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

Finally, the 20 cases that did not have either

risk factorX 1 ¼0 andX 2 ¼0 are

discordant with 2þ 48 þ 50 ¼100 noncases

and

tiedwith 100 noncases.

We now sum up all the above concordant and

tied pairs, respectively, to obtain

w¼11,480 total concordant pairs and

z¼5,420 total tied pairs.

We then usew,z, andnpto calculate the area

under the ROC curve using the AUC formula,

as shown on the left.

To describe how to obtain this result geometri-

cally, we first point out that with 100 cases

and 200 noncases, the total number of case/

noncase pairs (i.e., 100200) can be geomet-

rically represented by the rectangular area with

height 100 and width 200 shown at the left.

A scaled-up version of the ROC curve is super-

imposed within this area. Also, the values listed

on the Y-axis (i.e., for cases) correspond to the

number of cases testing positive at the cut-

points used to plot the ROC curve. Similarly,

the values listed on the X-axis (i.e., for non-

cases) correspond to the number of noncases

testing positive at these same cut-points.

EXAMPLE (continued)

WhenX 1 ¼0 andX 2 ¼0:

20 cases have lower^PðXÞthan

2 þ 48 þ 50 ¼ 100 noncases

i:e:,20 100 ¼2,000 discordantpairs

20 cases and 100 noncases have same

^PðXÞ,i:e:,20 100 ¼2,000 tied pairs

8

>>>

>>>

>>>

<

>>>

>>>

>>>

:

total no. of concordant pairs:

w¼1,980þ7,500þ2,000¼11,480

total no. of tied pairs:

z¼ 20 þ2,400þ1,000þ2,000¼5,420

AUC¼

wþ 0 : 5 z

np

¼

11,480þ 0 : 5 ð5,420Þ

20,000

¼

14,190

20,000

¼ 0 : 7095

Geometrical Approach for

Calculating AUC

100

80

60

10

Cases

Discordant + ½ ties

Concordant + ½ ties

(^0250100200)
Noncases
ROC curve: scaled-up
(from 100%100% axes to 100
200 axes)
Y-axis:no. ofcases testingþ
X-axis:no. ofnoncases testingþ
362 10. Assessing Discriminatory Performance of a Binary Logistic Model