Rank Correlation of Observed Responses and Predicted Probabilities

The LOGISTIC Procedure

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:

$&&{c } =(n_c+0.5(t-n_c-n_d))/t \&&{Somers' D } =(n_c-n_d)/t \&&{Goodman-Kruskal Gamma } =(n_c-n_d)/(n_c+n_d) \&&{Kendall's Tau-a } =(n_c-n_d)/(0.5N(N-1))$

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

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