Proportions: Correlated and Paired
Calculate the number of subjects required for Cochran's Q test when the same subjects are evaluated under three or more related conditions with a binary outcome. Enter the expected success proportion for each condition, alpha, and desired power. The calculation uses an asymptotic noncentral-χ2 planning approximation.
Cochran's Q test compares the success proportions of k related binary measurements made on the same subjects. It is the repeated-measures extension of McNemar's test and tests whether the marginal success proportions are equal across the conditions.
For binary responses, let \(Y_{ij}\) denote the response from subject \(i\) in condition \(j\), with success coded as 1 and failure as 0. If \(G_j=\sum_iY_{ij}\) is the number of successes in condition \(j\), \(T=\sum_jG_j\), and \(L_i=\sum_jY_{ij}\) is the number of successes contributed by subject \(i\), Cochran's Q statistic is
For sufficiently large effective samples, Q is approximately distributed as a chi-square random variable with \(k-1\) degrees of freedom. NCSS's Cochran's Q documentation gives the same statistic and large-sample chi-square approximation. The documented rule of thumb is that at least four subjects should have non-identical responses across conditions and that \(nk\) should be at least 24.
For planning, this calculator specifies the expected marginal success probabilities \(p_1,\ldots,p_k\). Under the transparent working approximation used here, the expected Cochran Q statistic is obtained from the corresponding expected values of the numerator and denominator of the Q statistic. The resulting expected Q is treated as the noncentrality parameter of the asymptotic chi-square distribution.
The calculator searches upward over integer values of N and reports the smallest number of paired subjects for which the calculated power reaches the requested target.
Because marginal proportions alone do not completely specify the joint distribution of repeated binary outcomes, this implementation uses a working independence assumption when calculating the planning noncentrality parameter. The subjects are still treated as matched repeated observations for the Cochran Q statistic itself. If substantial within-subject dependence beyond this working assumption is expected, simulation based on a specified joint binary distribution is preferable.
Cochran, W. G. (1950). The comparison of percentages in matched samples. Biometrika, 37(3–4), 256–266.
Ramsey, P. P., & Ramsey, P. H. (1981). Minimum sample sizes for Cochran's test. Proceedings of the American Statistical Association, Survey Research Methods Section.
NCSS, LLC. Cochran's Q Test, Chapter 521. NCSS Statistical Software documentation. The documentation describes Cochran's Q for matched binary responses, its test statistic, its chi-square approximation, and practical large-sample requirements.