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

Within the above rectangle, the concordant

pairs are represented by the areaunder the

ROC curve while thediscordant pairsare repre-

sented by the areaoverthe ROC curve. Thetied

pairsand are split equally over and under the

ROC curve (using the trapezoid rule).

To compute the actual area within the rectan-

gle under the ROC curve, we can partition this

area using sub-areas of rectangles and trian-

gles as shown at the left. The areas denoted by

Crepresent concordant pairs. The triangular

areas denoted byTrepresent ½ of tied pairs.

Using the grids provided for the Y- and X-axes,

the actual areas can be calculated as shown at

the left. Note that an area labeled asTis calcu-

lated as ½ the corresponding rectangular area

above and below the hypotenuse of a triangle

that connects two consecutive cut points.

The sum of all the subareas under the curve is

14,190, whereas the total area in the rectangle

of width 200 and height 100 is 200100, or

20,000 (np). Therefore, the proportion of the

total area taken up by the area under the ROC

curve is 14,190 divided by 20,000 or 0.7095,

which is the value calculated using the AUC

formula.

EXAMPLE (continued)

Concordant pairs: within area under
ROC

Discordant pairs: within area over
ROC

Tied pairs, split equally over and under
ROC

2

100

80

60

10

(^050100200)
Cases
Noncases
C C C
T
T
T
C C
C
T
2
100
80
60
10
(^050100200)
Cases
Noncases
1000
500 2000
2500 5000
1200
10 480 500 1000
Sum of subareas under ROC
¼ 10 þ 480 þ 500 þ1,000þ1,200
þ þ1,000
¼14,190
Proportion of total rectangular area
under ROC
¼14,190=20,000¼ 0 : 7095 ð¼AUCÞ
Presentation: IV. Computing the Area Under the ROC (AUC) 363