McNemar's Test Calculator

What Is McNemar's Test?

McNemar's test compares paired proportions from dichotomous outcomes, such as before-and-after results on the same subjects. Use it when each pair contributes one cell in a 2×2 table and you want to know whether the marginal proportions differ. Related tools include the Fisher's Exact Test Calculator and the Hypothesis Testing Calculator.

Contingency Table Layout

Label rows as Test 1 outcomes and columns as Test 2 outcomes:

• a: Test1+ and Test2+
• b: Test1+ and Test2− (discordant)
• c: Test1− and Test2+ (discordant)
• d: Test1− and Test2−

Only the discordant cells b and c enter the McNemar statistic. Concordant cells a and d do not change the test statistic.

Formulas

Standard McNemar chi-squared statistic:

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

Edwards continuity correction:

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

Yates continuity correction:

$$\chi^2 = \frac{(|b - c| - 0.5)^2}{b + c}$$

For small samples ($b + c < 25$), prefer the exact binomial or mid-p versions. Under the null, the discordant count follows a binomial distribution with $p = 0.5$.

Example

With $a=60$, $b=70$, $c=50$, $d=40$:

$$\chi^2 = \frac{(70 - 50)^2}{70 + 50} = \frac{400}{120} \approx 3.333$$

The chi-squared p-value is about 0.068, so at $\alpha = 0.05$ you would not reject the null of equal marginal proportions.