In statistics, Somers’ D, sometimes incorrectly referred to as Somer’s D, is a measure of ordinal association between two possibly dependent random variables X and Y. Somers’ D takes values between − 1 {\displaystyle -1} when all pairs of the variables disagree and 1 {\displaystyle 1} when all pairs of the variables agree. Somers’ D is named after Robert 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 binary choice or ordinal regression (e.g., logistic regressions) and credit scoring models.
Somers’ D for sample We say that two pairs ( x i , y i ) {\displaystyle (x_{i},y_{i})} and ( x j , y j ) {\displaystyle (x_{j},y_{j})} are concordant if the ranks of both elements agree, or x i > x j {\displaystyle x_{i}>x_{j}} and y i > y j {\displaystyle y_{i}>y_{j}} or if x i < x j {\displaystyle x_{i}<x_{j}} and y i < y j {\displaystyle y_{i}<y_{j}} . We say that two pairs ( x i , y i ) {\displaystyle (x_{i},y_{i})} and ( x j , y j ) {\displaystyle (x_{j},y_{j})} are discordant, if the ranks of both elements disagree, or if x i > x j {\displaystyle x_{i}>x_{j}} and y i < y j {\displaystyle y_{i}<y_{j}} or if x i < x j {\displaystyle x_{i}<x_{j}} and y i > y j {\displaystyle y_{i}>y_{j}} . If x i = x j {\displaystyle x_{i}=x_{j}} or y i = y j {\displaystyle y_{i}=y_{j}} , the pair is neither concordant nor discordant. Let ( x 1 , y 1 ) , ( x 2 , y 2 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),\ldots ,(x_{n},y_{n})} be a set of observations of two possibly dependent random vectors X and Y. Define Kendall tau rank correlation coefficient τ {\displaystyle \tau } as
τ = N C − N D n ( n − 1 ) / 2 , {\displaystyle \tau ={\frac {N_{C}-N_{D}}{n(n-1)/2}},}
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