On the Q statistic with constant weights for standardized mean difference

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, τ 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 τ 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 τ 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 whenτ 2 = 0 , but otherwise the Q ‐profile interval performs very well.