Notes on conditional confidence limits under inverse sampling

When the number of subjects in a two-by-two table is small or moderate, we may commonly use the exact conditional distribution with all marginals fixed to derive the conditional confidence limits on the underlying parameter. Under inverse sampling, in which we continue to sample subjects until we obtain exactly a pre-determined number of subjects failing into a specific category, this paper notes that derivation of a confidence interval, which has the coverage probability equal to or larger than a nominal 1-alpha confidence level, for relative risk and relative difference in cohort studies is straightforward. This paper further finds that, when the underlying disease is rare, we can similarly apply an inverse sampling to produce an approximate 1-alpha conditional confidence limits on attributable risk in case-control studies as well. When the number of subjects is small and the test statistic derived on the basis of large sample theory is not strictly adequate for use, this paper also presents an exact hypothesis testing procedure for the above parameters in the corresponding study designs.