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

For testing "univariate" binomial proportions, it has been proven that, under mild conditions, there exist group sequential designs which satisfy the pre-specified Type I error and power of the single-stage design while the sample size is bounded above by that of the single-stage design (Kepner and Chang, 2003). In this article, we extend this result and prove the existence of such group sequential designs for various decision rules in the space of bivariate binomial variables. We also demonstrate how to obtain the actual group sequential designs for detecting changes in bivariate binomial variables.