# Interpreting the Somers' D statistics

I'm running a logistic regression model by using SAS PROC LOGISTIC.

In the Association Statistics table that SAS provides as analysis output there is the Somers' D stat.

Could you provide an interpretation of such statistics and a reference about it, please?

## 2 Answers

Somers’ $D$ is an index that you want to be closer to 1 and farther from $-1$.  First, you use your model to generate the predicted scores for your dependent variable, $\hat{y}_i$.  Then you think about every possible pairing of data points.  A pair of predicted scores are “in agreement” if the rank order of the predicted scores match the rank order of the observed scores.  That is, if $y_1 < y_2$ and $\hat{y}_1 < \hat{y}_2$, then the pair of points is considered to agree with the prediction.  (To make life easy for this explanation, we will assume that ties are not possible.)  If $\hat{y}_1 > \hat{y}_2$, then the predicted values are not in agreement with the observed values.  After considering all possible pairs of points in your data set, $n_c$ is the number of agreements and $n_d$ is the number of disagreements (I'm using the same mathematical notation as the SAS PROC page, but using slightly different vocabulary.)

Let $p=\frac{n(n-1)}{2}$, the total number of possible pairs for a data set with sample size $n$.  The formula for Somers” $D$ is $$D = \frac{n_c-n_d}{p}$$

Hope this explanation is clear.

• Thanks for the answer @Gregg! Could you provide the document/link where you found this explanation? For instance the SAS PROC page. Thanks again! Mar 29, 2018 at 7:28
• I believe a google search of "sas proc logistic somer's d" will bring up the sas manual page in the first few results Mar 29, 2018 at 12:01
• Why would this be a correct interpretation: "Somers’ D is an index that you want to be closer to 1 and farther from −1" ??? Mar 29, 2018 at 13:55
• You (ideally) want perfect agreement, $n_c=p$, as opposed to perfect disagreement, $n_d=p$. Mar 29, 2018 at 13:58
• @GreggH going through the correspondung wikipedia article, I believe your formula might be wrong. I think you are showing kendall's tau and sommers d is a ration of two different kendall's tau calculations. Please check again and correct me if I am wrong. Jun 11, 2018 at 9:15

i had a problem with the denominator of Somers D formula. And I find it was explained clearly with an example in this paper(link below). Hopefully it will be helpful with those who have the same concern like me. Basically, the denominator is the ( number of concordant pairs + number of discordant pairs + number of pairs with tied ranks on the dependent or dependent variable). http://uregina.ca/~gingrich/gamma.pdf

• Welcome to the site. This isn't quite a complete answer by our standards. Maybe it should be a comment. Can you expand on this to make it a complete account of how to interpret Somers' D? Dec 27, 2020 at 17:18