McNemar Test for Significance of Changes

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The dichotomous responses from a sample of n′ individuals over two periods of time may be tallied into a 2 × 2 table of cross-classifications as follows:

With respect to the population from which the aforementioned sample was taken, let pij be the probability of responses to the ith category before the treatment condition was imposed and the jth category after. The pairs of marginal probabilities before and after treatment sum to unity; that is, p1. + p2. = 1 and p.1 + p.2 = 1.

Testing for Significance of Changes in Related Proportions

In order to investigate changes in repeated dichotomous measurements, the null hypothesis is that of symmetry:

That is, the null hypothesis tested is conditioned on those n = x12 + x21 individuals whose responses change, where the probability (p21) of a switch to a more favorable position is equal to the probability (p12) of a switch to a less favorable position, 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 p = 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.

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

For a one-tailed test, the null hypothesis may be rejected if

Using Microsoft Excel's BINOMDIST function, exact p values of the McNemar test are obtained for all n ≤ 1,000.

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For studies where n > 1,000 a simple normal approximation formula for the test statistic M is given by

where Z = N(0,1), the standardized normal distribution. The decision rule for testing the null hypothesis (H0: pij = pji for i ≠ j) depends on whether the test is two-tailed or one-tailed. Based on its definition, M cannot exceed 0.5n, so the test statistic Z can be rejected only in the left tail of a standardized normal distribution. For a two-tailed test against the alternative H1: pij ≠ pji, the decision rule is to reject H0 if Z ≤ Zα/2. For a one-tailed test, the decision rule is to reject H0 if Z ≤ Zα.

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