On Steel's Test with Ties

TL;DR

This paper revisits Steel's multiple comparison test using Wilcoxon statistics, deriving new theoretical properties under ties, establishing asymptotic normality, and expanding the test's applicability with practical implementation in R.

Contribution

It provides new formulas for means, variances, and covariances of Wilcoxon statistics under ties, and extends the test to unequal sample sizes and broader scenarios.

Findings

Asymptotic multivariate normality of Wilcoxon statistics is established.

Normal approximation with quadrature is effective for significance testing.

The method is implemented in the R package kSamples.

Abstract

This note revisits Steel's multiple comparison test which uses Wilcoxon statistics in pairwise comparisons of several treatment samples with a common control sample. It derives means, variances and covariances of the Wilcoxon statistics under the conditional randomization distribution, given the tie pattern in the pooled samples. Sample sizes do not have to be equal. Under the randomization distribution asymptotic multivariate normality of these Wilcoxon statistics is established. This widens the scope of normal approximations to conditional tests, assuming independent samples of respective sizes $n_0, n_1,\ldots,n_K$ from any common population or randomized treatment assignment to $N=n_0 + n_1+\ldots+n_K$ experimental subjects. Significance probabilities are obtained using a normal approximation and a single quadrature. In the continuous shift model the simultaneous tests are converted to simultaneous confidence bounds and intervals. This is implemented in the {\sf R} package {\tt kSamples}. Extensions to all pairwise Wilcoxon test comparisons are discussed.