Bayesian model selection for logistic regression models with random intercept

Data, collected to model risk of an interesting event, often have a multilevel structure as patients are clustered within larger units, e.g. clinical centers. Risk of the event is usually modeled using a logistic regression model, with a random intercept to control for heterogeneity among clusters. Model specification requires to decide which regressors have a non-negligible effect, and hence, should be included in the final model and whether risk is actually heterogeneous among centers, i.e. whether the model should include a random intercept or not. In a Bayesian approach, these questions can be answered by combining variable selection with variance selection of the random intercept. Bayesian model selection is performed for a reparameterized version of the logistic random intercept model using spike and slab priors on the parameters subject to selection. Different specifications for these priors are compared on simulated data as well as on a data set where the goal is to identify risk factors for complications after endoscopic retrograde cholangiopancreatography (ERCP).