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
Contents
- Somers D for sample
- Somers D for distribution
- Somers D for binary dependent variables
- Example
- References
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
Let
where
As
Somers’ D for distribution
Let two independent bivariate random variables
or the difference between the probabilities of concordance and discordance. Somers’ D of Y with respect to X is defined as
If X and Y are both binary with values 0 and 1, then Somers’ D is the difference between two probabilities:
Somers' D for binary dependent variables
In practice, Somers' D is most often used when the dependent variable X 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. Somers’ D is related to the area under the receiver operating characteristic curve (AUC),
In the case where the independent (predictor) variable Y is discrete and the dependent (outcome) variableX is binary, Somers’ D equals
where
Example
Suppose that the predictor variable Y takes three values, 6999250000000000000♠0.25, 6999500000000000000♠0.5, or 6999750000000000000♠0.75, and outcome variable X takes two values, 5000000000000000000♠0 or 7000100000000000000♠1. The table below contains observed combinations of X and Y:
The number of concordant pairs equals
The number of discordant pairs equals
The number of pairs tied on Y but not on X equals
Thus, Somers’ D equals