Excellent question. You've connected the dots perfectly. The short answer is yes, you are correct.

The most general and accurate way to characterize the optimality of the Wilcoxon-Mann-Whitney (WMW) test is to say that it is the most powerful (or more precisely, the locally most powerful rank) test for detecting alternatives under the proportional odds (PO) model.

Let's break down why this is the case, building on the points you've already made.

1. Formalizing the Proportional Odds Assumption

Let be a continuous random variable. Let and be the cumulative distribution functions (CDFs) for the two groups.

The proportional odds assumption states that the odds of the outcome being less than or equal to in one group is proportional to the odds in the other group. Mathematically, for some constant effect size :

Applying the logit transformation, where , this becomes:

This means that if you plot the logit-transformed CDFs for the two groups, they will be parallel, separated by a vertical distance of .

2. The Connection to the Score Test (Your Key Insight)

Your second bullet point is the formal proof of this concept.

• The semiparametric proportional odds ordinal logistic model is a powerful framework that models the relationship above without making any assumptions about the baseline CDF, . It is semiparametric because it has a parametric component () and a non-parametric component (the baseline distribution).
• For hypothesis testing, a standard and theoretically optimal approach is the score test. Score tests are derived from the derivative (the "score") of the log-likelihood function and are known to be Locally Most Powerful (LMP). This means they have the best power for detecting small deviations from the null hypothesis (i.e., for close to 0).
• When you derive the score test for a simple two-group comparison (i.e., a single binary predictor) within the proportional odds model, the resulting test statistic's numerator is precisely the Wilcoxon-Mann-Whitney rank-sum statistic () centered at its null expectation.

Because the WMW test is algebraically equivalent to the score test for the PO model, it inherits the score test's optimality properties. Therefore, the WMW test is the LMP rank test against proportional odds alternatives.

3. Why is This More General than the Logistic Location-Shift?

Your first bullet point is a specific instance of the more general PO case.

If follows a logistic distribution with location and scale , its CDF is . The logit of this CDF is:

Now, consider two groups with a location shift:

• Group 0:
• Group 1:

The logit-transformed CDFs are:

The difference is , which is a constant. This perfectly fits the PO model where .

However, the PO model is more general. It does not require the underlying distributions to be logistic. For example, two normal distributions with different means but equal variances will approximately satisfy the PO assumption, especially near the center of the distributions. The WMW test is often nearly as powerful as the t-test in this a case and much more powerful for heavy-tailed distributions. The PO model correctly identifies the structural relationship for which the WMW test is optimal, regardless of the baseline distribution's shape.

4. The Perfect Analogy: Log-Rank Test and Proportional Hazards

Your analogy to the log-rank test is spot on and highlights a beautiful symmetry in semiparametric statistics.

Summary

To conclude, your intuition is correct.

1. The WMW test is the score test for the semiparametric proportional odds model.
2. Score tests are locally most powerful for the alternatives defined by their underlying model.
3. Therefore, the most general and precise statement is that the Wilcoxon-Mann-Whitney test is the locally most powerful rank test for detecting proportional odds alternatives.

This is a more profound understanding than the common textbook explanation that "it tests for differences in medians" (which is only true under additional assumptions) or that "it's optimal for the logistic distribution" (which is just one special case).