Differential analysis & tests / Rank-sum test (Mann–Whitney)
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The Mann–Whitney U test (Wilcoxon rank-sum test) compares the location of two distributions without assuming normality; it suits skewed data, counts or data with outliers.
P values use the exact null distribution for small samples without ties; otherwise the normal approximation with tie correction and, optionally, continuity correction (the usual default).
U is the statistic of the first group. The rank-biserial r = 2U/(n₁n₂) − 1 is the probability that a value from the first group exceeds one from the second minus the reverse (−1 to 1). The Hodges–Lehmann shift is the median of all between-group differences, with a Moses interval.
Batch mode: upload a feature × sample matrix and a sample sheet; every row is tested and P values are adjusted. With only two or three samples per group the smallest attainable rank-sum P is limited, so few results survive adjustment — a property of the method.
Mann H. B. & Whitney D. R. (1947) Ann Math Stat 18:50–60; Hodges J. L. & Lehmann E. L. (1963) Ann Math Stat 34:598–611. U and P match scipy.stats.mannwhitneyu; the Hodges–Lehmann interval follows Moses (normal approximation to the U quantile, no tie correction).
Up to 100,000 rows; the Hodges–Lehmann shift is skipped when n₁ × n₂ exceeds 4,000,000.
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