Approximate Bayesian model selection with the deviance statistic

Bayesian model selection poses two main challenges: the specification of parameter priors for all models, and the computation of the resulting Bayes factors between models. There is now a large literature on automatic and objective parameter priors in the linear model, which unburden the statistician from eliciting them manually in the absence of substantive prior information. One important class are $g$-priors, which were recently extended from linear to generalized linear models (GLMs). We show that the resulting Bayes factors can be approximated by test-based Bayes factors using the deviance statistics of the models. To estimate the hyperparameter $g$, we propose empirical and fully Bayes approaches and link the former to minimum Bayes factors and shrinkage estimates from the literature. Furthermore, we describe how to approximate the corresponding posterior distribution of the regression coefficients based on the standard output from a maximum likelihood analysis of the GLM considered. We extend the proposed methodology to the Cox proportional hazards model and illustrate the approach with the development of a clinical prediction model for 30-day survival in the GUSTO-I trial using logistic regression, and with variable and function selection in Cox regression for the survival times of primary biliary cirrhosis patients. Keywords: Bayes factor, Cox model, deviance, generalized linear model, $g$-priors, model selection, shrinkage