# McNemar's test

Short description: Statistical test used on paired nominal data

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.[1] An application of the test in genetics is the transmission disequilibrium test for detecting linkage disequilibrium.[2]

The commonly used parameters to assess a diagnostic test in medical sciences are sensitivity and specificity. Sensitivity (or recall) is the ability of a test to correctly identify the people with disease. Specificity is the ability of the test to correctly identify those without the disease.

Now presume two tests are performed on the same group of patients. And also presume that these tests have identical sensitivity and specificity. In this situation one is carried away by these findings and presume that both the tests are equivalent. However this may not be the case. For this we have to study the patients with disease and patients without disease (by a reference test). We also have to find out where these two tests disagree with each other. This is precisely the basis of McNemar's test. This test compares the sensitivity and specificity of two diagnostic tests on the same group of patients.[3]

## Definition

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.

 Test 2 positive Test 2 negative Row total Test 1 positive a b a + b Test 1 negative c d c + d Column total a + c b + d N

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[1]

\displaystyle{ \begin{align} H_0 & :~p_b=p_c \\ H_1 & :~p_b \ne p_c \end{align} }

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

The McNemar test statistic is:

$\displaystyle{ \chi^2 = {(b-c)^2 \over b+c}. }$

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 pb ≠ pc, which would mean that the marginal proportions are significantly different from each other.

### Variations

If either b or c is small (b + c < 25) then $\displaystyle{ \chi^2 }$ is not well-approximated by the chi-squared distribution.[citation needed] An exact binomial test can then be used, where b is compared to a binomial distribution with size parameter n = b + c and p = 0.5. Effectively, the exact binomial test evaluates the imbalance in the discordants b and c. To achieve a two-sided P-value, the P-value of the extreme tail should be multiplied by 2. For bc:

$\displaystyle{ \text{exact-P-value} = 2 \sum_{i=b}^{n} {n\choose i}0.5^i(1-0.5)^{n-i}, }$

which is simply twice the binomial distribution cumulative distribution function with p = 0.5 and n = b + c.

Edwards[4] proposed the following continuity corrected version of the McNemar test to approximate the binomial exact-P-value:

$\displaystyle{ \chi^2 = {(|b-c|-1)^2 \over b+c}. }$

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:[5][6]

$\displaystyle{ \text{mid-p-value} = 2 \left( \sum_{i=b}^n {n\choose i} 0.5^i (1-0.5)^{n-i} - 0.5 {n\choose b} 0.5^b (1-0.5)^{n-b} \right) }$

This is equivalent to:

$\displaystyle{ \text{mid-p-value} = \text{exact-p-value} - {n\choose b}0.5^b(1-0.5)^{n-b} }$

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.[6]

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.[6] 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).

 After: present After: absent Row total Before: present 101 121 222 Before: absent 59 33 92 Column total 160 154 314

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:

$\displaystyle{ \chi^2 = {(121 - 59)^2 \over {121 + 59}} }$

has the value 21.35, which is extremely unlikely to form the distribution implied by the null hypothesis (p < 0.001). Thus the test provides strong evidence to reject the null hypothesis of no treatment effect.

A second example illustrates differences between the asymptotic McNemar test and alternatives.[6] The data table is formatted as before, with different numbers in the cells:

 After: present After: absent Row total Before: present 59 6 65 Before: absent 16 80 96 Column total 75 86 161

With these data, the sample size (161 patients) is not small, however results from the McNemar test and other versions are different. The exact binomial test gives p = 0.053 and McNemar's test with continuity correction gives $\displaystyle{ \chi^2 }$ = 3.68 and p = 0.055. The asymptotic McNemar's test gives $\displaystyle{ \chi^2 }$ = 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.

