Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Notice that at cut-point 0.100, 36 of 45 true

events were correctly classified as events, and

9 of 45 were incorrectly classified as nonevents;

also 184 of 303 true nonevents were correctly

classified as nonevents, and 119 of 303 were

incorrectly classified as events. The sensitivity

(Se) for this row is 36/45, or 80%, the specificity

(Sp) is 184/303, or 60.7%, so that 1Sp is

39.3%. Thus, in this row (cut-pt 0.100), the Se

is larger than 1Sp, which indicates good dis-

crimination (for this cut point).

We can also see from this table that Se is at

least as large as 1Sp for all cut-pts. Notice,

further, that once the cut-pt reaches 0.5 (and

higher), none of the 45 true cases are correctly

classified as cases (Se¼0) whereas all 303 true

noncases are correctly classified as noncases

(Sp¼1 and 1Sp¼0).

Additional output obtained from SAS’s Logis-

tic procedure is shown at the left. This output

contains information and statistical measures

related to the ROC curve for the fitted model.

The “c” statistic of 0.745 in this output gives the

area under the ROC curve, i.e., AUC, that we

described earlier. The Somers’ D, Gamma, and

Tau-a are other measures of discrimination

computed for the fitted model.

Each of these measures involves different ways

to compute a correlation between ranked (i.e.,

ordered) observed outcomes (Yi¼0 or 1) and

ranked predicted probabilitiesðP^ðXiÞÞ. A high

correlation indicates that higher predicted

probabilities obtained from fitting the model

correspond to true cases (Yi¼1) whereas

lower predicted probabilities correspond to

true noncases (Yi¼0), hence good discrimina-

tion.

The formulae for each measure are derived

from the information provided on the left side

of the above output. The definitions of each of

the latter items are shown at the left. Note that

w,z, andnpwere defined in the previous sec-

tion for the formula for the AUC (i.e.,c).

EXAMPLE (continued)

cp¼0.100:

Se¼ 36 = 45 ¼ 0 : 80

Sp¼ 184 = 303 ¼ 0 : 607

1 Sp¼ 0 : 393

Se¼ 0 : 80 > 1 Sp¼ 0 : 393

ðgood discriminationÞ

Se 1 Sp for all cut-points,

where

Se¼ 1 Sp¼ 0 for cp 0 : 500

Edited Output (SAS ProcLogistic)-

Association of Predicted Probabil-

ities and Observed Responses

Percent Concordant 71.8 Somers’ D 0.489

Percent Discordant 22.9 Gamma 0.517

Percent Tied 5.3 Tau-a 0.111

Pairs 13635 c 0.745

c¼AUC
Somer’s D, Gamma, and Tau-a:
other measures of discrimination

c, Somer’s D, Gamma, and Tau-a:

(ranked) correlations between

observed outcomes (Yi¼0or1)

and

predicted probabilitiesðP^ðXiÞÞ

Percent Concordant: 100w/np

Percent Discordant: 100d/np

Percent Tied: 100z/np

Pairs:np¼n 1 n 0 ,

Presentation: V. Example from Study on Screening for Knee Fracture 367