On the equivalence of prospective and retrospective likelihood methods in case-control studies

We present new approaches to analyzing case-control studies using prospective likelihood methods. In the classical framework, we extend the equality of the profile likelihoods to the Barndorff-Nielsen modified profile likelihoods for prospective and retrospective models. This enables simple and accurate approximate conditional inference for stratified case-control studies of moderate stratum size. In the Bayesian framework, we provide sufficient conditions on priors for the prospective model parameters to yield a prospective marginal posterior density equal to its retrospective counterpart. Our results extend the prospective-retrospective equivalence in the Bayesian paradigm with a more general class of priors than has previously been investigated.