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A variety of problems in random-effects meta-analysis arise from theconventional $Q$ statistic, which uses estimated inverse-variance (IV) weights.In previous work on standardized mean difference and log-odds-ratio, we foundsuperior performance with an estimator of the overall effect whose weights useonly group-level sample sizes. The $Q$ statistic with those weights has theform proposed by DerSimonian and Kacker. The distribution of this $Q$ and the$Q$ with IV weights must generally be approximated. We investigateapproximations for those distributions, as a basis for testing and estimatingthe between-study variance ($\tau^2$). Some approximations require the varianceand third moment of $Q$, which we derive. We describe the design and results ofa simulation study, with mean difference as the effect measure, which providesa framework for assessing accuracy of the approximations, level and power ofthe tests, and bias in estimating $\tau^2$. Use of $Q$ with sample-size-basedweights and its exact distribution (available for mean difference and evaluatedby Farebrother's algorithm) provides precise levels even for very small andunbalanced sample sizes. The corresponding estimator of $\tau^2$ is almostunbiased for 10 or more small studies. Under these circumstances thisperformance compares favorably with the extremely liberal behavior of thestandard tests of heterogeneity and the largely biased estimators based oninverse-variance weights.

Simulations for a Q statistic with constant weights to assess heterogeneity in meta-analysis of mean difference