McNemar's test

McNemar's test analyzes paired nominal data presented in two-by-two contingency tables. It applies to situations where individuals are evaluated twice under different conditions, such as pre-test and post-test designs. The test is ideal for binary outcomes like yes/no or success/failure responses measured in matched pairs.

Use McNemar's test for paired samples with binary outcomes where the same subjects are measured under two different conditions. It is particularly suited for pre-post study designs, matched-pair studies, and clinical trials comparing treatment effectiveness. McNemar's test is a nonparametric alternative that does not assume normal distribution of data.

McNemar's test requires paired samples where each subject in one group corresponds to a subject in the other group. The outcome must be dichotomous (binary). Each pair must be independent of other pairs. Additionally, a minimum of 10 discordant pairs—where outcomes differ between conditions—is recommended for reliable results and adequate statistical power.

McNemar's test statistic is computed using a specific formula based on the values in the two-by-two contingency table. The resulting test statistic approximates a chi-square distribution with one degree of freedom. If the calculated statistic exceeds the critical value, the null hypothesis is rejected, indicating a significant change in proportions between the two trials.

The null hypothesis in McNemar's test states that the proportions in the two trials are the same. In other words, there is no significant difference in the binary outcome between the paired measurements. If the change in proportion between two trials is significant, the null hypothesis is rejected, suggesting a meaningful difference in the paired conditions.

McNemar's test is relatively robust for small sample sizes compared to parametric tests. However, reliability decreases with very small samples. A minimum of 10 discordant pairs is generally recommended for meaningful results. With fewer discordant pairs, the test may lack sufficient statistical power to detect a true difference between the paired conditions.

A practical example involves evaluating ant behavior in response to artificial prey odor. Thirty ants are tested in two arena sizes, with each arena divided into an odor-infused section and a control section. Ants moving toward the odor are scored positive, while those moving toward the control are scored negative. McNemar's test determines whether behavioral response proportions differ significantly between the two arena sizes.