McNemar's test

McNemar's test is to analyze contingency tables where subjects are matched or paired. This test is built-in to GraphPad QuickCalcs, but not Prism or InStat.

This example is adapted from the first edition of Intuitive Biostatistics. 

In a standard case-control study, the investigator compares a group of controls with a group of cases. As a group, the controls are supposed to be similar to the cases (except for the absence of disease). Another way to perform a case-control study is to match individual cases with individual controls based on age, gender, occupation, location and other relevant variables. This is the kind of data McNemar's test is designed for.

Displaying and analyzing data from matched case-control studies on an ordinary contingency table obscures the fact that the cases and controls were matched. Matching makes the experiment stronger, so the analysis ought to take it into account. Here are some sample data:

		Control
		+	-	Total
Case	+	13	25	38
	-	4	92	96
	Total	17	117	134

 

The investigators studied 134 cases and 134 matched controls, for a total of 268 subjects. Each entry in the table represents one pair (a case and a control). This is not a contingency table, so the usual analyses of contingency tables would not be helpful. It turns out that the odds ratio can be computed quite simply. The 13 pairs in which both cases and controls were exposed to the risk factor provide no information about the association between risk factor and disease. Similarly, the 92 pairs in which neither case nor control were exposed to risk factor provide no information. The odds ratio is calculated as the ratio of the other two values: pairs in which the case was exposed to the risk factor but the control was not divided by pairs in the control was exposed to the risk factor but the case was not. In this example, the odds ratio for the association between risk factor and disease is 25/4 = 6.25. The equation for the confidence interval is complicated (see page 286 of S. Selvin, Statistical Analysis of Epidemiologic Data, 2nd edition). The 95% confidence interval for the odds ratio ranges from 2.158 to 24.710.  

You can interpret the odds ratio from a matched case-control study just as you would interpret the odds ratio from an ordinary case-control study. If we assume that the disease is fairly rare, then we can conclude that exposure to the risk factor increases ones' risk 6.25 fold.

McNemar's test calculates a P value. This test uses only the number of discordant pairs, that is, the number of pairs for which the control was exposed to the risk factor but the case was not (4 in this example) and the number of pairs where the case was exposed to risk factor but the control was not (25 in this example). Call these two numbers R and S. Calculate chi-square using this equation: 

 

For this example, chi-square=13.79, which has one degree of freedom. The two-tailed P value is 0.0002. If there were really no association between risk factor and disease, there is a 0.02 percent chance that the observed odds ratio would be so far from 1.0 (no association).

 

Using GraphPad's McNemar test calculator

Enter the sample data into GraphPad's free web calculator like this:

Risk Factor?
Control	Case	# of pairs
No	Yes	25
Yes	No	4
Yes	Yes	13
No	No	92

More information

This article ("Changing Students' Perspectives of McNemar's Test of Change") gives an in depth explanation.