# McNemar Chi-square and Exact test for Matched Pairs

Analysis_Exact_Matched Pairs;

Analysis_Chi-square_McNemar.

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.

DATA INPUT:

Observed frequencies should be entered as a paired fourfold table:

 Control/reference category: outcome present outcome absent Case/index category: outcome present: a b outcome absent: c d

Example

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: Growth No Growth Medium A: Growth: 20 12 No Growth: 2 16

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.