McNemar Test Calculator — Paired 2×2 Analysis

The McNemar Test. Some patients lose a symptom after treatment, while others gain it. This paired binary test asks whether those two directions of change are equally probable. Build one paired table. Cell a is yes then yes, b is yes then no, c is no then yes, and d is no then no. The diagonal cells never changed. Only b and c disagree across occasions, so only those discordant pairs can reveal a direction of change. Under marginal homogeneity, each discordant pair is equally likely to move either way. Conditional on b plus c, one direction follows a fair binomial distribution. When fewer than twenty-five pairs are discordant, Distri Scope uses the exact two-sided binomial result. It doubles the inclusive tail through the smaller count. At twenty-five discordant pairs or more, the page uses b minus c squared over b plus c, compared with chi square on one degree of freedom. That large-sample branch has no continuity correction. The report names which branch ran, so the calculation can be reproduced without guessing.

Pairing is the design, not a table decoration. Each row and column classification must come from the same person or one deliberately matched unit. Different pairs must remain independent. Households, clinics, or repeated pairs from one person require a model that represents that dependence. This procedure tests marginal change, not agreement. A large stable diagonal can coexist with asymmetric change, while poor agreement can still have symmetric directions. The discordant odds ratio is b divided by c. Values below one mean no-to-yes changes outnumber yes-to-no changes under this orientation. Reverse the occasions or the change labels and that ratio reciprocates. Keep the table orientation visible before interpreting its direction. If c equals zero, the page flags the ratio as undefined. It supplies no confidence interval, so a finite ratio still has unquantified precision. Without pairing, use an independent-group test. More occasions, more categories, clustered pairs, and agreement questions each need a different method.

Consider fifty patients measured before and after treatment. Twenty are yes both times, three move yes to no, nine move no to yes, and eighteen stay no. Thirty-eight pairs are concordant, but the directional evidence is the three-versus-nine discordant split. Their sum is twelve. Because twelve is below twenty-five, let X follow a binomial distribution with twelve trials and probability one half. The smaller count is three. The inclusive probability from zero through three is zero point zero seven two nine nine eight. Doubling it gives zero point one four five nine nine six. That exact p-value exceeds alpha point zero five, so fail to reject marginal homogeneity. The data do not establish unequal directions of change. The discordant odds ratio is three divided by nine, or zero point three three three. It describes the observed direction but does not rescue weak evidence. For contrast, twenty-five versus forty discordant pairs triggers the uncorrected chi-square branch. The statistic is three point four six two and p is about zero point zero six two eight.

The method switch is strict: twenty-four discordant pairs use exact binomial probabilities, while twenty-five use the uncorrected chi-square approximation. With no discordant pairs, the calculator stops. A perfectly stable diagonal contains no information about which change direction is more likely. Validate four nonnegative whole counts, at least one discordant pair, and alpha strictly between zero and one. Count pairs, not separate measurements. Specify the two occasions, positive category, direction labels, analysis rule, and alpha before looking at the split. Reversing labels afterward changes the claim. Route three or more occasions to a repeated binary method, larger square tables to a paired categorical method, and clustered pairs to dependence-aware modeling. Report all four cells, total pairs, b and c, selected branch, alpha, p-value, discordant odds ratio, design limits, and guarded conclusion. Map paired binary changes, inspect the discordant evidence, and run the matching calculator free at Distri Scope dot com.