Extension of the relationship between ROCAUC/c-index and Wilcoxon-Mann-Whitney U

Question

There is a relationship between ROCAUC and the Wilcoxon Mann-Whitney U test of the probability predictions of the two groups.

If our $y$ variable has more than two classes, can we extend this to a Kruskal-Wallis test? Can we go further and do something with a continuous $y$ and the test statistic of a general proportional odds ordinal logistic regression (of which Wilcoxon and Kruskal-Wallis are special cases)?

Answer 1

In extending If you like the Wilcoxon test you must like the proportional odds model I have simulations studying the concordance probabilities arising from the Kruskal-Wallis multi-group rank ANOVA setup. If you take from the PO model a $\beta$ that captures the comparison of group $i$ with group $j$, or a difference in $\beta$s that does that, the $j:i$ odds ratio (antilog of $\beta$) is a simple transformation of the concordance probability (scaled Wilcoxon statistic) from comparing $i$ to $j$ dropping the other groups from the data. The transformation is the same as for the two-group setup, which is approximately $\frac{OR^{0.65}}{1 + OR^{0.65}}$. This is true to a high level of approximation. The approximation starts to fail if the cumulative distribution functions for the two groups cross (i.e., under extreme non-proportional odds).

To my knowledge the only analytic results in the literature are from Agresti and pertain only to the probit ordinal model and not to the logistic (PO).

Note that this topic has nothing to do with machine learning, hence my edits to your tags.