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Res Synth Methods. 2011 Dec;2(4):254-70. doi: 10.1002/jrsm.54.

On the moments of Cochran's Q statistic under the null hypothesis, with application to the meta-analysis of risk difference.

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School of Computing Sciences, University of East Anglia, Norwich, NR4 7TJ, UK.
Department of Mathematics, Pacific Lutheran University, Tacoma, WA, USA.
Faculty of Engineering and Science, University of Agder, Kristiansand, Norway.


W. G. Cochran's Q statistic was introduced in 1937 to test for equality of means under heteroscedasticity. Today, the use of Q is widespread in tests for homogeneity of effects in meta-analysis, but often these effects (such as risk differences and odds ratios) are not normally distributed. It is common to assume that Q follows a chi-square distribution, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. In this paper, the effect and weight for an individual study may depend on two parameters: the effect and a nuisance parameter. We present expansions for the first two moments of Q without any normality assumptions. Our expansions will have wide applicability in testing for homogeneity in meta-analysis. As an important example, we present a homogeneity test when the effects are the differences of risks between treatment and control arms of the several studies-a test which is substantially more accurate than that currently used. In this situation, we approximate the distribution of Q with a gamma distribution. We provide the results of simulations to verify the accuracy of our proposal and an example of a meta-analysis of medical data.


gamma distribution; heterogeneity test; nuisance parameter; weighted analysis of variance; weighted sum of squares


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