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

Observing the above ROCs, we see that, for

Model 1, TPR (i.e., Se) is consistently higher

than its corresponding FPR (i.e., 1Sp); so,

this indicates that Model 1 does well in differ-

entiating the true cases from the true noncases.

In contrast, for Model 2 corresponding true

positive and false positive rates are always

equal, which indicates that Model 2 fails to

differentiate true cases from true noncases.

The two ROCs we have shown actually repre-

sent two extremes of what typically results for

such plots. Model 1 gives perfect discrimina-

tion whereas Model 2 gives no discrimination.

We show in the figure at the left several differ-

ent types of ROCs that may occur. Typically, as

shown by the two dashed curves, the ROC plot

will lie above the central diagonal (45) line

that corresponds to Se¼ 1 Sp; for such

curves, the AUC is at least 0.5.

It is also possible that the ROC may lie

completely below the diagonal line, as shown

by the dotted curve near the bottom of the

figure, in which case the AUC is less than 0.5.

This situation indicates negative discrimina-

tion, i.e., the model predicts true noncases bet-

ter (i.e., higher predicted probability) than it

predicts true cases.

An AUC of exactly 0.5 indicates that the model

provides no discrimination, i.e., predicting the

case/noncase status of a randomly selected

subject is equivalent to flipping a fair coin.

A rough guide for grading the discriminatory

performance indicated by the AUC follows the

traditional academic point system, as shown

on the left.

Model 1 : TPRFPR always

+

Excellent discrimination

Model 2 : TPR¼FPR always

+

No discrimination

Two extremes:

Model 1: perfect discrimination

Model 2: no discrimination

Extremes

Se (=TPR)

1

ROC Types

(^0) 1 – Sp (= FPR)
Legend:
perfect discrimination
(Area = 1.0)
positive discrimination
(0.5 < Area ≤ 1.0)
negative discrimination
(0.0 ≤ Area < 0.5)
no discrimination
(Area = 0.5)
1
Grading Guidelines for AUC
values:
0.90–1.0¼excellent
discrimination (A)
0.80–0.90¼good discrimination
(B)
0.70–0.80¼fair discrimination (C)
0.60–0.70¼poor discrimination
(D)
0.50–0.60¼failed discrimination
(F)
Presentation: III. Receiver Operating Characteristic (ROC) Curves 357