  Cochran's $Q$ statistic is routinely used for testing heterogeneity in
meta-analysis. Its expected value is also used for estimation of between-study
variance $\tau^2$. Cochran's $Q$, or $Q_{IV}$, uses estimated inverse-variance
weights which makes approximating its distribution rather complicated. As an
alternative, we are investigating a new $Q$ statistic, $Q_F$, whose constant
weights use only the studies' effective sample sizes. For 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 the between-study variance
$\tau^2$. These include new DerSimonian-Kacker (2007)-type moment estimators
based on the first moment of $Q_F$, and novel median-unbiased estimators of
$\tau^2$.
