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

Why does the area under the ROC measure

discriminatory performance(DP)? We discuss

this question and other characteristics of ROC

curves inSection IIIof this chapter.

In the previous section, we illustrated how a

cut-pointcould be used with a fitted logistic

model to assign a subject X based on the

predicted valueP^ðXÞto be a “predicted” case

or noncase.

Denoting the general cut-point ascp, we typi-

cally predict a subject to be a case ifP^ðXÞ

exceedscp vs. a noncase ifP^ðXÞdoesn not

exceedcp.

Given a cut-pointcp, the observed and pre-

dicted outcomes can then be combined into a

classification (diagnostic) table, the general

form of which is shown here. The cell frequen-

cies within this table give the number oftrue

positives(nTP) andfalse negatives(nFN) out of

the number oftrue cases(n 1 ), and the number

offalse positives(nFP) andtrue negatives(nTN)

out of the number of true noncases (n 0 ).

From the classification table, we can compute

thesensitivity(Se)and thespecificity(Sp).

Ideally, perfect discrimination would occur if

both sensitivity and specificity are equal to 1,

which would occur if there were no false nega-

tives (nFN¼0) and no false positives (nFP¼0).

Why does area under ROC measure
DP? See Section III.

II. Assessing
Discriminatory
Performance Using
Sensitivity and
Specificity Parameters

Cut-point can be used withP^ðXÞto
predict whether subject is case or
noncase.

IfP^ðXÞ>cp, predict subjXto be
case.
If^PðXÞcp, predict subjXto be
noncase.

Table 10.1

General Classification/Diagnostic

Table

True (Observed) Outcome

cp Y¼ 1

(case)

Y¼ 0
(noncase)
Predicted Y¼ 1 nTP nFP
Outcome Y¼ 0 nFN nTN
n 1 n 0

Se¼Prðtrue positivejtrue caseÞ

¼nTP=n 1

Sp ¼Prðtrue negativejtrue noncaseÞ

¼nTN=n 0

Perfect Discrimination (Se¼Sp¼1)
True (Observed) Outcome
cp Y¼ 1 Y¼ 0
Predicted Y¼ 1 nTP 0
Outcome Y¼ 00 nTN
n 1 n 0

350 10. Assessing Discriminatory Performance of a Binary Logistic Model