When should I use the McNemar test?

Use the McNemar test when comparing binary outcomes from the same subjects before and after an intervention. For example, comparing pass/fail rates before and after a training program, or symptom presence before and after treatment. The key requirements are paired (matched) data and binary outcomes.

What is the difference between McNemar and chi-square?

The chi-square test compares categorical data from independent samples, while the McNemar test is designed for paired samples (same subjects measured twice). The McNemar test analyzes only the discordant pairs (b and c), ignoring concordant pairs.

What is the continuity correction?

The continuity correction adjusts the McNemar chi-square statistic to reduce error when approximating the discrete binomial distribution with the continuous chi-square distribution. The formula is chi-sq = (|b-c| - 1)^2 / (b+c). When discordant pairs are below 25, the exact binomial test is more appropriate.

How do I report McNemar results in APA format?

For asymptotic: 'McNemar's test, chi-sq(1) = X.XX, p = .XXX.' For exact (small samples): 'McNemar's exact test, p = .XXX.' Include the odds ratio and confidence interval when possible. StatMate generates this format automatically.

McNemar Test Calculator

Test for changes in proportions in paired nominal data. Used for before-after studies or matched-pair designs with binary outcomes.

What is the McNemar Test?

The McNemar test is a non-parametric statistical test used to analyze paired binary data—situations where the same subjects are measured twice on a dichotomous outcome. Developed by Quinn McNemar in 1947, it determines whether the proportion of subjects who changed from one category to another is significantly different from what would be expected by chance. It is essentially a test of symmetry in a 2×2 contingency table for matched pairs.

When to Use the McNemar Test

Use the McNemar test when you have paired or matched binary data. Common scenarios include: comparing a diagnostic test result before and after treatment, evaluating whether a training program changes pass/fail rates, testing if an intervention changes behavior (yes/no), or comparing two diagnostic methods applied to the same patients. The test focuses exclusively on the discordant pairs—subjects who changed their response between the two measurements.

McNemar Test vs. Chi-Square vs. Cochran's Q

Feature	McNemar	Chi-Square	Cochran's Q
Data type	Paired binary	Independent categorical	Paired binary (≥3 timepoints)
Samples	Matched pairs	Independent	Matched (3+ measures)
Groups compared	2 (before/after)	2 or more	3 or more
Focuses on	Discordant pairs	All cells	Changes across conditions

Worked Example: McNemar Test

A researcher tests whether a training program changes employee certification pass rates. 100 employees take the certification exam before and after training.

	After: Pass	After: Fail	Total
Before: Pass	40	12	52
Before: Fail	5	43	48
Total	45	55	100

The discordant pairs are b = 12 (passed before, failed after) and c = 5 (failed before, passed after). Since b + c = 17 < 25, the exact binomial test is used.

Results

McNemar's exact test, p = .143

There was no statistically significant change in pass rates after the training program. While 5 employees improved and 12 worsened, this difference was not significant at the .05 level.

Assumptions of the McNemar Test

The McNemar test requires the following assumptions to be met:

1. Paired/Matched Binary Data

Each subject must be measured twice (e.g., before and after), and the outcome must be binary (e.g., yes/no, pass/fail, positive/ negative). The data form a 2×2 table of matched pairs.

2. Mutually Exclusive Categories

Each subject must fall into exactly one of the four cells of the 2×2 table. The categories must be exhaustive and mutually exclusive at each time point.

3. Random Sampling

Subjects should be randomly selected from the population of interest, or randomly assigned to conditions. The matched pairs should be independent of each other (one pair's outcome should not influence another pair's outcome).

Understanding the Continuity Correction

The standard McNemar test uses a chi-square statistic with one degree of freedom. Because the binomial distribution (which underlies the test) is discrete, a continuity correction of 1 is applied: χ² = (|b − c| − 1)² / (b + c). This correction makes the chi-square approximation more accurate for moderate sample sizes. For small samples (discordant pairs < 25), StatMate automatically uses the exact binomial test instead, which does not require any approximation.

How to Report McNemar Results in APA Format

Report the test statistic, degrees of freedom, and p-value. If the exact test was used, note this in the report:

Asymptotic Test Template

McNemar's test indicated [a significant/no significant] change in [outcome] from [time 1] to [time 2], χ²(1) = X.XX, p = .XXX.

Exact Test Template

McNemar's exact test indicated [a significant/no significant] change in [outcome] from [time 1] to [time 2], p = .XXX.

Note: Use the exact test when the total number of discordant pairs (b + c) is less than 25. Report p-values to three decimal places, using p < .001 when below that threshold. Include descriptive statistics about the discordant pairs.

Common Mistakes to Avoid

Using chi-square for paired data: The standard chi-square test assumes independent observations. When the same subjects are measured twice, McNemar's test must be used instead because the data are dependent.
Ignoring small-sample requirements: When the total number of discordant pairs is less than 25, the chi-square approximation may be unreliable. Use the exact binomial test instead, which StatMate selects automatically.
Misinterpreting the table layout: The rows represent the "before" condition and columns the "after" condition. The discordant cells (b and c) are what the test analyzes. Concordant cells (a and d) do not affect the test result.
Applying to non-binary outcomes: McNemar's test is for binary outcomes only. For ordinal paired data, use the Wilcoxon signed-rank test. For three or more related binary measurements, use Cochran's Q test.
Forgetting the continuity correction: The uncorrected McNemar statistic (b − c)² / (b + c) tends to overestimate significance. Always use the corrected version (|b − c| − 1)² / (b + c) for more accurate results.

Calculation Accuracy

StatMate's McNemar test calculations have been validated against R's mcnemar.test() function and SPSS output. The implementation uses the continuity- corrected chi-square statistic and the jstat library for probability distributions. For small samples (discordant pairs < 25), the exact two-tailed binomial test is used. All results match R output to at least 4 decimal places.

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