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

We now illustrate how this numerical formula

translates into the geometrical area under the

curve.

The method we illustrate is often referred to as

thetrapezoid method; this is because the area

directly under the curve requires the computa-

tion and summation of several trapezoidal

sub-areas, as shown in the sketch at the left.

As in our previous example, we consider 100

cases and 200 noncases and the fitted logistic

regression model shown at the left involving

two binary predictors, in which both^b 1 and^b 2

are positive.

A classification table for these data shows four

covariate patterns of the predictors that define

exactly four cut points for classifying a subject

as positive or negative in the construction of an

ROC curve. A fifth cut point is included

(c 0 ¼1) for which nobody tests positive since

P^ðXÞ 1 always. The ROC curve will be deter-

mined from a plot of these five points.

Note that the cut points are listed in decreasing

order.

Also, as the cut point lowers, both the sensitiv-

ity and 1specificity will increase.

More specifically, at cutpoint c 1 , 10 of 100

cases (10%) test positive and 2 out of 200 (1%)

noncases test positive. At cutpointc 2 , 60% of

the cases and 25% of the noncases test positive.

At cutpointc 3 , 80% of the cases and 50% of the

noncases test positive. At cutpointc 4 , all 100

cases and 200 noncases test positive because

P^ðXÞis equal to the cut point even for subjects

without any risk factor (X 1 ¼0 andX 2 ¼0).

Se

100%

0%

T

T P

P

T

P

T

P

ROC curve

1 – Sp 100%

Tp = trapezoidal

sub-area

defined by 2

cut-pts

EXAMPLE

P^ðXÞ¼^1

1 þexp½ð^aþ^b 1 X 1 þ^b 2 X 2 ފ

^b 1 >0,^b 2 > 0

Classification information for

different cut points (cp)

--c 0 =1

c 1

c 2

c 3

c 4

0

00

0

0

0 0 000

1

1

1

1

10

50

20

20

2

48

50

100

10

60

80

100

2

50

100

200

10

60

80

100

1

25

50

100

X 1 X 2 P(X) C NC C+NC+Se%1 – Sp%

Covariatepatterns

c 0 = 1 ⇒ 0 cases (C) and0 non-cases (NC)

test +

c 1 ¼P^ðXc 1 Þ>c 2 ¼P^ðXc 2 Þ> >c 4 ¼P^ðXc 4 Þ

cp#)Se and 1 Sp"

Se 1 – Sp

c 1

c 2

c 3

10% cases test +

60% cases test +

80% cases test +

c 4 100% cases test +100% noncases test +

50% noncases test +

1% noncases test +

25% noncases test +

360 10. Assessing Discriminatory Performance of a Binary Logistic Model