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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 dataare on different scales or come from different instruments, the customarymeasure of effect is the standardized mean difference (SMD). The SMD is definedas the difference between the means in the treatment and control arms,standardized by the assumed common standard deviation, $\sigma$. However, ifthe variances in the two arms differ, there is no consensus on a definition ofSMD. Thus, we propose a new effect measure, the difference of standardizedmeans (DSM), defined as $\Delta = \mu_T/\sigma_T - \mu_C/\sigma_C$. Theestimated DSM can easily be used as an effect measure in standardmeta-analysis.  For random-effects meta-analysis of DSM, we introduce new point and intervalestimators 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-knownestimators based on the $Q$ statistic with inverse-variance weights, $Q_{IV}$,such as the Mandel-Paule, DerSimonian-Laird, and restricted-maximum-likelihoodestimators.

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