Designs of superiority and noninferiority trials for binary responses are noninterchangeable

To better understand the design of noninferiority trials for binary data, we identify analogies and contrasts between this and the more familiar superiority trial design. We restrict attention to the problem of detecting a difference between experimental and control response rates in the setting where there is no difference (piE - piC = 0) under the noninferiority alternative hypothesis and under the superiority null, and a matching difference between groups under the complementary hypotheses (/piE - piC/ = delta). Our derivation of the constrained maximum likelihood estimates (MLEs) reveals that superiority and noninferiority trials have different nuisance parameters--the marginal response rate and the control-group response rate, respectively. Our empirical results show that when individuals are allocated to treatment groups in the ratio that minimizes the overall sample size, balanced allocation is optimal only for superiority trials when the error rates are equal; otherwise imbalanced allocation is optimal. Different allocation ratios between trial types lead to different variances, and thus to different sample sizes. Finally, since the value of the marginal response rate--a design parameter in noninferiority trials--typically cannot be obtained from preliminary or published studies, we suggest a means of identifying a value that can be used. We conclude that full documentation of the design of a trial requires specification not only of the design parameters but also of the allocation ratio and the nuisance parameter, the value of which is not obvious under unequal allocation.