Estimating exposure-specific disease rates from case-control studies using Bayes' theorem

The methods used for selecting subjects yield three types of case-control studies: 1) incident cases are compared to non-cases chosen to be representative of the exposure distribution among the person-years which produced the cases. In this type of study the exposure-odds ratio equals the incidence density ratio; 2) incident cases are compared to residual non-cases at the end of the risk period (exposure-odds ratio = cumulative incidence-odds ratio); 3) prevalent cases are compared to non-cases (exposure-odds ratio = prevalence odds ratio). In study type 1 the equivalence of odds ratio to rate ratio requires no "rare disease assumption;" this permits estimation of exposure-specific illness rates when the overall rate is known. In study types 2 and 3 the exposure-odds ratio equals the corresponding rate ratios only when exposure-specific rates are low. Nonetheless, exposure-specific rates can be calculated without making any rare disease assumption using Bayes' theorem and information on the overall disease rate. A method for obtaining approximate confidence limits around the exposure-specific rates is presented.