Is the New Analgesic More Effective than Aspirin?

Twelve subjects were each supplied with two analgesic tablets for chronic headache. One tablet was Aspirin (Tablet A) and the other was a new analgesic (Tablet N). The subjects were instructed to take the two tablets 8 hours apart, and to record if a tablet provided relief from headache. The results are tabulated below:

Since 9 of the 12 subjects responded to Tablet N but not to Tablet A, while only one subject responded to Tablet A and not to Tablet N, it would appear that Tablet N is a more effective analgesic. These numbers are small, however, and could also have arisen by chance.

McNemar's test can determine if the observed difference in the two analgesics represents a real difference. Using StatXact we note that the exact p-value for McNemar's test is 0.0215, statistically significant at the 5% level.

The above statistical analysis did not take into consideration the order in which the two tablets were administered to the patient. Yet tablet order could influence the patient's response, since the tablet taken second might interact in some way with the first tablet, taken eight hours earlier. The data on tablet order and patient response are displayed below. A "+" after the treatment symbol represents relief from headache while a "-" represents no relief:

The problem now is to compute the p-value for the treatment effect while simultaneously adjusting for the order in which the two treatments are administered. This problem can be solved by fitting a conditional logistic regression model to the data, with "type of tablet" and "order of tablet" as the two independent variables. (The conditional approach is preferred to the unconditional approach since it takes care of the patient specific variability.) LogXact is the only commercial software package offering conditional logistic regression with exact inference for small samples. It reveals that the exact p-value for "type of tablet" is 0.0476. Thus, even after we adjust for "order of tablet", the new tablet is seen to be statistically more effective than Aspirin at the 5% significance level. In this example the usual large-sample method of inference provided by conventional logistic regression packages failed to converge, thus resulting in "?" symbols instead of numbers on the LogXact output screen shown below:

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