Simulations for the Q statistic with constant and inverse variance weights for binary effect measures

Cochran's Q statistic is routinely used for testing heterogeneity in meta-analysis. Its expected value (under an incorrect null distribution) is part of several popular estimators of the between-study variance, $\tau^2$. Those applications generally do not account for estimation of the 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 are investigating a $Q$ statistic, $Q_F$, whose constant weights use only the studies' effective sample sizes. For log-odds-ratio, log-relative-risk, and risk 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. Results of our simulations are provided and in 114 A4 Figures, 133 pages in total.