Testing for equivalence in k sample models

Apl. Prof. Dr. Helmut Finner
Deutsches Diabetes-Zentrum
Institut für Biometrie und Epidemiologie


        Hodges and Lehmann (1954) proposed to distinguish between statistical and material significance in hypotheses testing. They discussed several testing problems where the hypotheses are defined in terms of certain threshold values. Among others, they derived a new test related to Student's t-test in terms of a complicated two-dimensional rejection region. Many years later, their method was adopted for the construction of a test for equivalence of two normal means. A good source for two-sample equivalence tests is Wellek (2003).

        In this talk we are concerned with the more advanced problem of establishing equivalence of treatment means in k sample models, where equivalence is defined in terms of a threshold value. We give a review of developments in this field. Besides the problem of establishing equivalence between all treatment means we consider the problem of selecting treatments which are equivalent to the best treatment. We reformulate the original selection problem as a multiple testing problem and propose some step-down and step-up procedures which can be constructed by applying the closure principle and partitioning principles. The new step-down procedure provides a uniform improvement over procedures currently in use.


Giani, G. and Finner, H. (1991). Some general results on least favourable parameter configurations with special reference to equivalence testing and the range statistic. J. Statist. Planning Inf. 28, 33-47.

Finner, H. and Strassburger, K. (2006). On delta-equivalence with the best in k-sample models. J. Am. Statist. Assoc. 101, 737-74.

Finner, H. & Strassburger, K. (2002). The partitioning principle: A powerful tool in multiple decision theory. Ann. Statist. 30, 1194 -1213.

Hodges, J. L.  and Lehmann, E. L. (1954). Testing the approximate validity of statistical hypotheses. J. R. Stat. Soc., B 16, 261-268.

Wellek, S. (2003). Testing statistical hypotheses of equivalence. Boca Raton, Chapman and Hall.