## 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.[7] 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.[8]

### Information in the pairings

In the 1970s, it was conjectured that retaining one's tonsils might protect against Hodgkin's lymphoma. John Rice wrote:[9]

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:

$\displaystyle{ \begin{array}{c|c|c} \hline & \text{Tonsillectomy} & \text{No tonsillectomy} \\ \hline\text{Hodgkins} & 41 & 44 \\ \hline\text{Control} & 33 & 52 \end{array} }$

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:

$\displaystyle{ \begin{array}{cc} & \text{Sibling} \\ \text{Patient} & \begin{array}{c|c|c} \hline & \text{No tonsillectomy} & \text{Tonsillectomy} \\ \hline\text{No tonsillectomy} & 37 & 7 \\ \hline\text{Tonsillectomy} & 15 & 26 \end{array} \end{array} }$

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

• 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.[10][11]
• 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.[12][13][14]
• The Bhapkar's test (1966) is a more powerful alternative to the Stuart–Maxwell test,[15][16] but it tends to be liberal. Competitive alternatives to the extant methods are available.[17]
• 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.[18]

## References

1. McNemar, Quinn (June 18, 1947). "Note on the sampling error of the difference between correlated proportions or percentages". Psychometrika 12 (2): 153–157. doi:10.1007/BF02295996. PMID 20254758.
2. Spielman RS; McGinnis RE; Ewens WJ (Mar 1993). "Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM)". Am J Hum Genet 52 (3): 506–16. PMID 8447318.
3. Hawass, N E (April 1997). "Comparing the sensitivities and specificities of two diagnostic procedures performed on the same group of patients.". The British Journal of Radiology 70 (832): 360–366. doi:10.1259/bjr.70.832.9166071. ISSN 0007-1285. PMID 9166071.
4. Edwards, A (1948). "Note on the "correction for continuity" in testing the significance of the difference between correlated proportions". Psychometrika 13 (3): 185–187. doi:10.1007/bf02289261. PMID 18885738.
5. Lancaster, H.O. (1961). "Significance tests in discrete distributions.". J Am Stat Assoc 56 (294): 223–234. doi:10.1080/01621459.1961.10482105.
6. Fagerland, M.W.; Lydersen, S.; Laake, P. (2013). "The McNemar test for binary matched-pairs data: mid-p and asymptotic are better than exact conditional". BMC Medical Research Methodology 13: 91. doi:10.1186/1471-2288-13-91. PMID 23848987.
7. Yang, Z.; Sun, X.; Hardin, J.W. (2010). "A note on the tests for clustered matched-pair binary data". Biometrical Journal 52 (5): 638–652. doi:10.1002/bimj.201000035. PMID 20976694.
8. Durkalski, V.L.; Palesch, Y.Y.; Lipsitz, S.R.; Rust, P.F. (2003). "Analysis of clustered matched-pair data". Statistics in Medicine 22 (15): 2417–28. doi:10.1002/sim.1438. PMID 12872299. Retrieved April 1, 2009.
9. Rice, John (1995). Mathematical Statistics and Data Analysis (Second ed.). Belmont, California: Duxbury Press. pp. 492–494. ISBN 978-0-534-20934-6.
10. Liddell, D. (1976). "Practical Tests of 2 × 2 Contingency Tables". Journal of the Royal Statistical Society 25 (4): 295–304. doi:10.2307/2988087.
11. Sun, Xuezheng; Yang, Zhao (2008). "Generalized McNemar's Test for Homogeneity of the Marginal Distributions". SAS Global Forum.
12. Stuart, Alan (1955). "A Test for Homogeneity of the Marginal Distributions in a Two-Way Classification". Biometrika 42 (3/4): 412–416. doi:10.1093/biomet/42.3-4.412.
13. Maxwell, A.E. (1970). "Comparing the Classification of Subjects by Two Independent Judges". The British Journal of Psychiatry 116 (535): 651–655. doi:10.1192/bjp.116.535.651. PMID 5452368.
14. "McNemar Tests of Marginal Homogeneity". John-uebersax.com. 2006-08-30.
15. Bhapkar, V.P. (1966). "A Note on the Equivalence of Two Test Criteria for Hypotheses in Categorical Data". Journal of the American Statistical Association 61 (313): 228–235. doi:10.1080/01621459.1966.10502021.
16. Yang, Z.; Sun, X.; Hardin, J.W. (2012). "Testing Marginal Homogeneity in Matched-Pair Polytomous Data". Therapeutic Innovation & Regulatory Science 46 (4): 434–438. doi:10.1177/0092861512442021.
17. Agresti, Alan (2002). Categorical Data Analysis. Hooken, New Jersey: John Wiley & Sons, Inc.. p. 413. ISBN 978-0-471-36093-3.