Statistical Methods in Medical Research 4ed

quite acceptable as being within the tolerated range of equivalence, but also
because a non-significant result gives no indication as to whether or not the true
difference lies within the tolerated interval.
Suppose the parameter u measures the difference in efficacy between T and S,
with high values favouring T. In a two-sided equivalence trial, the limits of
equivalence may be denoted by uL < 0 and uU > 0. The investigator will be in
a position to claim equivalence if it can be asserted that u lies within the range
(uL, uU). A simple approach is to assert equivalence if an appropriate confidence
interval for u lies wholly within this range.
Suppose that, for this purpose, we use 100(1 2a)% confidence limits; e.g.
95% limits would have a ˆ 0 025. An equivalent approach would be to carry out
significance tests using each of the equivalence limits for the null hypothesis:
1 Test the null hypothesis HL that u ˆ uL against the one-sided alternative that
u > uL, with one-sided significance level a.
2 Test the null hypothesis HU that u ˆ uU against the one-sided alternative that
u < uU, with one-sided significance level a.
Equivalence is asserted if and only if both these null hypotheses are rejected. For
a non-inferiority trial, only test 1 is necessary, and this corresponds to the
requirement that a one-sided 100(1 a)% confidence interval, extending to
plus infinity, falls wholly to the right of uL.
A Bayesian interpretation
18.9 Special designs 637
A Bayesian formulation for equivalence might require that the posterior
probabilities that u < uL and that u > uU are both less than some small value
a. With the usual assumptions of a non-informative prior and a normally
distributed estimator, this is equivalent to the frequentist approach given
above. For a non-inferiority trial, only the first of these two requirements is
needed.
A slightly less rigorous Bayesian requirement would be that the posterior
probability that u lies within the equivalence range …uL, uU† is 1 2a. This is
slightly more lenient towards equivalence than the previous approach, since an
estimate near the upper limit of uU might give a posterior probability for
exceeding that bound which was rather greater than a, and yet equivalence
would be conceded because the probability that u < uL was very small and the
total probability outside the range was still less than 2a. The frequentist analogue
of this approach is a proposal by Westlake (1979) that a 100(1 2a)%
confidence interval should be centred around the mid-point of the range, rather
than about the parameter estimate. This would be equivalent to doing separate
tests of HL and HU at the one-sided a level when the estimate is near the centre of
the range, but a test of the nearest limit at the one-sided 2a level when the
estimate is near that limit.