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, 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, QIV) rather complicated. As an alternative, we investigate a new Q statistic, QF, 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 QIV and QF, 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 QF, 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 QF reasonably well, whereas the usual chi-squared approximation for the null distribution of QIV and the Biggerstaff-Jackson approximation to its alternative distribution are poor; in estimating tau 2, our moment estimator based on QF 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.