Test-compatible confidence intervals for adaptive two-stage single-arm designs with binary endpoint

Inference after two-stage single-arm designs with binary endpoint is challenging due to the nonunique ordering of the sampling space in multistage designs. We illustrate the problem of specifying test-compatible confidence intervals for designs with nonconstant second-stage sample size and present two approaches that guarantee confidence intervals consistent with the test decision. Firstly, we extend the well-known Clopper-Pearson approach of inverting a family of two-sided hypothesis tests from the group-sequential case to designs with fully adaptive sample size. Test compatibility is achieved by using a sample space ordering that is derived from a test-compatible estimator. The resulting confidence intervals tend to be conservative but assure the nominal coverage probability. In order to assess the possibility of further improving these confidence intervals, we pursue a direct optimization approach minimizing the mean width of the confidence intervals. While the latter approach produces more stable coverage probabilities, it is also slightly anti-conservative and yields only negligible improvements in mean width. We conclude that the Clopper-Pearson-type confidence intervals based on a test-compatible estimator are the best choice if the nominal coverage probability is not to be undershot and compatibility of test decision and confidence interval is to be preserved.