Short answer Yes—if by “optimal” you mean locally (asymptotically) most powerful for contiguous alternatives within the natural semiparametric model—then the Wilcoxon–Mann–Whitney (WMW) test is optimal for proportional-odds (PO) alternatives. In the two-sample problem with a single binary covariate and a continuous outcome (no ties), the WMW statistic is exactly the efficient score test for β=0 in the semiparametric proportional-odds cumulative logit model. Consequently, when PO holds, the WMW test is semiparametrically efficient and locally most powerful for detecting departures from the null.

Details and context

• Proportional-odds model: For two groups with cdfs F and G, PO means logit G(y) = logit F(y) + β for all y, or equivalently G(y) = Hβ(F(y)) with Hβ(u) = e^β u / [1 + (e^β − 1)u].

• Score test equivalence: In the semiparametric PO cumulative logit model with a single binary covariate, the score test for H0: β=0 reduces (with no ties) to the Wilcoxon rank-sum statistic. This is a standard result in ordinal regression; it’s the cumulative-logit analogue of “log-rank = score test” in the Cox proportional hazards model.

• Optimality sense: In parametric models, the (efficient) score test is locally most powerful. In semiparametric models, after projecting out the infinite-dimensional nuisance (here, the unspecified baseline α(y) or, equivalently, the unknown marginal F), the resulting efficient score test is locally asymptotically maximin (i.e., semiparametrically efficient). Thus, under true PO, WMW has maximal local asymptotic power.

• Why Wilcoxon is the right rank test for PO: Within the class of linear rank tests, the locally most powerful scores for alternatives of the form G(y) = Kθ(F(y)) are determined by ψ(u) ∝ ∂/∂θ log K′θ(u) at θ=0 (Hájek–Šidák–Sen). For PO, Hβ(u) = e^β u / [1 + (e^β − 1)u] has derivative H′β(u); differentiating log H′β(u) at β=0 gives a score proportional to (1 − 2u), i.e., linear in u. That yields the Wilcoxon scores, so WMW is the locally most powerful rank test for PO alternatives.

• Logistic location-shift is a special case: If Y in each group is logistic with the same scale and different locations, logit F(y) is an affine function of y, so the difference in logits between groups is constant in y. Hence logistic location-shift implies proportional odds, and the Wilcoxon optimality in that case is contained within the broader PO-optimality.

Caveats and limits

• “Most powerful” here is local and asymptotic. There is generally no uniformly most powerful (UMP) test for two-sample problems with infinite-dimensional nuisances. The result is about local (Pitman) efficiency in the semiparametric PO model.

• Outside PO, another rank test can be more powerful. For example, under normal location-shift, the normal-scores (van der Waerden) test is locally optimal among linear rank tests; under proportional hazards, the log-rank test is the efficient score test.

• With ties/discrete outcomes, the literal identity with the WMW numerator needs standard tie adjustments; the optimality statement is asymptotic.

References

• Hájek, J., Šidák, Z., and Sen, P. K. (1999). Theory of Rank Tests, 2nd ed. (see the treatment of optimal linear rank tests and Lehmann-type alternatives).
• McCullagh, P. (1980). Regression models for ordinal data. JRSS B, 42, 109–142.
• Agresti, A. (2010). Analysis of Ordinal Categorical Data, 2nd ed. (links between cumulative logit models and rank tests).
• Harrell, F.E. (2015/2020). Regression Modeling Strategies, 2nd ed. (Section on Wilcoxon as the score test for the PO model).
• Bickel, P. J., Klaassen, C. A., Ritov, Y., and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models (semiparametric efficiency framework).