UMMS Affiliation

Department of Population and Quantitative Health Sciences

Publication Date


Document Type



Epidemiology | Statistics and Probability


Cochran's Q statistic is routinely used for testing heterogeneity in meta-analysis. Its expected value is also used in several popular estimators of the between-study variance, tau 2 . Those applications generally have not considered the implications of its use of estimated variances in the inverse-variance weights. Importantly, those weights make approximating the distribution of Q (more explicitly, Q IV ) rather complicated. As an alternative, we investigate a new Q statistic, Q F , whose constant weights use only the studies' effective sample sizes. For the standardized mean difference as the measure of effect, we study, by simulation, approximations to distributions of Q IV and Q F , as the basis for tests of heterogeneity and for new point and interval estimators of tau 2 . These include new DerSimonian-Kacker-type moment estimators based on the first moment of Q F , and novel median-unbiased estimators. The results show that: an approximation based on an algorithm of Farebrother follows both the null and the alternative distributions of Q F reasonably well, whereas the usual chi-squared approximation for the null distribution of Q IV and the Biggerstaff-Jackson approximation to its alternative distribution are poor; in estimating tau 2 , our moment estimator based on Q F is almost unbiased, the Mandel - Paule estimator has some negative bias in some situations, and the DerSimonian-Laird and restricted maximum likelihood estimators have considerable negative bias; and all 95% interval estimators have coverage that is too high when tau 2 = 0 , but otherwise the Q-profile interval performs very well.


effective sample sizes, heterogeneity, inverse-variance weights, random effects

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© 2022 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI of Published Version



Bakbergenuly I, Hoaglin DC, Kulinskaya E. On the Q statistic with constant weights for standardized mean difference. Br J Math Stat Psychol. 2022 Jan 30. doi: 10.1111/bmsp.12263. Epub ahead of print. PMID: 35094381. Link to article on publisher's site

Journal/Book/Conference Title

The British journal of mathematical and statistical psychology

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Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.