Sample sizes for trials involving multiple correlated must-win comparisons

In Clinical trials involving multiple comparisons of interest, the importance of controlling the trial Type I error is well-understood and well-documented. Moreover, when these comparisons are themselves correlated, methodologies exist for accounting for the correlation in the trial design, when calculating the trial significance levels. However, less well-documented is the fact that there are some circumstances where multiple comparisons affect the Type II error rather than the Type I error, and failure to account for this, can result in a reduction in the overall trial power. In this paper, we describe sample size calculations for clinical trials involving multiple correlated comparisons, where all the comparisons must be statistically significant for the trial to provide evidence of effect, and show how such calculations have to account for multiplicity in the Type II error. For the situation of two comparisons, we provide a result which assumes a bivariate Normal distribution. For the general case of two or more comparisons we provide a solution using inflation factors to increase the sample size relative to the case of a single outcome. We begin with a simple case of two comparisons assuming a bivariate Normal distribution, show how to factor in correlation between comparisons and then generalise our findings to situations with two or more comparisons. These methods are easy to apply, and we demonstrate how accounting for the multiplicity in the Type II error leads, at most, to modest increases in the sample size.