I have two measures from each of 20 studies. Random-effects meta analysis has been performed for each of the two measures, and for each measure I have two values to indicate data variability, cross-study (random effect) variance $\tau^2$ and its chi-square test statistic $Q$. Now I'd like to compare the cross-study variability between the two measures. $\tau^2$ does not seem to be a good choice because it is dimensional (having different units) and carries the same physical meaning as its corresponding measure. Since the chi-square value $Q$ is dimensionless, I was wondering whether I could construct the ratio of the two $Q$ values as a pseudo $F$-statistic (with the numerator and denominator degrees of freedom being the same) to test for equal variability across subjects between the two measures. Or any better approach?
I guess you could calculate the ratio of the two $Q$ values as a way of quantifying the comparative variability of the two measures, but it won't have an $F$-distribution as there's no reason to think the numerator and denominator $Q$ values have chi-squared distributions or that they are independent. The only alternative I can think of is bootstrapping the studies and looking at the bootstrap distribution of this ratio, though 20 studies is perhaps a rather small sample size for bootstrapping.