The Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians

To illustrate and document the tenuous connection between the Wilcoxon–Mann–Whitney (WMW) procedure and medians, its relationship to mean ranks is first contrasted with the relationship of a <i>t</i>-test to means. The quantity actually tested: Pr ^(X1&lt;X2)+ Pr ^(X1=X2)/2 is then described and recommended as the basis for an alternative summary statistic that can be employed instead of medians. In order to graphically represent an estimate of the quantity: Pr(<i>X</i>₁ &lt; <i>X</i>₂) + Pr(<i>X</i>₁ = <i>X</i>₂)/2, use of a bubble plot, an ROC curve and a dominance diagram are illustrated. Several counter-examples (real and constructed) are presented, all demonstrating that the WMW procedure fails to be a test of medians. The discussion also addresses another, less common and perhaps less clear cut, but potentially even more important misconception: that the WMW procedure requires continuous data in order to be valid. Discussion of other issues surrounding the question of the WMW procedure and medians is presented, along with the authors' teaching experience with the topic. SAS code used for the examples is included as supplementary material.

为阐释并验证威尔科克森-曼-惠特尼（Wilcoxon–Mann–Whitney procedure，WMW）检验与中位数之间的微弱关联，本文首先将该检验与平均秩的关系，同t检验与均值的关系进行对比。随后对实际待检验的统计量：$Pr(X_1 < X_2) + frac{1}{2}Pr(X_1 = X_2)$进行阐释，并推荐其作为替代中位数的另一类汇总统计量的构建基础。为以图形化方式展示该统计量的估计结果，本文分别演示了气泡图、受试者工作特征（Receiver Operating Characteristic，ROC）曲线以及优势图的应用方法。本文共展示了多个反例（包含真实案例与人工构建案例），所有案例均证明WMW检验并非针对中位数的假设检验。讨论部分还涉及另一类相对少见、或许界定不够清晰，但潜在重要性更高的认知误区：即认为WMW检验必须使用连续数据方能保证其有效性。本文还探讨了围绕WMW检验与中位数关系的其他相关议题，并分享了作者在该主题上的教学实践经验。文中示例所用的SAS代码已作为补充材料提供。