In statistics, Somers’ D, sometimes incorrectly referred to as Somer’s D, is a measure of ordinal association between two possibly dependent random variables and. Somers’ D takes values between when all pairs of the variables disagree and 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 and credit scoring models.
Somers’ ''D'' for sample
We say that two pairs and are concordant if the ranks of both elements agree, or and or if and. We say that two pairs and are discordant, if the ranks of both elements disagree, or if and or if and. If or, the pair is neither concordant nor discordant. Let be a set of observations of two possibly dependent random vectors and. Define Kendall tau rank correlation coefficient as where is the number of concordant pairs and is the number of discordant pairs. Somers’ D of with respect to is defined as. Note that Kendall's tau is symmetric in and, whereas Somers’ D is asymmetric in and. As quantifies the number of pairs with unequal values, Somers’ D is the difference between the number of concordant and discordant pairs, divided by the number of pairs with values in the pair being unequal.
Somers’ ''D'' for distribution
Let two independent bivariate random variables and have the same probability distribution. Again, Somers’ D, which measures ordinal association of random variables and in, can be defined through Kendall's tau or the difference between the probabilities of concordance and discordance. Somers’ D of with respect to is defined as. Thus, is the difference between the two corresponding probabilities, conditional on the values not being equal. If has a continuous probability distribution, then and Kendall's tau and Somers’ D coincide. Somers’ D normalizes Kendall's tau for possible mass points of variable. If and are both binary with values 0 and 1, then Somers’ D is the difference between two probabilities:
Several statistics can be used to quantify the quality of such models: area under the receiver operating characteristic curve, Goodman and Kruskal's gamma, Kendall's tau, Somers’ D, etc. Somers’ D is probably the most widely used of the available ordinal association statistics. Somers’ D is related to the area under the receiver operating characteristic curve, In the case where the independent variable is and the dependent variable is binary, Somers’ D equals where is the number of neither concordant nor discordant pairs that are tied on variable and not on variable.
Example
Suppose that the independent variable takes three values,,, or, and dependent variable takes two values, or. The table below contains observed combinations of and :
The number of concordant pairs equals The number of discordant pairs equals The number of pairs tied is equal to the total number of pairs minus the concordant and discordant pairs Thus, Somers’ D equals
