Short answer
Yes—if we formulate the null and the alternative inside the semiparametric proportional-odds (PO) model the Wilcoxon–Mann–Whitney (WMW) statistic is the efficient (hence, asymptotically most powerful) test for a two–sample comparison. No other regular test—rank-based or not—can have larger first-order (Pitman) power against contiguous PO alternatives. Outside that model, however, the WMW test is not uniformly most powerful; as soon as the PO assumption is violated another test can be more powerful.
Why the Wilcoxon statistic appears
Proportional-odds model
For a continuous response Y and a 0/1 group indicator Z
logit{P(Y ≤ y | Z=z)} = α(y) + βz, (1)
with an unrestricted baseline function α(·) and scalar parameter β.
Efficient score for β
Let F₀(y)=P(Y≤y |Z=0). At β=0 the efficient influence function (the derivative of the log-likelihood after orthogonal projection on the nuisance tangent space) can be written, after some algebra (Tsiatis, 1981; Anderson, 1984), as
φ(y,z,F₀) = (z–π) ∫_{-∞}^{y} dF₀(u), π=P(Z=1). (2)
Replacing the unknown integral by the empirical distribution of the pooled sample and summing over the observations gives
S = Σ_{i} (Z_i–π) R_i (3)
where R_i is the mid-rank of Y_i in the combined sample. After centring and scaling (3) is exactly the Wilcoxon rank–sum statistic.
Connection with the “logistic-location-shift’’ statement
If Y|Z=0 has a logistic c.d.f. F(x)=1/[1+e^{-(x−μ)}], a simple location shift yields
F₁(x)=1/[1+e^{-(x−μ−Δ)}],
and
logit F₁(x)= (x−μ−Δ) = logit F₀(x) −Δ,
so the two c.d.f.’s are in proportional odds with constant β=-Δ.
Thus the well-known fact that the WMW test is optimal for detecting logistic location shifts is a special case of the general PO result; the logistic distribution is merely a situation in which the (local) PO alternative corresponds to a location shift.
How far does the optimality go?
• Local / first-order only. The efficiency statement is asymptotic and local (β→0). For fixed, non-small β another statistic might have higher power even when (1) is true.
• Model dependent. If model (1) is wrong (e.g. hazards are proportional instead of odds, or the two c.d.f.’s cross) there is in general no uniformly most powerful rank test; a log-rank score, a normal-score test, etc., can dominate WMW for some data-generating mechanisms.
• Among all tests, not just rank tests. Because (3) is the efficient score it cannot be improved by using the actual data values—any such test would coincide with the efficient likelihood-score in the semiparametric model.
So the most general correct statement is
“In the two-sample problem, under the semiparametric proportional-odds model (1) with an unspecified baseline distribution, the Wilcoxon rank-sum test is the efficient score test for the parameter β and is therefore asymptotically the most powerful test against proportional-odds alternatives. It is not guaranteed to be optimal outside that model.”
References
Andersen, P. K. (1984) Score tests in generalized linear models. J. R. Statist. Soc. B 46, 194–202.
Tsiatis, A. A. (1981) A large-sample study of hypotheses testing for competing-risks models. JASA 76, 72–83.
Lehmann, E. L. & D’Abrera, H. J. M. (2006) Nonparametrics: Statistical Methods Based on Ranks, 2nd ed., §9.4.
van der Vaart, A. W. (1998) Asymptotic Statistics, §25.
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version: o3-2025-04-16
Status: UQ Validated
Validated: 8 months ago
Status: Failed Human Verification
Verified: 7 months ago
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