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

The resulting ROC curve is shown at the

left. The AUC for this curve is 0.7095. We now

show the calculation of this area using the AUC

formula given earlier and using the trapezoid

approach.

We first apply the above AUC formula. In our

example, there are 100 cases and 200 noncases

yielding 100 200 ¼20,000 total pairs.

The ten cases withX 1 ¼1 andX 2 ¼1 have the

same predicted probability (tied) as the two

noncases who also haveX 1 ¼1 andX 2 ¼1.

But those same ten cases have a higher pre-

dicted probability (concordant) than the other

48 þ 50 þ100 noncases.

Similarly, the 50 cases with X 1 ¼1 and

X 2 ¼0 are

discordantwith the 2 noncases that have a

higher predicted probability,

tiedwith 48 noncases, and

concordantwith 50þ 100 ¼150 noncases.

The 20 cases withX 1 ¼0 andX 2 ¼1 are

discordantwith 2þ 48 ¼50 noncases,

tiedwith 50 noncases, and

concordantwith 100 noncases.

EXAMPLE (continued)

100%

80%

60%

10%

0%1% 25% 50% 1 – Sp 100%

Se

ROC curve (AUC = 0.7095)

np¼ 100 cases 200 noncases

¼20,000case=noncase pairs

WhenX 1 ¼1 andX 2 ¼1:
10 cases and 2 noncases have same^PðXÞ,
i:e:,10 2 ¼20 tied pairs

10 cases have higher^PðXÞthan

48 þ 50 þ 100 ¼ 198 noncases

i:e:,10 198 ¼1,980 concordant pairs

8

<

:

WhenX 1 ¼1 andX 2 ¼0:

50 cases have lower^PðXÞthan 2 noncases

i:e:,50 2 ¼100 discordantpairs

50 cases and 48 noncases have same^PðXÞ,

i:e:,50 48 ¼2,400 tied pairs

50 cases have higher^PðXÞthan

50 þ 100 ¼ 150 noncases

i:e:,50 150 ¼7,500 concordantpairs

8

<

:

WhenX 1 ¼0 andX 2 ¼1:
20 cases have lower^PðXÞthan
2 þ 48 ¼ 50 noncases
i:e:,20 50 ¼1,000 discordantpairs

20 cases and 50 noncases have sameP^ðXÞ,

i:e:,20 50 ¼1,000 tied pairs

20 cases have higher^PðXÞthan 100 noncases

i:e:,20 100 ¼2,000 concordantpairs

8

<

:

Presentation: IV. Computing the Area Under the ROC (AUC) 361