UMMS Affiliation

Department of Population and Quantitative Health Sciences

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Document Type



Statistics and Probability


The conventional Q statistic, using estimated inverse-variance (IV) weights, underlies a variety of problems in random-effects meta-analysis. In previous work on standardized mean difference and log-odds-ratio, we found superior performance with an estimator of the overall effect whose weights use only group-level sample sizes. The Q statistic with those weights has the form proposed by DerSimonian and Kacker. The distribution of this Q and the Q with IV weights must generally be approximated. We investigate approximations for those distributions, as a basis for testing and estimating the between-study variance (tau(2) ). A simulation study, with mean difference as the effect measure, provides a framework for assessing accuracy of the approximations, level and power of the tests, and bias in estimating tau(2) . Two examples illustrate estimation of tau(2) and the overall mean difference. Use of Q with sample-size-based weights and its exact distribution (available for mean difference and evaluated by Farebrother's algorithm) provides precise levels even for very small and unbalanced sample sizes. The corresponding estimator of tau(2) is almost unbiased for 10 or more small studies. This performance compares favorably with the extremely liberal behavior of the standard tests of heterogeneity and the largely biased estimators based on inverse-variance weights.


effective-sample-size weights, exact distribution, inverse-variance weights, mean difference, random effects

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© 2021 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



Kulinskaya E, Hoaglin DC, Bakbergenuly I, Newman J. A Q statistic with constant weights for assessing heterogeneity in meta-analysis. Res Synth Methods. 2021 Nov;12(6):711-730. doi: 10.1002/jrsm.1491. Epub 2021 Jul 6. PMID: 33969638. Link to article on publisher's site

Journal/Book/Conference Title

Research synthesis methods

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