Somers' D

In statistics, Somers’ D, sometimes incorrectly referred to as Somer’s D, is a measure of ordinal association between two variables $X$ and $Y$. Somers’ D takes values between $-1$ when all pairs of the variables disagree and $1$ when all pairs of the variables agree. Somers’ D is named after R. H. Somers, who proposed it in 1962.

Somers’ D plays a central role in rank statistics and is the parameter behind many nonparametric methods. It is also used as a quality measure of logistic regressions and credit scoring models.

Somers’ D for sample

We say that two pairs $(x_{i},y_{i})$ and $(x_{j},y_{j})$ are concordant, if the ranks of both elements agree, or $x_{i}>x_{j}$ and $y_{i}>y_{j}$ or if $x_{i}<x_{j}$ and $y_{i}<y_{j}$. We say that two pairs $(x_{i},y_{i})$ and $(x_{j},y_{j})$ are discordant, if the ranks of both elements disagree, or if $x_{i}>x_{j}$ and $y_{i}<y_{j}$ or if $x_{i}<x_{j}$ and $y_{i}>y_{j}$. If $x_{i}=x_{j}$ or $y_{i}=y_{j}$, the pair is neither concordant nor discordant.

Let $(x_{1},y_{1}),(x_{2},y_{2}),\ldots ,(x_{n},y_{n})$ be a set of observations of two possibly dependent random variables $X$ and $Y$. Define Kendall tau rank correlation coefficient $\tau$ as

$\tau ={\frac {N_{S}-N_{D}}{n(n-2)/2}},$

where $N_{S}$ is the number of concordant pairs and $N_{D}$ is the number of discordant pairs. Somes’ D of $Y$ with respect to $X$ is defined as $D_{YX}=\tau (X,Y)/\tau (X,X)$.

Note that Kendall's tau is symmetric in $X$ and $Y$, whereas Somers’ D is asymmetric in $X$ and $Y$.

Somers’ D for distribution

Let two bivariate random variables $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$ are independently drawn from the same probability distribution $\operatorname {P} _{XY}$. Again, Somers’ D can be defined through Kendall's tau

$\tau (X,Y)=\operatorname {E} (\operatorname {sgn}(X_{1}-X_{2})\operatorname {sgn}(Y_{1}-Y_{2}))=\operatorname {P} (\operatorname {sgn}(X_{1}-X_{2})\operatorname {sgn}(Y_{1}-Y_{2})=1)-\operatorname {P} (\operatorname {sgn}(X_{1}-X_{2})\operatorname {sgn}(Y_{1}-Y_{2})=-1),$

or the difference between the probabilities of concordance and discordance. Somes’ D of $Y$ with respect to $X$ is defined as $D_{YX}=\tau (X,Y)/\tau (X,X)$. Thus, $D_{YX}$ is the difference between the two corresponding probabilities, conditional on the $X$ values not being equal. If $X$ has continuous СDF, then $\tau (X,X)=1$ and Kendall's tau and Somers’ D coincide. Somers’ D normalizes Kendall's tau for possible mass points of variable $X$.

If $X$ and $Y$ are both binary with values 0 and 1, then Somers’ D is the difference between two probabilities:

$D_{YX}=\operatorname {P} (Y=1\mid X=1)-\operatorname {P} (Y=1\mid X=0).$

Somers’ D for logistic regression

Several statistics can be used to measure quality of logistic regressions: AUC or c-statistic, Goodman and Kruskal's gamma, Kendall's tau (Tau-a), Somers’ D, etc. Somers’ D is probably the most widely used of the available rank order correlation statistics. For $X$ being predicted probability of the outcome and $Y$ being the outcome, Somers’ D for logistic regression can be rewritten as

$D_{YX}={\frac {N_{S}-N_{D}}{N_{S}+N_{D}+T_{Y}}},$

where $T_{Y}$ is the number of pairs tied on variable $Y$.

In logistic regressions, Somers’ D is related to the well-known area under the receiver operating characteristic curve (AUA), $AUC=D_{YX}/2+0.5$.