# Correct interpretation of Tau and Tau^2 in meta-analysis of different measures

Simple question. Given that Tau is defined as the estimated standard deviation of underlying true effects across studies (and, accordingly, Tau^2 as the variance), how can I interpret this data in different meta-analysis (i.e., meta-analysis of odds ratio and meta-analysis of mean differences)?

E.g.: if I perform a meta-analysis of mean differences, obtaining a pooled MD of 1.00 [0.50-2.00], is the tau interpretable as the standard deviation of the pooled MD?

if I perform a meta-analysis of OR, obtaining a pooled estimate of 1.80 [1.40-2.00], how can I interpret the tau?

In a random-effects model, $$\tau$$ is the standard deviation of the underlying true outcomes in the 'population of studies', that is, we assume that there is an essentially infinite collection of studies that could have been conducted and in this population of studies, the true outcomes are normally distributed with a certain expected value, $$\mu$$, and a certain standard deviation, $$\tau$$. The studies we actually have and that are included in the meta-analysis are assumed to be a random sample from this hypothetical population of studies from which we obtain estimates $$\hat{\mu}$$ and $$\hat{\tau}$$. The latter is the estimated standard deviation in the true outcomes. What 'outcomes' refers to depends on the outcome measure used for the meta-analysis. Hence, if you conduct a meta-analysis with raw mean differences, then $$\hat{\tau}$$ is the estimated standard deviation of the true mean differences. If you conduct a meta-analysis with log odds ratios, then $$\hat{\tau}$$ is the estimated standard deviation of the true log odds ratios (note: when meta-analyzing odds ratios, this is done on the log scale).
It is not correct to say that $$\hat{\tau}$$ is the estimated 'standard deviation of the pooled outcome'. The pooled outcome is $$\hat{\mu}$$. It also has a standard deviation, or more commonly referred to as a standard error (i.e., $$\mbox{SE}[\hat{\mu}]$$). This reflects the uncertainty in our estimate of $$\mu$$. This is something different than $$\hat{\tau}$$ (which, as noted above, is the estimated standard deviation of the distribution of true outcomes).