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

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.