What is McNemar's test used for?

It tests whether the marginal row and column proportions in a paired 2×2 table are equal. Common uses: before/after studies, comparing two diagnostic tests on the same patients, and assessing attitude changes in panel surveys.

Why do only discordant pairs matter?

Concordant pairs (both positive or both negative) don't help distinguish between the two conditions. Only subjects who changed (discordant pairs) provide information about whether the conditions differ.

When should I use the exact binomial test?

When the number of discordant pairs (b + c) is small — typically under 25. The exact binomial version is safer for sparse tables, while the chi-square approximation is more comfortable once the discordant total is larger.

What is Yates' continuity correction?

It subtracts 1 from |b − c| before squaring, making the chi-square approximation more accurate for small samples by reducing the statistic slightly. This makes the test conservative.

How is McNemar's test different from a paired t-test?

McNemar's test is for binary (yes/no) outcomes, while the paired t-test is for continuous data. Both handle paired observations, but McNemar's uses a chi-square framework for categorical data.

Can McNemar's test be extended to more than two conditions?

Yes, Cochran's Q test generalizes McNemar's test to three or more related groups with binary outcomes. For ordered categories, the Stuart-Maxwell test or Bhapkar's test may be appropriate.

McNemar's Test Calculator

Perform McNemar's test for paired binary data. Compute chi-square statistic, exact binomial p-value, odds ratio, and compare correction methods on matched pairs.

McNemar\'s Test Calculator

Continuity Correction

Significance Level (α)

Paired 2×2 Table

Rows = Condition 1 (Before), Columns = Condition 2 (After)

	After +	After −
Before +
Before −

χ² Statistic

4.0500

McNemar's chi-square with Yates' correction, 1 df

p-Value (Asymptotic)

0.0442

Significant at α = 0.05

p-Value (Exact Binomial)

0.0414

Exact two-sided test based on discordant pairs

Decision

Reject H₀

Marginal proportions differ significantly

Discordant Pairs

b = 5, c = 15 (total = 20)

Only discordant pairs (b and c cells) inform the test

Odds Ratio (b/c)

0.3333

Ratio of discordant pair counts

Marginal Proportions

Measure	Value
Proportion positive — Before (a+b)/n	0.2500 (25.0%)
Proportion positive — After (a+c)/n	0.3500 (35.0%)
Difference (p₁ − p₂)	-0.1000 (-10.0%)
Concordant +/+ pairs (a)	20
Concordant −/− pairs (d)	60
Discordant +/− pairs (b)	5
Discordant −/+ pairs (c)	15
Total observations	100

Comparison of Corrections

Method	χ²	p-value
No Correction	5.0000	0.0253
Yates\' Correction	4.0500	0.0442
Edwards\' Correction	4.5125	0.0336
Exact Binomial	—	0.0414

Visual: Discordant Pairs

b = 5

+/−

c = 15

−/+

Planning notes, formulas, and examples

About the McNemar's Test Calculator

McNemar's test is designed for paired binary data — situations where the same subjects are measured under two conditions (before/after, test A/test B) with a yes/no outcome. Unlike the chi-square test of independence, which assumes independent observations, McNemar's test properly accounts for the paired structure of the data.

This calculator takes a paired 2×2 contingency table and computes the McNemar statistic with optional continuity corrections (Yates' or Edwards'), the exact binomial p-value, and marginal proportion differences. It also compares results across all correction methods in a single summary table.

Common applications include evaluating treatment effects in before-after studies, comparing diagnostic test accuracy, assessing attitude changes in matched surveys, and testing agreement patterns in reliability studies.

When This Page Helps

When your data consists of matched pairs with binary outcomes, an ordinary chi-square test is inappropriate because it ignores the pairing structure. McNemar's test focuses only on the discordant pairs — the cases where the two conditions disagree — providing a valid test of whether the marginal proportions differ. This calculator handles all the math and lets you compare correction methods side by side.

How to Use the Inputs

Enter the four cells of the paired 2×2 table: a (both +), b (before +, after −), c (before −, after +), d (both −).
Or click a preset example for common scenarios.
Select a continuity correction method (Yates', Edwards', or none).
Set your significance level alpha.
Review the chi-square statistic and p-values (asymptotic and exact).
Examine marginal proportions to see the direction of change.
Compare all correction methods in the summary table.

Formula used

McNemar's Test (no correction):
  χ² = (b − c)² / (b + c)

With Yates' correction:
  χ² = (|b − c| − 1)² / (b + c)

Exact Binomial Test:
  Under H₀, b ~ Binomial(b + c, 0.5)
  p-value = 2 × P(X ≤ min(b, c))

Where:
  a = concordant +/+ pairs
  b = discordant +/− pairs
  c = discordant −/+ pairs
  d = concordant −/− pairs

Example Calculation

Result: χ² = 4.05 (Yates'), p = 0.0442

With 5 discordant +/− pairs and 15 discordant −/+ pairs, McNemar's test with Yates' correction gives χ² = 4.05 (1 df), p = 0.044. This is significant at α = 0.05, indicating the marginal proportions differ — more subjects changed from negative to positive than the reverse.

Tips & Best Practices

Only the discordant pairs (b and c cells) contribute to the test. Concordant pairs (a and d) are irrelevant for testing marginal differences.
Use the exact binomial test when the total number of discordant pairs (b + c) is small (under 25).
Yates' continuity correction makes the test more conservative, reducing the chance of a Type I error at the cost of power.
The odds ratio b/c quantifies the direction and magnitude of the discordance.
McNemar's test is a special case of Cochran's Q test for two related groups (k = 2).
Check that your data is truly paired — the same subjects measured twice. If subjects differ between conditions, use chi-square instead.

Understanding the Paired 2×2 Table

In McNemar's setup, each subject contributes to exactly one cell. Cell a contains subjects who were positive on both occasions, d those negative on both, b those who changed from positive to negative, and c those who changed from negative to positive. The test asks whether b and c differ more than expected by chance alone.

Choosing a Correction Method

The uncorrected McNemar statistic can give slightly liberal p-values with small discordant counts. Yates' correction subtracts 1 from |b − c| to better approximate the discrete binomial distribution with a continuous chi-square. Edwards' correction subtracts 0.5 instead. For very small discordant counts (under 10), the exact binomial test is recommended regardless.

Applications in Medical Research

McNemar's test is ubiquitous in diagnostic accuracy studies. When comparing two diagnostic tests applied to the same patients, it determines whether the tests have different sensitivity or specificity. In clinical trials with crossover designs, it tests treatment effects on binary endpoints. In epidemiology, it's used for matched case-control studies.

Sources & Methodology

Last updated: March 8, 2026

Frequently Asked Questions

It tests whether the marginal row and column proportions in a paired 2×2 table are equal. Common uses: before/after studies, comparing two diagnostic tests on the same patients, and assessing attitude changes in panel surveys.