McNemar test

In statistics, McNemar's test is a statistical test used on paired nominal data. It is applied to 2 × 2 contingency tables with a dichotomous trait, with matched pairs of subjects, to determine whether the row and column marginal frequencies are equal (that is, whether there is "marginal homogeneity"). It is named after Quinn McNemar, who introduced it in 1947. An application of the test in genetics is the transmission disequilibrium test for detecting linkage disequilibrium.

The test is applied to a 2 × 2 contingency table, which tabulates the outcomes of two tests on a sample of n subjects, as follows.

The null hypothesis of marginal homogeneity states that the two marginal probabilities for each outcome are the same, i.e. p_a + p_b = p_a + p_c and p_c + p_d = p_b + p_d.

Thus the null and alternative hypotheses are

Here p_a, etc., denote the theoretical probability of occurrences in cells with the corresponding label.

The McNemar test statistic is:

Under the null hypothesis, with a sufficiently large number of discordants (cells b and c), ${\displaystyle \chi ^{2}}$ has a chi-squared distribution with 1 degree of freedom. If the ${\displaystyle \chi ^{2}}$ result is significant, this provides sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that p_b ≠ p_c, which would mean that the marginal proportions are significantly different from each other.

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