Optimal restricted two-stage designs

The fixed sample p value at the final stage of a group sequential design may be "significant" while the overall p value for the trial as designed may be "nonsignificant." This can lead to confusion in reporting and interpreting final stage results. Two-stage designs are presented that allow acceptance, rejection, or continuation at the first stage while retaining the fixed sample critical value at the second stage. The designs are optimized under this restriction to minimize the expected sample size under the null hypothesis, the expected sample size under a specific alternative hypothesis, or the maximum expected sample size. These restricted designs are almost fully efficient when compared to optimal unrestricted two-stage designs. Additionally, one can interpret the results of hypothesis tests at the final stage as though a fixed sample design had been used. Examples are given which illustrate the use of these restricted plans in the design and analysis of clinical trials.