McNemar Chi-square and Exact test for Matched Pairs


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Analysis_Exact_Matched Pairs;



Paired proportions have traditionally been compared using McNemar's test but an exact alternative due to Liddell (1983) is preferable. StatsDirect gives you both.


The exact test is a special case of the sign test. The b count in the table below is treated as a binomial variable from the sample b+c. Using the ratio R' (R' = b/c) as a point estimate of relative risk, a two sided probability is calculated that R' = 1 (the null hypothesis). The test statistic F=b/(c+1).


Confidence limits for R' are calculated as follows:

- where F(α/2, n, d) is a quantile from the F distribution with n and d degrees of freedom.


You should use the exact test for analysis; McNemar's test is included for interest only.


If you need the exact confidence interval for the difference between a pair of proportions then please see paired proportions.



Observed frequencies should be entered as a paired fourfold table:




Control/reference category:



outcome present

outcome absent

Case/index category:

outcome present:



outcome absent:





From Armitage and Berry (1994, p. 127).


The data below represent a comparison of two media for culturing Mycobacterium tuberculosis. Fifty suspect sputum specimens were plated up on both media and the following results were obtained:




Medium B:




No Growth

Medium A:





No Growth:




To analyse these data in StatsDirect you should select the McNemar function from the chi-square section of the analysis menu. Select the default 95% confidence interval. Enter the counts into the table as shown above.


For this example:

McNemar's test:

Yates' continuity corrected Chi² = 5.785714 P = 0.0162


After Liddell (1983)Refs:

Point estimate of relative risk (R') = 6

Exact 95% confidence interval = 1.335744 to 55.197091


F = 4, P (two sided) = 0.0129

R' is significantly different from unity


Here we can conclude that the tubercle bacilli in the experiment grew significantly better on medium A than on medium B. With 95% confidence we can state that the chances of a positive culture are between 1.34 and 55.20 times greater on medium A than on medium B.


P values

confidence intervals



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