ABSTRACT
If the trial run in order to compare the two treatments A and B follows an ordinary two-arm design, the data set to be analyzed consists ofm+n mutually independent observations X1, . . . , Xm, Y1, . . . , Yn. For definiteness, we keep assigning the X’s to treatment group A, whereas the Y ’s are assumed to belong to the group of subjects or patients given treatment B. The statistical model which is assumed to hold for these variables is the same as in the ordinary two-sample t-test for conventional one-and two-sided hypotheses. Accordingly, we suppose throughout this section that both the Xi and the Yj follow a normal distribution with some common variance and possibly different expected values where all three parameters are unknown constants allowed to vary unrestrictedly over the respective part of the parameter space. Formally speaking, this means that the observations are assumed to satisfy the basic parametric model
Xi ∼ N (ξ, σ2) ∀ i = 1, . . . ,m , Yj ∼ N (η, σ2) ∀ j = 1, . . . , n , (6.1)
with ξ, η ∈ IR, σ2 ∈ IR+ . In § 1.6, we have argued that the most natural measure of dissimilarity
of two homoskedastic Gaussian distributions is the standardized difference of their means. Accordingly, under the present parametric model, we define equivalence of treatments A and B through the condition that the true value of this measure falls into a sufficiently narrow interval (−ε1, ε2) around zero. In other words, we formulate the testing problem as
H : (ξ − η)/σ ≤ −ε1 or (ξ − η)/σ ≥ ε2 versus K : −ε1 < (ξ − η)/σ < ε2 (ε1, ε2 > 0) , (6.2)
in direct analogy to the setting of the paired t-test for equivalence [recall (5.27), (5.28)].
