McNemar's Test

Definition

McNemar’s test is used to analyze paired nominal data (same samples are used for both conditions), particularly in before-and-after studies or matched-pair designs.

The test statistic is defined as $Q=\frac{(n_{12}-n_{21}{)}^2}{n_{12}+n_{21}}\sim {\chi }_2(1)$

Examples

	After treatment / No insomnia	After treatment / Insomnia
Before treatment / No insomnia	45	15
Before treatment / Insomnia	25	15

Consider a null hypothesis $H_0:$ The treatment is not effective. $Q=\frac{(15-25{)}^2}{15+25}=2.5<{\chi }_{0.05}^2(1)$ We can not reject the null hypothesis.