Estimating a Proportion when Sampling from a Subpopulation

The problem of estimating a population proportion when only a subpopulation may be sampled is considered. This is a simplified version of a problem which arises when testing for the association between a risk factor and a disease by analyzing a contingency table constructed from hospital data. The proportion for the subpopulation, is expressed in terms of the population proportion and a nuisance parameter in such a way that the model is unfortunately not identifiable. A Bayesian procedure is developed which may be used, however, if one is willing to make certain assumptions about the nuisance parameter. As an illustration the procedure is used with a particular prior to obtain a posterior confidence interval for the population proportion. The coverage probabilities of the interval for various fixed values of the parameters are then evaluated.