Bayesian analysis of case-control studies with categorical covariates

In a case-control study the appropriate likelihood is the 'retrospective' likelihood, i.e. the likelihood of exposure given disease. For the classical frequentist analysis, the 'prospective' likelihood, i.e. the likelihood of disease given exposure, and the retrospective likelihood produce the same odds-ratio estimators for exposure, and so logistic regression may be used for both. The Bayesian analysis is not so simple, but the Bayesian framework for case-control studies offers flexible possibilities for the hierarchical modelling that is needed in many contexts. We review the Bayesian approaches to the analysis of case-control studies developed so far, show how to extend these approaches to the situation of a study with any number of categorical or discretised continuous exposure variables, and identifying suitable priors. We then show how the resulting models may be fitted using Markov chain Monte Carlo methods, and provide an illustration based on genotype data.