 Somers' D

Here $C,D$ count concordant pairs and discordant pairs, and $T_Y$ counts pairs tied on response $Y$ but not on predictor $X$. Somers' D divides by the number of pairs with distinct $X$ values, including pairs that the predictor separates but the response ties. Thus it is directional and generally changes when predictor and response are interchanged. For an $r\times c$ contingency table with row predictor totals $n_{i+}$, that denominator is $[n^2-{\sum }_i{n}_{i+}^2]/2$. The double-counting convention for concordances and discordances cancels the factor two.^[ref]