= Somers' D
{c}
{title2=$D_{Y\mid X}=(C-D)/(C+D+T_Y)$}
{wiki}
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.https://andrisignorell.github.io/DescTools/reference/SomersDelta.html[DescTools documentation, Details: column response given row predictor and the row-total denominator]{ref}
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