Optimal futility stopping boundaries for binary endpoints

Group sequential designs incorporating the option to stop for futility at the time point of an interim analysis can save time and resources. Thereby, the choice of the futility boundary importantly impacts the design's resulting performance characteristics, including the power and probability to correctly or wrongly stop for futility. Several authors contributed to the topic of selecting good futility boundaries. For binary endpoints, Simon's designs (Control Clin Trials 10:1-10, 1989) are commonly used two-stage designs for single-arm phase II studies incorporating futility stopping. However, Simon's optimal design frequently yields an undesirably high probability of falsely declaring futility after the first stage, and in Simon's minimax design often a high proportion of the planned sample size is already evaluated at the interim analysis leaving only limited benefit in case of an early stop.