## Rank Correlation of Observed Responses and Predicted
Probabilities

Define an event response as the response having Ordered Value of 1.
A pair of observations with different responses is said to be concordant
(discordant) if the observation with the response that has the larger
Ordered Value has the lower (higher) predicted event probability.
If a pair of observations with different responses
is neither concordant nor discordant, it is a tie. Enumeration of the total
numbers of concordant and discordant pairs is carried out by categorizing the
predicted probabilities into intervals of length 0.002 and accumulating the
corresponding frequencies of observations.
Let *N* be the sum of observation frequencies in the data.
Suppose there is a total of *t* pairs
with different responses, *n*_{c} of them are concordant,
*n*_{d} of them are discordant,
and *t*-*n*_{c}-*n*_{d} of them are tied.
PROC LOGISTIC computes the following four indices of rank correlation
for assessing the predictive ability of a model:

Note that *c* also gives the area under the receiver operating
characteristic (ROC) curve when the response is binary (Hanley and
McNeil 1982).

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.