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McNemar's Test
McNemar's test, also known as a test of correlated proportions, is a nonparametric test used with dichotomous nominal or ordinal data to determine whether two sample proportions based on the same individuals are equal. McNemar's test is used in many fields, including the behavioral and biomedical sciences. In short, it is a test of symmetry between two related samples based on the chi-square distribution with 1 degree of freedom (df).
McNemar's test is unique in that it is the only test that can be used when one or both conditions being studied are measured using the nominal scale. It is often used in before–after studies, in which the same individuals are measured at two times, a pretest–posttest, for example. McNemar's test is also often used in matched-pairs studies, in which similar people are exposed to two different conditions, such as a case–control study. This entry details the McNemar's test formula, provides an example to illustrate the test, and examines its application in research.
Formula
McNemar's test, in its original form, was designed only for dichotomous variables (i.e., yes–no, right–wrong, effect–no effect) and therefore gives rise to proportions. McNemar's test is a test of the equality of these proportions to one another given the fact that they are based in part on the same individual and therefore correlated. More specifically, McNemar's test assesses the significance of any observed change while accounting for the dependent nature of the sample. To do so, a fourfold table of frequencies must be set up to represent the first and second sets of responses from the same or matched individuals. This table is also known as a 2 × 2 contingency table and is illustrated in Table 1.
In this table, Cells A and D represent the discordant pairs, or individuals whose response changed from the first to the second time. If an individual changes from + to −, he or she is included in Cell A. Conversely, if the individual changes from – to +, he or she is tallied in Cell D. Cells B and C represent individuals who did not change responses over time, or pairs that are in agreement. The main purpose of McNemar's test is determine whether the proportion of individuals who changed in one direction (+ to −) is significantly different from that of individuals who changed in the other direction (− to +).
When one is using McNemar's test, it is unnecessary to calculate actual proportions. The difference between the proportions algebraically and conceptually reduces to the difference between the frequencies given in A and D. McNemar's test then assumes that A and D belong to a binomial distribution defined by
Based on this, the expectation under the null hypothesis would be that ½(A + D) cases would change in one direction and ½(A + D) cases would change in the other direction. Therefore, Ho: A = D. The χ2 formula,
where Oi = observed number of cases in the ith category and Ei = expected number of cases in the ith category under H0, converts
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