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

We then fit a logistic model P(X) to this data

set, and we compute the predicted probability

of being a case, i.e.,P^ðXiÞ, for each of the 300

subjects.

For this dataset, thetotal number of possible

case/noncase pairs(i.e.,np) is the product 100

200, or 20,000.

We now letwdenote the number of these pairs

for which^PðXÞfor the case is larger thanP^ðXÞ

for the corresponding control. Suppose, for

example, thatw¼11,480, which means that

in 57.4% of the 20,000 pairs, the case had a

higher predicted probability than its noncase

pair.

Now letzdenote the number of case/noncase

pairs in which both case and noncase had

exactly the same predicted probability. Con-

tinuing our example, we supposez¼5,420, so

that this result occurred for only 27.1% of the

20,000 pairs.

Then, for our example, the proportion of the

20,000 case-control pairs for which the case

has at least as large a predicted probability as

the control is (wþz)/np, which is 16,900/

20,000, or 0.8450.

A modification of this formula (called “c”)

involves weighting by 0.5 any pair with equal

predicted probabilities; that is, the numerator is

modified to “wþ0.5z”, so thatcbecomes 0.7095.

It is the latter modified formula that is equiva-

lent to the area under the ROC, i.e., AUC.

Based on the grading guidelines for AUC that

we provided in the previous section, the AUC of

0.7095 computed for this hypothetical example

would be considered to provide fair discrimi-

nation (i.e., grade C).

In our presentation of the above AUC formula,

we have not explicitly demonstrated why this

formula actually works to provide the area

under the ROC curve.

Example: (Continued)

Fit logistic model PðXÞ

and

computeP^ðXiÞfori¼1,..., 300

np¼n 1 n 0 ¼ 100  200 ¼20,000

w = no. of case/noncase pairs for which

Pˆ(Xcase)>Pˆ(Xnoncase)

EXAMPLE

Example: Supposew¼11,480

(i.e., 57.4% of 20,000)

Z = no. of case/noncase pairs for which

ˆP(Xcase) =P(ˆXnoncase)

EXAMPLE

Example: Supposez¼5,420.

(i.e., 27.1% of 20,000)

pd¼

wþz

np

¼

11,480þ5,420

20,000

¼ 0 : 8450

Modified formula:

20,000

=14,190

=

c =11,48020,000+ 0.5(5,420) =0.7095

c w+n0.5z=AUC

p

Interpretation from guidelines:

AUC¼ 0 : 7095 )Fair discrimination

ðgrade CÞ

How does AUC formula provide
geometrical area under curve?
Ilustrative Example below.

Presentation: IV. Computing the Area Under the ROC (AUC) 359