 # Matched analysis for paired binary data (McNemar test)

Sometimes we are interested in comparing pairs of observations within the same subjects. Then we should bear in mind that those observations are not independent and should be treated accordingly.

Let us consider the scenario where we are planning a study to assess the effectiveness of a new cone-beam computed tomography (CBCT) device with a lower radiation dose (device A) compared with the standard panoramic radiography (device B) in identifying early lateral incisor root resorption in subjects with impacted canines. Table I shows the results of the 2 devices for the same 30 subjects in terms of identifying root resorption (+) or no resorption (−). We can see that in some subjects, a , both devices identify root resorption (+ +); b , device A identifies root resorption (+) but device B does not (−); c , device B identifies root resorption (+) but device A does not (−); and d, neither device identifies root resorption (− −).

Table I
Results of 2 devices identifying root resorption (+) or no resorption (−) in 30 subjects
Patient Device A Device B Patient Device A Device B
1 16
2 + 17
3 + + 18 +
4 19 +
5 + 20
6 + + 21
7 + + 22 + +
8 + 23 +
9 + 24
10 + 25
11 26 +
12 27 +
13 28
14 29 + +
15 + + 30

Summing the numbers of a, b, c, and d results in the tabulation in Table II .

Table II
Sums of the numbers of devices A+ and B+, devices A+ and B−, devices A− and B+, and devices A− and B− from Table I
Device B Total
+
Device A
+ a = 6 b = 9 a + b = 15
c = 1 d = 14 c + d = 15

a , Device A+ and device B+; b , device A+ and device B−; c , device A− and device B+; d , device A − and device B−.

The pairs that agree and give either a positive or a negative result for each patient are called concordant pairs. In this example, we have 20 (a + d) concordant pairs. There are 10 pairs that disagree in this example (b + c), called discordant pairs. It is obvious that the pairs that disagree can give us information about which of the 2 devices is better in identifying root resorption. So, the statistical analysis will be focused on the discordant pairs.

An easy and quick way to assess the difference between these 2 devices in terms of proportions would be:

b − c N