability, notonly to the model that has both additive and multiplicative eects for each predictor, but also to submodels corresponding to no association, to only additive eects, and to only proportional eects. Theadditive component of the model is constrained to ensure non-negative hazards, a condition often violatedby current methods. After augmenting the data with Poisson latent variables, the prior is conditionallyconjugate, and posterior computation can proceed via an ecient Gibbs sampling algorithm. Simulationstudy results are presented, and the methodology is illustrated using data from the Framingham heartstudy.Key words: additive hazards, Cox model, gibbs sampler, order restricted inference, posterior probability,proportional hazards, survival analysis, variable selection1. IntroductionCox proportional hazards regression (Cox, 1972) is by far the most widely used

approach for the analysis of time to event data. Unfortunately, the proportional

hazards assumption is often violated, and there is a need for alternative and more

exible models. Many possibilities have been proposed, including accelerated life

models, additive hazards models, and additive-multiplicative models, which have

both additive and proportional hazards components. When the form of the model

and the important predictors are known, a variety of methods are available for

parameter estimation. Unfortunately, the regression structure is typically not known

a priori, and one must account for uncertainty in the model to avoid biased inferences (e.g., about the eect of an important predictor).This article focuses on the problem of variable selection and inference, rst in the

additive hazards model (Aalen, 1980; Cox and Oakes, 1984; Lin and Ying, 1994) and

then in the more general additive-multiplicative model (Lin and Ying, 1995;Bayesian Model Selection and...