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I received an email asking how to test two correlated proportions to see if one proportion is significantly larger than another. This is a classic application of McNemar's test.

McNemar's test arises when a pair of raters both evaluate the same set of objects and provide a binary (yes/no) rating. This test focuses on the discordant pairs--pairs where the raters disagree. Pairs are discordant when the first rater says "yes" and the second says "no" or when the first rater says "no" and the second rater says "yes." When the discordant pairs are skewed in one direction (for example, more yes/no than no/yes), this is evidence that the overall proportion of yeses is higher for one rater than the other. When the discordant pairs are split evenly, then this is evidence that the overall proportion is about the same for both raters.

In a study of the cytomegalovirus antigenemia assay (see citation below), formaldehyde fixation was compared to acetone fixation. In 405 samples, cytomegalovirus was detecting 36 times (8.8%) using formaldehyde and 22 times (5.4%) using acetone. The table below illustrates how both assays performed.

Is the rate of detection significantly greater for formaldehyde? To answer this, we need to know how often the tests disagreed and if the disagreements were significantly skewed towards formaldehyde. Indeed they were. For 18 cases formaldehyde detected cytomegalovirus when acetone did not and only for 4 cases did acetone detect cytomegalovirus when formaldehyde did not.

If both fixations were equally efficient then we would expect that the discrepancies would be split 50-50. But the probability that a discrepancy favors formaldehyde is actually 82% (18 / 22). This proportion differs significantly from 50%, since the 95% confidence interval is 67% to 98%.

It's interesting that McNemar's test ignores the 18 times that both tests detect cytomegalovirus and the 365 times that neither test detected cytomegalovirus. This has always bothered me somewhat, but you can safely ignore the concordant pairs (the pairs where both raters agree) because they represent non-informative data. They don't provide any information about the degree to which one proportion is greater than the other. Only those discordant pairs provide information about the possible discrepancy in proportions.

**Comparison of several fixation methods for cytomegalovirus antigenemia assay.** Perez
JL, De Ona M, Niubo J, Villar H, Melon S, Garcia A, Martin R. J Clin Microbiol. 1995
Jun;33(6):1646-9.
[Medline] [Abstract]
[PDF]

Here are some websites that discuss McNemar's test: