Simulations for estimation of heterogeneity variance and overall effect with constant and inverse-variance weights in meta-analysis of difference in standardized means (DSM)

When the individual studies assembled for a meta-analysis report means ($\mu_C$, $\mu_T$) for their treatment (T) and control (C) arms, but those data are on different scales or come from different instruments, the customary measure of effect is the standardized mean difference (SMD). The SMD is defined as the difference between the means in the treatment and control arms, standardized by the assumed common standard deviation, $\sigma$. However, if the variances in the two arms differ, there is no consensus on a definition of SMD. Thus, we propose a new effect measure, the difference of standardized means (DSM), defined as $\Delta = \mu_T/\sigma_T - \mu_C/\sigma_C$. The estimated DSM can easily be used as an effect measure in standard meta-analysis. For random-effects meta-analysis of DSM, we introduce new point and interval estimators of the between-studies variance ($\tau^2$) based on the $Q$ statistic with effective-sample-size weights, $Q_F$. We study, by simulation, bias and coverage of these new estimators of $\tau^2$ and related estimators of $\Delta$. For comparison, we also study bias and coverage of well-known estimators based on the $Q$ statistic with inverse-variance weights, $Q_{IV}$, such as the Mandel-Paule, DerSimonian-Laird, and restricted-maximum-likelihood estimators.