Approximate Bayesian Model Selection with the Deviance Statistic

TL;DR

This paper introduces a method to approximate Bayesian model selection in generalized linear models using deviance statistics, providing practical approaches for hyperparameter estimation and posterior inference, demonstrated on a clinical survival prediction model.

Contribution

It extends test-based Bayes factors to generalized linear models with $g$-priors, offering empirical and fully Bayesian methods for hyperparameter estimation and posterior approximation.

Findings

Effective approximation of Bayes factors using deviance statistics.

Proposed empirical and Bayesian methods for hyperparameter g estimation.

Application to a clinical prediction model for 30-day survival.

Abstract

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. 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 (Johnson [Scand. J. Stat. 35 (2008) 354-368]) 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 GLM output. We illustrate the approach with the development of a clinical prediction model for 30-day survival in the GUSTO-I trial using logistic regression.