McNemar Test Calculator

Assess paired nominal responses with options for asymptotic chi-square, Edwards continuity correction, and exact binomial p-values. Enter the 2x2 table for pre/post or matched designs and receive the test statistic, p-value, effect size, and interpretation.

1. Enter Paired 2x2 Table

Focus on discordant pairs

Row and column labels (optional)

Discordant focus

McNemar's test compares the off-diagonal counts \(b\) and \(c\) in the paired table:

\\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \\]

The null hypothesis assumes \(b = c\), meaning the treatment or time shift has no effect on the binary outcome.

Cell a (Row 1, Col 1)

Cell b (Row 1, Col 2)

Cell c (Row 2, Col 1)

Cell d (Row 2, Col 2)

Method

Asymptotic Continuity Exact binomial

Alternative hypothesis

Two-sided Right tail (b > c) Left tail (b < c)

Significance level (alpha)

Test Results

Results ready

Key Values

Method
Statistic
p-value
Alpha
Alternative

Effect Size

Discordant difference
Proportion difference
Phi coefficient

Decision

Observed Table

	Column 1	Column 2	Row total
Row 1
Row 2
Column total

Step-by-Step Workflow

Why the exact test?

When \(b + c\) is small, the chi-square approximation becomes liberal. The exact binomial test uses \(X \sim \text{Binomial}(b + c, 0.5)\) to deliver valid p-values even with very small samples.

Formula Reference

Asymptotic chi-square

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

Use when \(b + c \ge 25\) or the sample size is otherwise large enough for the chi-square approximation.

Edwards correction

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

Applies a continuity correction to reduce Type I error when discordant counts are modest.

Exact binomial

\\[ X \sim \text{Binomial}(b + c, 0.5) \\]

Two-sided p-values are computed by doubling the smaller tail probability and capping at 1.

Effect metrics

\\[ \text{Diff} = b - c, \quad \Delta_p = \frac{b - c}{n}, \quad \phi = \frac{b - c}{\sqrt{n (b + c)}} \\]

Here \(n = a + b + c + d\) is the total number of pairs.

Step-by-Step Guide

Collect paired responses and tabulate the 2x2 contingency table.
Identify discordant counts \(b\) and \(c\); the null expects them to be equal.
Select the desired method (asymptotic, continuity corrected, or exact) and alternative hypothesis.
Compute the test statistic and corresponding p-value, comparing to the chosen alpha level.
Report the effect size (difference and phi) alongside the decision for practical interpretation.

References

McNemar, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12(2), 153-157.
Edwards, A. L. (1948). Note on the "correction for continuity" in testing the significance of the difference between correlated proportions. Psychometrika, 13(3), 185-187.
Agresti, A. (2018). Statistical Methods for the Social Sciences (5th ed.). Pearson.