An exact test for comparing matched proportions in crossover designs

Abstract

SUMMARY An exact test for comparing proportions in matched samples is derived for the situation where order within pairs may be important. The analysis is based on an extension of the logistic model of Cox (1958 c). The test is formally equivalent to Fisher's exact test for 2 x 2 tables. The McNemar test (1947) has long been used in comparing matched or paired responses each of which take only two values, 0 or 1. The test consists of counting only those pairs which are unlike, (0, 1) or (1, 0) and testing whether the distribution of these two types is consistent with a binomial distribution with a probability of 2. Cox (1958c) showed that this test may be derived from a logistic model, and found a confidence interval for the relevant parameter of non-centrality. Cochran (1950) extended the test to more than two proportions. In many experimental situations, matching entails an ordering within the pairs. For instance, in testing certain drugs the matching involves a patient receiving the two drugs in two different time periods. Sometimes the order of the drug administration has appreciable effect on the patient's response; see, for instance, Meier, Free & Jackson (1958). In such instances, the McNemar test may ignore pertinent and important information on order within pairs. To fix ideas, we recall an example cited by Mosteller (1952). Two drugs, A and B, are used on each of 100 subjects, the response being either 'nausea' or 'not nausea'. Of these subjects, 81 never had nausea, denoted by (0, 0); 9 subjects had nausea with A but not with B, (1, 0); 1 had nausea with B but not with A, (0, 1); and 9 had nausea with both drugs, (1, 1). Mosteller considered only the unlike pairs, and calculated from the binomial distribution the onetailed significance level,