Boundary problem in Simon's two-stage clinical trial designs

The activity of a new treatment in clinical trials with binary endpoints can be assessed by comparing the observed response rate to the target response rate. Traditionally, a one-sided hypothesis is used to make statistical inference, and the actual Type I error rate has to be computed over the parameter space of the null hypothesis. The monotonicity property is a fundamental property that guarantees the actual Type I error rate occurring at the boundary. One-arm two-stage designs are considered in this article. We theoretically proved this important property when the final threshold value of a design is less than the first-stage sample size together with another weak condition being satisfied. The method used in this article may finally lead to the final complete proof of this property in the future. We also numerically proved that the monotonicity property is satisfied for designs with the first-stage and the second-stage sample sizes from 10 to 100.