Conditional regression analysis of the exposure-disease odds ratio using known probability-of-exposure values

Conditional inference methods are proposed for the odds ratio between binary exposure and disease variables when only the probability of exposure is known for each study subject. We develop a conditional likelihood approach that removes nuisance parameters and permits inferences to be made about important parameters in log odds ratio regression models. We also discuss a heuristic procedure based on estimating the (unknown) number of truly exposed individuals; this procedure provides a simple framework for interpreting our likelihood-based statistics, and leads to a Mantel-Haenszel-type estimator and a goodness-of-fit test. As an example of the use of this methodology, we present an analysis of some genetic data of Swift et al. (1976, Cancer Research 36, 209-215).