Samples of exact k-stage group sequential designs for Phase II and Pilot studies

That the test of H₀: p=p₀ versus H₁: p>p₀ can be based on a binomially distributed random variable is widely known among users of statistical methods. What is not generally known is that under certain very general conditions, it is possible to find an exact k-stage group sequential test whose total sample size is bounded above by the sample size for the single stage binomial test. That is, it is possible to find k-stage tests for detecting H₁ for which the sum of the sample sizes at each of the stages is bounded above by the sample size for the standard binomial test. This result is somewhat remarkable since the total sample size under the group sequential test setting can be strictly less than the sample size for the uniformly most powerful (UMP) one-stage binomial test. In other words, exact group sequential tests cannot only save the average sample size but can also save the maximum sample size when they are compared to the standard binomial test. In this paper, implications of existing theory are explored and a web application written by the authors is presented. No new theory is established. Applications are described and methods are demonstrated that use the web application to rapidly create efficient designs for Phase II and Pilot studies that put a minimum number of patients at risk and that facilitate the rapid progression through a scientific research agenda. While couched here in the context of clinical trials, the results may be used in any field of inquiry where inferences are made based on the size of a binomial random variable.