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A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann‐Whitney Statistic
Author(s) -
Wellek S.
Publication year - 1996
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710380608
Subject(s) - mathematics , statistics , equivalence (formal languages) , nonparametric statistics , estimator , test statistic , combinatorics , normal distribution , random variable , bioequivalence , sample size determination , statistical hypothesis testing , statistic , null hypothesis , discrete mathematics , bioavailability , bioinformatics , biology
By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P [ X ≤ x ], G(y) = P [ Y ≤ y ] with ( X, Y ) denoting an independent pair of real‐valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P [ X > Y ] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H o P[X > Y ] ≤ 1/2 ‐ ε 1 or P [ X > Y ] ≥ 1/2 + ε 2 versus H 1 : 1/2 − ε 1 < P [ X > Y ] < 1/2 + ∈ 2 , with sufficiently small ε 1 , ε 2 ∈ (0, 1/2). The testing procedure we derive for ( 0 , H 1 ) and propose to term Mann‐Whitney test for equivalence, consists of carrying out in terms of the U ‐statistics estimator of P [ X > Y ] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution‐free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann‐Whitney and the t ‐statistic in the conventional one or two‐sided setting, provided the power of the parametric test does not fall short of 80%.