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When is the Wilcoxon–Mann–Whitney procedure a test of location? Implications for effect‐size measures
Author(s) -
Parker Scott,
Jernigan Robert W.,
Lansky Joshua M.
Publication year - 2020
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/bmsp.12162
Subject(s) - monotone polygon , mathematics , wilcoxon signed rank test , mann–whitney u test , statistics , cumulative distribution function , probability density function , geometry
The Wilcoxon–Mann–Whitney procedure is invariant under monotone transformations but its use as a test of location or shift is said not to be so. It tests location only under the shift model, the assumption of parallel cumulative distribution functions (cdfs). We show that infinitely many monotone transformations of the measured variable produce parallel cdfs, so long as the original cdfs intersect nowhere or everywhere. Thus there are infinitely many effect sizes measured as shifts of medians, invalidating the notion that there is one true shift parameter and thereby rendering any single estimate dubious. Measuring effect size using the probability of superiority alleviates this difficulty.