Ullrich Munzel (University of Goettingen),

Ludwig Hothorn (University of Hannover, Germany)

Nonparametric Multiple Comparisons in the Presence of Ties

Multiple comparisons are considered in a general nonparametric one-way layout, which includes continuous distributions as well as discontinuous distributions. The so called normalized version of the distribution function is used to define generalizations of the well known Mann-Whitney effect for pairwise comparisons. The corresponding effect estimator is shown to be asymptotically equivalent to a sum of independent, uniformly bounded random variables. This asymptotic argument is used to show the asymptotic normality and to estimate the correlation matrix of the estimators under alternative. Thus, it is possible to derive simultaneous confidence intervals of the effects as well as multiple test procedures for a nonparametric generalization of the Behrens-Fisher-Problem. The application to the many-to-one problem and to the all-pairs problem are discussed. Moreover, the correlation structure of the effect estimators is examined under the hypothesis of homogenitiy, i.e. the pairwise equality of the underlying distributions. The resulting test procedures for the many-to-one problem and the all-pairs problem have a product correlation structure and are generalizations of Steel's asymptotic test for the many-to-one problem (Steel,1959) and of Steel's and Dwass' (Steel, 1960; Dwass, 1960) asymptotic test procedure for the all-pairs problem, respectively.

References:

- Brunner and Munzel (1999). The nonparametric Behrens-Fisher problem: asymptotic theory and a small-sample approximation. Biometrical Journal, to appear.
- Steel, R.D.G.(1959). A multiple comparison rank sum test: Treatments versus control. Biometrics 15, 560-572.
- Steel, R.D.G.(1960). A rank sum test for comparing all pairs of treatments. Technometrics 2, 197-207.
- Dwass, M (1960). Some k-sample rank-order tests. Contributions to Probability an Statistics (Eds. I. Olkin et al.) Stanford UniversityPress, 198-202.