Consider a clinical trial comparing two groups, a standard therapy (s) and a novel therapy (n). In each group, a proportion of subjects responds to the treatment: Ps and Pn. If the intention of the study is to show that the two groups are equivalent, the usual formulation of the null hypothesis (Ps=Pn) encounters logical difficulty. A statistical test may fail to reject this null hypothesis, but this will not mean that the two treatments are equivalent.
Blackwelder (Controlled Clinical Trials 1982; 3: 345-353) proposes a solution. If a difference between the two treatments, call it D, is specified that practically represents equivalence, then the null hypothesis can be restated to include the specified difference. In other words, that: Ps is greater than or equal to Pn + D. Rejection of this hypothesis implies that the difference between the standard and novel treatments is less than or equal to D, indicating equivalence.
The sample size needed to reject this hypothesis at alpha = 0.05 and beta = 0.20 is:
(Z 0.95 + Z 0.80)2 [Ps(1-Ps) + Pn(1-Pn)] / (Ps-Pn-D)2
Important: Note that Ps - Pn must be < D also, all calculations assume alpha = 0.05 and beta = 0.20 (power = 80%)