On testing simultaneously non-inferiority in two multiple primary endpoints and superiority in at least one of them.

In a clinical trial with an active treatment and a placebo the situation may occur that two (or even more) primary endpoints may be necessary to describe the active treatment's benefit. The focus of our interest is a more specific situation with two primary endpoints in which superiority in one of them would suffice given that non-inferiority is observed in the other. Several proposals exist in the literature for dealing with this or similar problems, but prove insufficient or inadequate at a closer look (e.g. Bloch et al. (2001, 2006) or Tamhane and Logan (2002, 2004)). For example, we were unable to find a good reason why a bootstrap p-value for superiority should depend on the initially selected non-inferiority margins or on the initially selected type I error alpha. We propose a hierarchical three step procedure, where non-inferiority in both variables must be proven in the first step, superiority has to be shown by a bivariate test (e.g. Holm (1979), O'Brien (1984), Hochberg (1988), a bootstrap (Wang (1998)), or Läuter (1996)) in the second step, and then superiority in at least one variable has to be verified in the third step by a corresponding univariate test. All statistical tests are performed at the same one-sided significance level alpha. From the above mentioned bivariate superiority tests we preferred Läuter's SS test and the Holm procedure for the reason that these have been proven to control the type I error strictly, irrespective of the correlation structure among the primary variables and the sample size applied. A simulation study reveals that the performance regarding power of the bivariate test depends to a considerable degree on the correlation and on the magnitude of the expected effects of the two primary endpoints. Therefore, the recommendation of which test to choose depends on knowledge of the possible correlation between the two primary endpoints. In general, Läuter's SS procedure in step 2 shows the best overall properties, whereas Holm's procedure shows an advantage if both a positive correlation between the two variables and a considerable difference between their standardized effect sizes can be expected.