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Biometrika 2004 91(1):15-25; doi:10.1093/biomet/91.1.15
© 2004 by Biometrika Trust
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Equivalence of prospective and retrospective models in the Bayesian analysis of case-control studies

Shaun R.Seaman1 and Sylvia Richardson2

1 MRC Biostatistics Unit, Institute of Public Health, Cambridge CB2 2SR, U.Kseaman{at}mrc-bsu.cam.ac.uk 2 Department of Epidemiology and Public Health, Imperial College, London W2 1PG, U.K.sylvia.richardson{at}ic.ac.uk

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.

Key Words: Bayesian inference; Case-control study; Dirichlet distribution; Markov chain Monte Carlo; Retrospective likelihood

Received April 2002. Revised April 2003


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