On construction of smallest one-sided confidence intervals for the response rate in adaptive two- or multi-stage designs

To assess the effectiveness of a treatment in phase II clinical trial for cancer study, an adaptive multi-stage design, especially a two-stage one, is commonly used. This type of design allows an on-going study to end early for the participant's safety and design's efficiency. Since a large value of the response rate p for the treatment is wanted, lower one-sided confidence intervals for p are of interest. Due to the limited sample size, exact intervals with a guaranteed confidence level are derived using a rank function that is based on the Clopper-Pearson lower confidence limit. When the sample size in stage 2 is a constant, two kinds of smallest intervals are constructed with or without using the sufficiency principle. The proposed intervals outperform the existing exact intervals, and the intervals not based on minimal sufficient statistic are recommended for practice due to their small expected lengths. When the sample size in stage 2 varies, the smallest interval is also proposed.