Note on Cochran's Q-Test for the Comparison of Correlated Proportions

Cochran's Q-test, proposed to test the equality of three proportions for correlated observations, may have a larger asymptotic significance level than the nominal one, unless the admissible family of distributions is restricted to ensure that under the hypothesis the correlations between the observations are also equal. For a large class of such restricted families, optimal C 2(α) tests of the hypothesis are obtained, and the family for which the Q-test is an optimal C 2(α) test is identified. The use of Q for testing a related hypothesis, suggested by Madansky, is also discussed: it is shown that the Q-test is not consistent against all alternatives to Madansky's hypothesis, while the classical χ2-test provides an optimal C 4(α) test for it.