Bayesian information criterion approximations to Bayes factors for univariate and multivariate logistic regression models

Schwarz's criterion, also known as the Bayesian Information Criterion or BIC, is commonly used for model selection in logistic regression due to its simple intuitive formula. For tests of nested hypotheses in independent and identically distributed data as well as in Normal linear regression, previous results have motivated use of Schwarz's criterion by its consistent approximation to the Bayes factor (BF), defined as the ratio of posterior to prior model odds. Furthermore, under construction of an intuitive unit-information prior for the parameters of interest to test for inclusion in the nested models, previous results have shown that Schwarz's criterion approximates the BF to higher order in the neighborhood of the simpler nested model. This paper extends these results to univariate and multivariate logistic regression, providing approximations to the BF for arbitrary prior distributions and definitions of the unit-information prior corresponding to Schwarz's approximation. Simulations show accuracies of the approximations for small samples sizes as well as comparisons to conclusions from frequentist testing. We present an application in prostate cancer, the motivating setting for our work, which illustrates the approximation for large data sets in a practical example.