McNemar's Test#

McNemar's test tests whether the distribution of a dichotomous variable (e.g., yes/no) has changed significantly between two related measurements. It is commonly used for before-and-after comparisons with binary data.

When to Use#

Use McNemar's test when you want to:

Compare paired binary data (e.g., the same subject at two time points)
Test whether a change in the frequency of a binary characteristic has occurred
You have a 2x2 contingency table with dependent samples

Assumptions#

Data are paired (e.g., before-after on the same subject)
The dependent variable is binary (dichotomous)
The observation pairs are independent of each other
Sufficient number of discordant pairs (b + c ≥ 25 for the normal approximation)

Formula#

The basis is the 2x2 contingency table of paired observations:

	After: Yes	After: No
Before: Yes	a	b
Before: No	c	d

The test statistic is based only on the discordant pairs (b and c):

\chi^2 = \frac{(b - c)^2}{b + c}

with 1 degree of freedom.

For small samples (b + c < 25), the continuity correction by Edwards is recommended:

\chi^2_{\text{corr}} = \frac{(|b - c| - 1)^2}{b + c}

Example#

Practical Example: Campaign Effectiveness

A company wants to know whether an advertising campaign changed purchase intention. 200 people are surveyed before and after the campaign (purchase intention: yes/no).

	After: Yes	After: No
Before: Yes	80	10
Before: No	30	80

Discordant pairs: b = 10, c = 30
$\chi^2 = (10 - 30)^2 / (10 + 30) = 400 / 40 = 10.0$
p < 0.01 (significant)

Interpretation: The campaign led to a significant change in purchase intention. More people switched from "No" to "Yes" (30) than vice versa (10).

Effect Size#

The effect size is calculated using the odds ratio of the discordant pairs:

OR = \frac{b}{c}

Alternatively, the phi coefficient or Cohen's g can be used:

g = \left| \frac{b}{b+c} - 0.5 \right|

Effect Size	Cohen's g
Small	0.05
Medium	0.15
Large	0.25

Note: McNemar's test only considers the discordant pairs (b and c). The concordant pairs (a and d) do not enter the test statistic.