Retrospective-prospective symmetry in the likelihood and Bayesian analysis of case-control studies

Prentice & Pyke (1979) established that the maximum likelihood estimate of an odds-ratio in a case-control study is the same as would be found by running a logistic regression: in other words, for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model. Similar results have been obtained for other models, and conditions have also been identified under which the corresponding Bayesian property holds, namely that the posterior distribution of the odds-ratio be the same, whether computed using the prospective or the retrospective likelihood. Here we demonstrate how these results follow directly from certain parameter independence properties of the models and priors, and identify prior laws that support such reverse analysis, for both standard and stratified designs.