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

III. Receiver Operating
Characteristic (ROC)
Curves

Denotes cut-point

for classification

1.0 ROC Example

1.0

1 – Sp (= FPR)

Se (= TPR)

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ROC history:

 Developed by engineers in WW
II to detect enemy objects
(signal detection),
i.e.,P^ðXÞis a radar signal

 Now used in medicine,
radiology, psychology,
machine learning, data mining

ROC provides answer to:

IfXtrue caseandXtrue noncaseare
covariate values for a randomly
chosen case/noncase pair,
will
P^ðXtrue caseÞ>P^ðXtrue noncaseÞ?

A Receiver Operating Curve (ROC)is a plot

ofsensitivity (Se) by 1 – specificity (1 – Sp)

values derived from several classification

tables corresponding to different cut-points

used to classify subjects into one of two-

groups, e.g., predicted cases and noncases of

a disease.

Equivalently, the ROC is a plot of thetrue posi-

tive rate(TPR¼Se)by the false positive rate

(FPR¼ 1 Sp).

As described in Wikipedia (a free Web-based

encyclopedia), “the ROC was first developed by

electrical engineers and radar engineers during

World War II for detecting enemy objects in

battle fields, also known as the signal detection

theory; in this situation, a signal represents the

predicted probability that a given object is an

enemy weapon.” ROC analysis is now widely

used in medicine, radiology, psychology and,

more recently in the areas of machine learning

and data mining.

When using an ROC derived from a logistic

model used to predict a binary outcome, the

ROC allows for an overall assessment of how

well the model predicts who will have the out-

come and who will not have the outcome.

Stated another way in the context of epidemio-

logic research, the ROC provides a measure of

how well the fitted model distinguishes true

cases (i.e., those observed to have the outcome)

from true noncases (i.e., those observed not to

have the outcome).

More specifically, an ROC provides an appro-

priate answer to the question we previously

asked when we compared classification tables

for two models: How often will a randomly

chosen (true) case have a higher probability

of being predicted to be a case than a randomly

chosen true noncase?

Presentation: III. Receiver Operating Characteristic (ROC) Curves 355