Exact sequential test of equivalence hypothesis based on bivariate non-central t-statistics

A critical step in group sequential designs is computation of the appropriate critical values for rejecting H0 at the interim look to keep the overall type I error rate at a prespecified level. When applying the sequential test in a study with an equivalence hypothesis, calculation of the critical values is complicated by the dependency between the dual test statistics at each interim look. Current methods for calculating critical values apply two primary approximations: z-statistics assuming a large sample size, and ignorance of the contribution to the overall type I error rate from rejecting one out of the two one-sided hypotheses under a null value. In the sequential testing, with smaller stagewise sample size and type I error rate, the first approximation would result in unsatisfactory inflation of the type I error rate, and the second approximation could lead to excessive conservatism. We establish a mathematical and computational framework of the exact sequential test based on bivariate non-central t statistics and propose several numerical approaches for computing the exact equivalence boundaries and futility boundaries. Examples and simulation studies are used to compare the operating characteristics between the exact test procedure and three other approximate test procedures.