Equivalence of prospective and retrospective models in the Bayesian analysis of case-control studies

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Abstract
The natural likelihood to use for a case‐control study is a ‘retrospective’ likelihood, i.e. a likelihood based on the probability of exposure given disease status. Prentice & Pyke (1979) showed that, when a logistic regression form is assumed for the probability of disease given exposure, the maximum likelihood estimators and asymptotic covariance matrix of the log odds ratios obtained from the retrospective likelihood are the same as those obtained from the ‘prospective’ likelihood, i.e. that based on probability of disease given exposure. We prove a similar result for the posterior distribution of the log odds ratios in a Bayesian analysis. This means that the Bayesian analysis of case‐control studies may be done using a relatively simple model, the logistic regression model, which treats data as though generated prospectively and which does not involve nuisance parameters for the exposure distribution.

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