The non-inferiority and superiority (NI/S) formulations to evaluate a new treatment are frequently used in active-controlled clinical trials. A key assumption for the NI/S statistical tests that compare 2 independent proportions is that corresponding critical regions are Barnard convex sets. This assumption allows significant reduction in the computation time required to calculate test sizes for these tests. This study presents arguments that both types of testing procedures (NI/S) require the corresponding critical regions to be Barnard convex sets. Otherwise, these tests may become meaningless. Notably, the critical regions of the well-known Blackwelder and Hauck-Anderson tests are not Barnard convex sets for many sample sizes. © The Author(s) 2013.