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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.

Author information

1
School of Computing Sciences, University of East Anglia, Norwich, NR4 7TJ, UK. e.kulinskaya@uea.ac.uk.
2
Department of Mathematics, Pacific Lutheran University, Tacoma, WA, USA.
3
Faculty of Engineering and Science, University of Agder, Kristiansand, Norway.

Abstract

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.

KEYWORDS:

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

PMID:
26061889
DOI:
10.1002/jrsm.54

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