Exact confidence limits for the response rate in two-stage designs with over- or under-enrollment in the second stage

Stat Methods Med Res. 2018 Apr;27(4):1045-1055. doi: 10.1177/0962280216650918. Epub 2016 Jul 7.


Simon's two-stage design has been widely used in early phase clinical trials to assess the activity of a new investigated treatment. In practice, the actual sample sizes do not always follow the study design precisely, especially in the second stage. When over- or under-enrollment occurs in a study, the original critical values for the study design are no longer valid for making proper statistical inference in a clinical trial. The hypothesis for such studies is always one-sided, and the null hypothesis is rejected when only a few responses are observed. Therefore, a one-sided lower interval is suitable to test the hypothesis. The commonly used approaches for confidence interval construction are based on asymptotic approaches. These approaches generally do not guarantee the coverage probability. For this reason, Clopper-Pearson approach can be used to compute exact confidence intervals. This approach has to be used in conjunction with a method to order the sample space. The frequently used method is based on point estimates for the response rate, but this ordering has too many ties which lead to conservativeness of the exact intervals. We propose developing exact one-sided intervals based on the p-value to order the sample space. The proposed approach outperforms the existing asymptotic and exact approaches. Therefore, it is recommended for use in practice.

Keywords: Clopper-Pearson approach; coverage probability; exact one-sided interval; two-stage design.

Publication types

  • Research Support, N.I.H., Extramural

MeSH terms

  • Algorithms
  • Clinical Trials as Topic* / statistics & numerical data
  • Confidence Intervals
  • Patient Safety
  • Research Design / statistics & numerical data*