Conditional and unconditional confidence intervals following a group sequential test

After a group sequential test, the naive confidence interval (CI) is usually biased in the sense that it does not cover the true parameter at the correct nominal level. Furthermore, when the stopping time is taken into account, the actual conditional confidence coverage probability can be much less accurate. In this article, we study the conditional coverage probability and other related properties of the naive CI and different versions of exact CI's. It is demonstrated that only correcting the overall confidence level does not necessarily improve the confidence level at any given stopping stage. Conditional inference can be applied to construct an exact conditional CI but it is not without serious undesirable properties. We propose a two-step restricted conditional confidence interval (RCCI) which considerably improves the conditional confidence level while minimizing the undesirable properties. Numerical comparisons are made between the proposed method and existing methods. The results show that the RCCI not only improves the conditional coverage probability considerably from the exact CI's but also is free of the major undesirable properties displayed by the pure conditional CI. Differences between the conditional and unconditional CI's and their respective strengths are also discussed.