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

Stratum 1

Compute Mantel – Haenszel c^2 and MOR

Stratum 2 Stratum 100

D

E E

D

E WY

E XZ

E E E E E E E E
D 101 D 101 D 011 D 011

D 101 D 011 D 101 D 011

WX Y Z

w^2 MH¼

ðXYÞ^2

XþY

;df¼ 1

McNemar’s test

McNemar’s test¼MH test for
pair-matching

MORd ¼X=Y, 95% CI:

MOR expd



 196

ffiffiffiffiffiffiffiffiffiffi

1

Xþ

1

Y

q 

One way is to carry out aMantel–Haenszel chi-

square testfor association based on the 100

strata and to compute aMantel–Haenszel odds

ratio, usually denoted as MOR, as a summary

odds ratio that adjusts for the matched vari-

ables. This can be carried out using any stan-

dard computer program for stratified analysis

e.g., PROC FREQUENCY, in SAS.

The other method of analysis, which is equiva-

lent to the above stratified analysis approach,

is to summarize the data in a single table, as

shown here. In this table, matched pairs are

counted once, so that the total number of

matched pairs is 100.

As described earlier, the quantityWrepresents

the number of matched pairs in which both the

case and the control are exposed. Similarly,X,

Y, andZare defined as previously.

Using the above table, the test for an overall

effect of exposure, controlling for the matching

variables, can be carried out using a chi-square

statistic equal to the square of the difference

X–Ydivided by the sum ofXandY. This chi-

square statistic has one degree of freedom in

large samples and is calledMcNemar’s test.

It can be shown that McNemar’s test statistic is

exactly equal to the Mantel–Haenszel (MH)

chi-square statistic obtained by looking at the

data in 100 strata. Moreover, the MOR esti-

mate can be calculated asX/Y, and a 95% con-

fidence interval for the MOR can also be

computed (shown on the left).

As an example of McNemar’s test, supposeW

equals 30,Xequals 30,Y equals 10, andZ

equals 30, as shown in the table here.

Then based on these data, the McNemar test

statistic is computed as the square of 30 minus

10 divided by 30 plus 10, which equals 400 over

40, which equals 10.

EXAMPLE

D

E E

D

E W¼ 30 Y¼ 10

E X¼ 30 Z¼ 30

D

E E

D

E 30 10

E 30 30

w^2 MH¼

ð 30  10 Þ^2

30 þ 10

¼

400

40

¼ 10 : 0

396 11. Analysis of Matched Data Using Logistic Regression