McNemar's 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 commonly used parameters to assess a diagnostic test in medical sciences are sensitivity and specificity. Sensitivity 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.85 Hodgkin's patients had a sibling of the same sexwho was free of the disease and whose age was within 5 years ofthe patient's. These investigators presented the following table: 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 commonly used parameters to assess a diagnostic test in medical sciences are sensitivity and specificity. Sensitivity 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. 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 Here pa, 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), χ 2 {displaystyle chi ^{2}} has a chi-squared distribution with 1 degree of freedom. If the χ 2 {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. If either b or c is small (b + c < 25) then χ 2 {displaystyle chi ^{2}} is not well-approximated by the chi-squared distribution. 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: which is simply twice the binomial distribution cumulative distribution function with p = 0.5 and n = b + c.