8.5 Correlated Proportions and McNemar’s Test 137were looking at mean differences, it was positive correlation that
helped. McNemar ’ s test is used for correlated categorical data. We can
use it to compare proportions when the data are correlated. Even if there
are more than two categories for the variables, McNemar ’ s test can be
used if there is a way to pair the observations from the groups.
As an example, suppose that we have subjects who are attempting
to quit smoking. We want to know which technique is more effective:
a nicotine patch or group counseling. So we take 300 subjects who get
the nicotine patch and compare them to 300 subjects who get the coun-
seling. We pair the subjects by characteristics that we think could also
affect successful quitting and pair the subjects accordingly.
For example, sex, age, level of smoking, and number of years you
have smoked may affect the diffi culty for quitting. So we match sub-
jects on these factors as much as possible. Heavy smokers who are
women and have smoked for several years would be matched with other
women who are heavy smokers and have smoked for a long time. We
denote by 0 as a failure to quit, where quitting is determined by not
smoking a cigarette for 1 year after the treatment. We denote by 1 a
success at quitting.
The possible outcomes for the pairs are (0, 0), (0, 1), (1, 0), and
(1, 1). We will let the fi rst coordinate correspond to the subject who
receives the nicotine patch, and the second coordinate his match who
gets counseling instead. The pairs (0, 0) and (1, 1) are called concordant
pairs because the subjects had the same outcome. The pair (0, 0) means
they both failed to quit, while the pair (1, 1) means that they both were
able to quit. The other pairs (0, 1) and (1, 0) are called discordant pairs
because the matched subjects had opposite outcomes.
The concordant pairs provide information indicating possible posi-
tive correlation between members of the pair without providing infor-
mation about the difference between proportions. Similarly, the
discordant observations are indicative of negative correlation between
the members of the pair. The number of 1s and 0s in each group then
provides the information regarding the proportions.
What we mean is that if I only tell you a pair is concordant and not
whether it is (0, 0) or (1, 1), you know that they are correlated but do
not know the actual outcome for either subject in the pair. The same
idea goes for the discordant observations. Although you know the
results are opposite, indicating a possible negative correlation, and we
know we have added a success and a failure to the total, we do not