Achieved power for the McNemar test

When to use it

Assess a before-and-after or matched comparison of a binary outcome, such as two diagnostic tests on the same patients.

Not for

With very few discordant pairs expected, the normal approximation is poor; an exact McNemar test is safer.

Notation

$n$: Number of pairs.
$p_{10}, p_{01}$: Shares of pairs that change one way and the other.
$f$: Total discordance, p₁₀ + p₀₁.
$d$: Difference, |p₁₀ − p₀₁|.

Formulas

Power, inverting the sample-size formula

\text{Power}=\Phi\!\left(\frac{d\sqrt n-z_{1-\alpha/\text{tails}}\sqrt f}{\sqrt{f-d^2}}\right)+I_{\text{tails}=2}\,\Phi\!\left(\frac{-d\sqrt n-z_{1-\alpha/\text{tails}}\sqrt f}{\sqrt{f-d^2}}\right)

z₁₋α/tails is the standard-normal critical value.

Parameters

Number of pairs $n$

Values: Integer, at least 2

Complete pairs available for analysis.

Discordant proportion p₁₀ $p_{10}$

Values: Between 0 and 1

Share of pairs that change from yes to no.

Discordant proportion p₀₁ $p_{01}$

Values: Between 0 and 1; p₁₀ + p₀₁ ≤ 1

Share of pairs that change from no to yes.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

Pairs are independent and enough discordant pairs are expected for the normal approximation [1].
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

For 100 pairs with discordant proportions 0.25 and 0.10 at α = 0.05: f = 0.35, d = 0.15, so d√n = 1.5 against z√f = 1.16.

Sample size: 100

Open this example in the calculator

Validation

Power is the McNemar sample-size formula solved for z_β, with the far tail added for a two-sided test. At the size that calculator returns it reaches the target, one fewer falls short, and it agrees with an independent calculation in R.

How to cite

StatWisely. Achieved power for the McNemar test. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-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.