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.

翻译：本文重新审视了Steel多重比较检验，该检验使用Wilcoxon统计量对多个处理样本与一个共同对照样本进行两两比较。在给定合并样本中结模式的条件随机化分布下，推导了Wilcoxon统计量的均值、方差和协方差。样本量不必相等。在随机化分布下，建立了这些Wilcoxon统计量的渐近多元正态性。这扩展了正态近似在条件检验中的应用范围，假设分别来自任意共同总体的容量为$n_0, n_1,\ldots,n_K$的独立样本，或对$N=n_0 + n_1+\ldots+n_K$个实验对象进行随机化处理分配。通过正态近似和单重求积获得显著性概率。在连续移位模型中，将同时检验转化为同时置信限和置信区间。该方法已在{\sf R}软件包{\tt kSamples}中实现。还讨论了向所有成对Wilcoxon检验比较的扩展。