McNemar's test

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. pa + pb = pa + pc and pc + pd = pb + pd. Thus the null and alternative hypotheses are proposed the following continuity corrected version of the McNemar test to approximate the binomial exact-P-value: :\chi^2 = {(|b-c|-1)^2 \over b+c}. The mid-P McNemar test (mid-p binomial test) is calculated by subtracting half the probability of the observed b from the exact one-sided P-value, then double it to obtain the two-sided mid-P-value: : \text{mid-p-value} = 2 \left( \sum_{i=b}^n {n\choose i} 0.5^i (1-0.5)^{n-i} - 0.5 {n\choose b} 0.5^b (1-0.5)^{n-b} \right) This is equivalent to: : \text{mid-p-value} = \text{exact-p-value} - {n\choose b}0.5^b(1-0.5)^{n-b} where the second term is the binomial distribution probability mass function and n = b + c. Binomial distribution functions are readily available in common software packages and the McNemar mid-P test can easily be calculated. The traditional advice has been to use the exact binomial test when b + c < 25. However, simulations have shown both the exact binomial test and the McNemar test with continuity correction to be overly conservative. When b + c < 6, the exact-P-value always exceeds the common significance level 0.05. The original McNemar test was most powerful, but often slightly liberal. The mid-P version was almost as powerful as the asymptotic McNemar test and was not found to exceed the nominal significance level. ==Examples==

In the first example, a researcher attempts to determine if a drug has an effect on a particular disease. There are 314 patients, and they are diagnosed (disease: present or absent) before and after using the drug, which means that each sample can be described using 1 out of 4 combinations. Counts of individuals are given in the table, with the diagnosis (disease: present or absent) before treatment given in the rows, and the diagnosis after treatment in the columns. The test requires the same subjects to be included in the before-and-after measurements (matched pairs). In this example, the null hypothesis of "marginal homogeneity" would mean there was no effect of the treatment. From the above data, the McNemar test statistic: :\chi^2 = {(121 - 59)^2 \over {121 + 59}} has the value 21.35, which is extremely unlikely to form the distribution implied by the null hypothesis (p \chi^2 = 3.68 and p = 0.055. The asymptotic McNemar's test gives \chi^2 = 4.55 and p = 0.033 and the mid-P McNemar's test gives p = 0.035. Both the McNemar's test and mid-P version provide stronger evidence for a statistically significant treatment effect in this second example. Example implementation in Python The following is an example implementation using the probability distributions provided by the SciPy package. The implementation uses the mid-P version if b + c  from scipy.stats import binom, chi2 def mcnemar(b: int, c: int, continuity_correction: bool = False) -> float: check_valid = lambda n: isinstance(n, int) or (isinstance(n, float) and n.is_integer()) if not all(map(check_valid, [b, c])): raise ValueError("b and c must be integers!") n_min, n_max = sorted([b, c]) corr = int(continuity_correction) if (n_min + n_max) It is also worth noting that the above implementation ensures that the smaller of b  and c  is passed as the first argument to the binomial probability distributions. See "Additional file 1" under the supplementary materials for further reading. ==Discussion==

An interesting observation when interpreting McNemar's test is that the elements of the main diagonal do not contribute to the decision about whether (in the above example) pre- or post-treatment condition is more favourable. Thus, the sum b + c can be small and statistical power of the tests described above can be low even though the number of pairs a + b + c + d is large (see second example above). An extension of McNemar's test exists in situations where independence does not necessarily hold between the pairs; instead, there are clusters of paired data where the pairs in a cluster may not be independent, but independence holds between different clusters. An example is analyzing the effectiveness of a dental procedure; in this case, a pair corresponds to the treatment of an individual tooth in patients who might have multiple teeth treated; the effectiveness of treatment of two teeth in the same patient is not likely to be independent, but the treatment of two teeth in different patients is more likely to be independent. Information in the pairings In the 1970s, it was conjectured that retaining one's tonsils might protect against Hodgkin's lymphoma. John Rice wrote: 85 Hodgkin's patients [...] had a sibling of the same sex who was free of the disease and whose age was within 5 years of the patient's. These investigators presented the following table: :: \begin{array}{c|c|c} \hline & \text{Tonsillectomy} & \text{No tonsillectomy} \\ \hline\text{Hodgkins} & 41 & 44 \\ \hline\text{Control} & 33 & 52 \end{array} They calculated a chi-squared statistic [...] [they] had made an error in their analysis by ignoring the pairings.[...] [their] samples were not independent, because the siblings were paired [...] we set up a table that exhibits the pairings: : \begin{array}{cc} & \text{Sibling} \\ \text{Patient} & \begin{array}{c|c|c} \hline & \text{No tonsillectomy} & \text{Tonsillectomy} \\ \hline\text{No tonsillectomy} & 37 & 7 \\ \hline\text{Tonsillectomy} & 15 & 26 \end{array} \end{array} It is to the second table that McNemar's test can be applied. Notice that the sum of the numbers in the second table is 85—the number of pairs of siblings—whereas the sum of the numbers in the first table is twice as big, 170—the number of individuals. The second table gives more information than the first. The numbers in the first table can be found by using the numbers in the second table, but not vice versa. The numbers in the first table give only the marginal totals of the numbers in the second table. McNemar's test allows the 15 and 7 pairs where the siblings had previously had differing treatment to their tonsils to be compared, as being relevant to the hypothesis, while ignoring the less informative 37 and 26 where the siblings had previously both had the treatment or to their tonsils or neither had. ==Related tests and methods==

• The binomial sign test gives an exact test for the McNemar's test. • The Cochran's Q test is an extension of the McNemar's test for more than two "treatments". • The Liddell's exact test is an exact alternative to McNemar's test. • The Stuart–Maxwell test is different generalization of the McNemar test, used for testing marginal homogeneity in a square table with more than two rows/columns. • The Bhapkar's test (1966) is a more powerful alternative to the Stuart–Maxwell test, but it tends to be liberal. Competitive alternatives to the extant methods are available. • The McNemar's test is a special case of the Cochran–Mantel–Haenszel test; it is equivalent to a CMH test with one stratum for each of the N pairs and, in each stratum, a 2x2 table showing the paired binary responses. • Multinomial confidence intervals are used for matched pairs binary data. ==See also==