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

IV. Computing the Area Under the ROC (pages
358–365)
A. General formula:

AUC¼

no:of case=noncase pairs in whichP^ðXcaseÞ^PðXnoncaseÞ

total#casecontrol pairs

B. Calculation formula:

cð¼AUCÞ¼

wþ 0 : 5 z

np

,

where

w¼no. of case/noncase pairs for which

P^ðXcaseÞ>^PðXnoncaseÞ

z¼no. of case/noncase pairs for which

P^ðXcaseÞ¼^PðXnoncaseÞ

np¼total no. of case/noncase pairs

C. Example of AUC calculation:

i. n¼300 subjects,n 1 ¼100 true cases,

n 0 ¼200 true noncases

ii. Fit logistic model P(X) and computeP^ðXiÞfor

i¼1,..., 300

iii. np¼n 1 n 0 ¼ 100  200 ¼20,000

iv. Supposew¼11,480 (i.e., 57.4% of 20,000)

andz¼5,420 (i.e., 27.1% of 20,000)

v. cð¼AUCÞ¼

wþ 0 : 5 z

np

¼

11,480þ 0 : 5 ð5,420Þ

20,000

¼ 0 : 7095 ,

grade C (fair) discrimination

V. Examples from Study on Screening for Knee

Fracture (pages 365–370)

A. Scenario:n¼348 ER patients; complaint: blunt

knee trauma; X-rayed for knee fracture.

B. Study purpose: use covariates to screen for

decision to perform X-ray.

C. Outcome variable: FRACTURE¼knee fracture

status (1¼yes, 0¼no)

Predictor variables:

FLEX¼ability to flex knee (0¼yes, 1¼no)

WEIGHT¼ability to put weight on knee (0¼yes,

1 ¼no)

AGECAT¼patients age (0¼age<55, 1¼age55)

HEAD¼injury to knee head (0¼no, 1¼yes)

PATELLAR¼injury to patella (0¼no, 1¼yes)

Detailed Outline 375