How is Somers' D calculated in PROC LOGISTIC?

[kashlik123]

Lets say we want to calculate Somers' D (SD) on the following data:

data have;
x = 1; y = 1; output;
x = 1; y = 1; output;
x = 1; y = 0; output;
x = 0; y = 0; output;
x = 1; y = 1; output;
run;

According to the definitions in the wiki article the following can be observed:

concordant pairs = 3

discordant pairs = 0

tied pairs = 3

all pairs = 10

Given these results, SD would be equal to (3 - 0)/(10 - 3) = 0.4285

Now if we execute the PROC LOGISTIC procedure we get the following:

proc logistic data=have;
model y = x;
run;

The results show a different value for SD. In addition if 50% of the pairs are tied, which means that the total number of pairs according to the output is equal to

6 + 6*0.5 = 9, which is incorrect.

Can someone please explain to me what I am missing?

[StatDave · Accepted solution]

All of the statistics in the "Association of Predicted Probabilities and Observed Responses" table can be computed as shown in this note. As noted in the Details: Model Fitting Information: Rank Correlation of Observed Responses and Predicted Probabilities section of the LOGISTIC documentation where these statistics are defined, each pair of observations must have differing responses. So, there are 6 pairs, not 10.