Somers' D

Somers’ D for distribution

Let two independent bivariate random variables ${\displaystyle (X_1,Y_1)}$ and ${\displaystyle (X_2,Y_2)}$ have the same probability distribution ${\displaystyle {\mathrm{P}}_{XY}}$. Again, Somers’ D, which measures ordinal association of random variables X and Y in ${\displaystyle {\mathrm{P}}_{XY}}$, can be defined through Kendall's tau

${\displaystyle \begin{array}{cc}\tau (X,Y)& =\mathrm{E}(\mathrm{sgn}(X_1-X_2)\mathrm{sgn}(Y_1-Y_2))\\ & =\mathrm{P}(\mathrm{sgn}(X_1-X_2)\mathrm{sgn}(Y_1-Y_2)=1)-\mathrm{P}(\mathrm{sgn}(X_1-X_2)\mathrm{sgn}(Y_1-Y_2)=-1),\end{array}}$

or the difference between the probabilities of concordance and discordance. Somers’ D of Y with respect to X is defined as ${\displaystyle D_{YX}=\tau (X,Y)/\tau (X,X)}$. Thus, ${\displaystyle D_{YX}}$ is the difference between the two corresponding probabilities, conditional on the X values not being equal. If X has a continuous probability distribution, then ${\displaystyle \tau (X,X)=1}$ 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:

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

Somers' D for binary dependent variables

In practice, Somers' D is most often used when the dependent variable Y is a binary variable, i.e. for binary classification or prediction of binary outcomes including binary choice models in econometrics. Methods for fitting such models include logistic and probit regression.

Several statistics can be used to quantify the quality of such models: area under the receiver operating characteristic (ROC) curve, Goodman and Kruskal's gamma, Kendall's tau (Tau-a), Somers’ D, etc. Somers’ D is probably the most widely used of the available ordinal association statistics. Identical to the Gini coefficient, Somers’ D is related to the area under the receiver operating characteristic curve (AUC),

${\displaystyle \mathrm{A}\mathrm{U}\mathrm{C}=\frac{D_{XY}+1}{2}}$.

In the case where the independent (predictor) variable X is discrete and the dependent (outcome) variable Y is binary, Somers’ D equals

${\displaystyle D_{XY}=\frac{N_C-N_D}{N_C+N_D+N_T},}$

where ${\displaystyle N_T}$ is the number of neither concordant nor discordant pairs that are tied on variable X and not on variable Y.

Example

Suppose that the independent (predictor) variable X takes three values, 0.25, 0.5, or 0.75, and dependent (outcome) variable Y takes two values, 0 or 1. The table below contains observed combinations of X and Y:

The number of concordant pairs equals

${\displaystyle N_C=3\times 7+3\times 6+5\times 6=69.}$

The number of discordant pairs equals

${\displaystyle N_D=1\times 5+1\times 2+7\times 2=21.}$

The number of pairs tied is equal to the total number of pairs minus the concordant and discordant pairs

${\displaystyle N_T=(3+5+2)\times (1+7+6)-69-21=50}$

Thus, Somers’ D equals

${\displaystyle D_{XY}=\frac{69-21}{69+21+50}\approx \mathrm{0.34.}}$