Exact and asymptotic inference in clinical trials with small event rates under inverse sampling

In this paper, we discuss statistical inference for a 2 × 2 table under inverse sampling, where the total number of cases is fixed by design. We demonstrate that the exact unconditional distributions of some relevant statistics differ from the distributions under conventional sampling, where the sample size is fixed by design. This permits us to define a simple unconditional alternative to Fisher's exact test. We provide an asymptotic argument including simulations to demonstrate that there is little power loss associated with the alternative test when the expected event rates are very small. We then apply the method to design a clinical trial in cataract surgery, where a rare side effect occurs in one in 1000 patients. The objective of the trial is to demonstrate that adjuvant treatment with an antibiotic will reduce this risk to one in 2000. We use an inverse sampling design and demonstrate how to set this up in a sequential manner. Particularly simple stopping rules can be defined when using the unconditional alternative to Fisher's exact test.