McNemar’s Test | Binomial Tests and 2×2 Tables

The McNemar test tests whether marginal proportions are equal in paired binary data, such as before and after measurements of the same subject or matched pairs. Unlike a regular 2x2 independence test, only the unmatched pairs that have changed results are informative.

Paired 2×2 Table#

Measurement 1\Measurement 2	positive	negative
positive	a: Positive → Positive	b: Positive → Negative
negative	c: Negative → positive	d: Negative → negative

a and d are unchanged matched pairs. b and c are mismatched pairs that change in opposite directions, and H₀ asserts that these two directions occur to the same extent.

Hypothesis and conditional binomial distribution#

null hypothesis

H_0:P_b=P_c

alternative hypothesis

H_1:P_b\ne P_c

Limited to mismatched pairs

H_0:\pi=P(b\mid b\text{ or }c)=\frac12

Total number of mismatched pairs

n=b+c

Distribution under H₀

B\mid(B+C=n)\sim\operatorname{Binomial}\left(n,\frac12\right)

Derivation of the Z Statistic#

Mean and variance under H₀

E(B)=\frac n2,\qquad\operatorname{Var}(B)=\frac n4

standardization

Z=\frac{b-n/2}{\sqrt{n/4}}

Assign n=b+c

Z=\frac{b-(b+c)/2}{\sqrt{(b+c)/4}}

simplification

Z=\frac{b-c}{\sqrt{b+c}}

McNemar chi-square statistic#

\chi_M^2=Z^2=\frac{(b-c)^2}{b+c},\qquad df=1

For a two-sided test, you can use χ², which is Z squared. The larger the difference between b and c compared to the total number of mismatched pairs, the more contrary to H₀ that the marginal ratios are equal.

Exact McNemar test#

p_{\mathrm{exact}}=2\min\left\{P(B\leq b_{\mathrm{obs}}),\ P(B\geq b_{\mathrm{obs}})\right\}\quad\text{(upper limit 1)}

If b+c is small, use B~Binomial(b+c,1/2) directly. Since it is a discrete distribution, specify the definition of a two-sided p value and whether to use mid-p.

Continuity correction#

Correction Z

Z_Y=\frac{|b-c|-1}{\sqrt{b+c}}

Corrected McNemar statistic

\chi_{M,Y}^2=\frac{\left(|b-c|-1\right)^2}{b+c}

If you subtract 0.5 on the binomial scale of b, you subtract 1 on the b - c difference scale. Although the correction is more conservative, the exact binomial method is more direct for small numbers of discrepancies.

Difference with independence test#

item	Pearson independence test	McNemar test
data	Two binary variables from independent observations	Binary results with matched and repeated measurements
H₀	two variables are independent	The two discordant transition probabilities, or marginal proportions, are equal
Cell to use	all cells	Mismatched cells b, c
large sample statistics	Σ(O−E)²/E	(b−c)²/(b+c)

Implementation and reporting procedures#

Check the definition of pairs and encoding of binary results
Make a, b, c, d and check especially the number of discrepancies b+c
Preset H₀, bilateral/unilateral, α
If the number of discrepancies is sufficient, select uncorrected or corrected χ², if small, select exact binomial method.
b and c, report proportions, changes, p-values, and usage for each time point
Consider the impact of missing data and dropped pairs on the results