﻿ McNemar chi-square and exact test for matched pairs

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# McNemar chi-square and exact test for 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(P,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 must select the matched pairs (McNemar, Liddell) 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