McNemar Test

Developing a Test for Significance of Changes in Related Proportions

The dichotomous responses from a sample of $n\prime$ individuals over two periods of time (i.e., before [Page 930]and after a treatment intervention such as a political debate) may be tallied into a 2 ×2 table of cross-classifications as shown in Table 1.

With respect to the population from which the aforementioned sample was taken, let $p_{ij}=x_{ij}/n^{\prime }$ be the probability of responses to the ith category before the treatment intervention was imposed and the jth category after.

To assess changes in repeated dichotomous responses, the null hypothesis is that of symmetry:

$$H_0:p_{12}=p_{21}.$$

That is, the null hypothesis tested is conditioned on those n = x12 + x21 individuals whose responses change, where the probability (p21) of a switch from B to A is equal to the probability (p12) of a switch from A to B, and that this probability is 0.5.

Under the null hypothesis, the random variable x12 is binomially distributed with parameters n and 0.5, as is the random variable x21 The expected value of each of these binomial distributions is 0.5n, and the variance for each is 0.25n. McNemar’s procedure enables an exact test of the null hypothesis using the binomial probability distribution with parameters n and 0.5.

The McNemar test statistic M, written as Min[x12,x21] is defined as the minimum of the response tallies x12 or x21 in the cross-classification table (i.e., Table 1).

For a two-tailed test, the null hypothesis can be rejected at the α level of significance if

$$\begin{array}{l}P\left(M\le Min\left[x_{12},x_{21}\right]|n,p_{12}=p_{21}=0.5\right)=\\ {\displaystyle \sum _{M=0}^{Min\left[x_{12},x_{21}\right]}\frac{n!}{M!\left(n-M\right)!}{\left(0.5\right)}^n}\le \alpha /2.\end{array}$$

...