On the maximum total sample size of a group sequential test about binomial proportions

It is well known that the standard single-stage binomial test is uniformly most powerful to detect an increase or decrease in a binomial proportion. The general perception is that, to achieve a fixed significance level and power, a group sequential test will require a larger maximum total sample size than required by the corresponding standard single-stage test because missing observations are possible under the group sequential test setting. In this article, it is proved that, under mild conditions, there exist group sequential tests which achieve the predesignated significance level and power with maximum total sample size bounded above by the sample size for the corresponding standard single-stage test.