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

Moreover, we will see that the answer to this

question can be quantified by obtaining the

area under an ROC curve (AUC): the larger

the area, the better the discrimination.

First, we provide the two ROCs derived from

hypothetical Models 1 and 2 that we consid-

ered in the previous section. Notice that the

ROC for each model is determined by connect-

ing the dots that plot pairs of Se and 1Sp

values obtained for several classification

cut-points.

For Model 1, the area under the ROC is 1.0.

In contrast, for Model 2, the area under the

ROC is 0.5.

Since the area under the ROC for Model 1 is

twice that for Model 2, we would conclude that

Model 1 has better discriminatory performance

than Model 2.

How can we explain this conceptually?

Our explanation:

The AUC measuresdiscrimination, that is, the

ability of the model to correctly classify those

with and without the disease. We would expect

a model that provides good discrimination to

have the property that true cases have a higher

predicted probability (of being classified as a

case) than true noncases. In other words, we

would expect the true positive rate (TPR¼Se)

to be higher than the false positive rate (FPR

¼ 1 Sp) for all cut-points.

Area under

ROC (AUC)

1 – Sp (= FPR)

Se (= TPR)

1.0

1.0

́

́

́

́

́

EXAMPLE

AUC = 1.0

Model 1

1 – Sp (= FPR)

Se (= TPR)

1.0

0.6 1.0

0.6

0.1

0.0

Cut-pt for prefect

prediction: Se=Sp= 1

AUC = 0.5

Model 2

1 – Sp (= FPR)

Se (= TPR)

0.1 0.6 0.91.0

0.0

0.1

0.6

0.8

0.9

1.0

So why is Model 1 a better dis-
criminator than Model 2?

Good discrimination
,
TPR>FPR
where

Se¼TPR,1Sp¼FPR

356 10. Assessing Discriminatory Performance of a Binary Logistic Model