Sample size for McNemar’s test

When to use it

The same people assessed twice with a binary outcome, or matched pairs, such as two diagnostic tests on the same patients.

Not for

Two independent groups. Use the two-proportion test.

Notation

$n$: Number of pairs
$p_{10},\ p_{01}$: Probabilities of the two discordant outcomes: yes then no, and no then yes
$f$: Total discordance, p₁₀ + p₀₁
$d$: Difference in discordance, p₁₀ − p₀₁
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Average number of participants per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters per group
$\ell$: Expected loss of whole clusters
$q$: Expected dropout

Formulas

Connor’s formula

n = \left\lceil \frac{\left[z_{1-\alpha/2}\sqrt{f} + z_{1-\beta}\sqrt{f - d^{2}}\right]^{2}}{d^{2}} \right\rceil

Only the discordant pairs carry information about a change, so the requirement depends on how often the two measurements disagree, not on the overall proportions [1].

Clustering

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho

When participants are recruited or randomised in clusters, observations within a cluster repeat information, and the sample is multiplied by the design effect. With equal clusters (CV = 0) this is 1 + (m − 1)ρ [2, 3].

Few clusters

F_t = \left(\frac{t_{2(C-1),\,1-\alpha/\tau} + t_{2(C-1),\,1-\beta}}{z_{1-\alpha/\tau} + z_{1-\beta}}\right)^{2}

Optional. A cluster trial is analysed on cluster-level summaries, so its critical values come from t on 2(C − 1) degrees of freedom rather than the normal distribution. With few clusters this costs participants; because C depends on the sample, it is iterated until it settles [4].

All adjustments together

n_{\text{adj}} = \left\lceil n \mathrm{DE}\;F_t \right\rceil

The adjustments multiply, in this order, and the result is rounded up once. Any adjustment not used contributes a factor of 1.

Dropout

n_{\text{recruit}} = \left\lceil \frac{n_{\text{adj}}}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Clusters to recruit

C = \left\lceil \frac{n_{\text{recruit}}}{m} \right\rceil, \qquad C_{\text{recruit}} = \left\lceil \frac{C}{1 - \ell} \right\rceil

Per group. Losing a whole cluster loses every participant in it, so an expected loss of clusters inflates the clusters recruited rather than the participants.

Parameters

Probability of yes then no $p_{10}$

Values: Between 0 and 1

The share of pairs expected positive on the first measurement and negative on the second.

Probability of no then yes $p_{01}$

Values: Between 0 and 1; with p₁₀ at most 1

The share of pairs expected negative first and positive second.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

For clustered designs only. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [5]; prefer an estimate from a comparable trial.

Average cluster size $m$ Optional

Values: At least 1

The average number of participants per cluster. Given with the ICC.

Variation in cluster size $\mathrm{CV}$ Optional

Values: 0 or more; 0 for equal clusters

The SD of cluster sizes divided by their mean. Unequal clusters cost power on their own account [3].

Expected loss of whole clusters $\ell$ Optional

Values: 0% up to, but not including, 100%

Clusters, such as practices or schools, expected to withdraw entirely.

Correction for few clusters $F_t$ Optional

Values: On or off

Worth applying when there are few clusters per group, roughly fewer than 15–20.

Assumptions

Pairs are independent apart from any clustering specified.
Enough discordant pairs are expected for the normal approximation; with very few, an exact McNemar test is safer.
With clustering: clusters are independent, with the same ICC throughout, and the analysis accounts for the clustering, for example with a mixed model or cluster-level summaries [2].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

A change in a yes/no outcome

p₁₀ = 0.2 and p₀₁ = 0.35, so f = 0.55 and d = 0.15: n = [1.96 × √0.55 + 0.8416 × √0.5275]² / 0.15² = 189.5, rounded up to 190 pairs.

Sample size: 190

Protocol statement
To detect a difference between discordant proportions of 20% and 35% with 80% power at a two-sided significance level of 0.05, using McNemar’s test, the required sample size is 190 pairs. Calculated with StatWisely (statwisely.com) using: n = [z_(1-α/tails)√f + z_(1-β)√(f-d²)]²/d²; f=p₁₀+p₀₁, d=p₁₀-p₀₁. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for missing second measurements

Expecting 10% not to complete: 190 / 0.9 = 211.1, so 212 are enrolled.

Needed for analysis: 190 · To recruit: 212

Protocol statement
To detect a difference between discordant proportions of 20% and 35% with 80% power at a two-sided significance level of 0.05, using McNemar’s test, the required sample size is 190 pairs. Allowing for 10% dropout, 212 pairs should be recruited. Calculated with StatWisely (statwisely.com) using: n = [z_(1-α/tails)√f + z_(1-β)√(f-d²)]²/d²; f=p₁₀+p₀₁, d=p₁₀-p₀₁. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.

How to cite

StatWisely. Sample size for McNemar’s test. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/proportions-mcnemar/

Cite the original references for the formulas your design uses as well.

References

1.Connor RJ. Sample size for testing differences in proportions for the paired-sample design. Biometrics. 1987;43(1):207–211.
2.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
3.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
4.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
5.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013