# Does a random effects meta analysis produce the mean of population effect sizes?

In a random effects meta analysis of correlation coefficients, is the outcome the mean of a distribution of population correlation coefficients? If so, is this denoted rho symbol (p) with a hat on? I have seen studies refer to it as just an "estimate of the population correlation coefficient," so does this assume the reader will know it refers to an estimate of an average and not a single population effect size? Given the random effects model assumes there is not just one population effect size to find but many different ones?

• population "correlation coefficient," can be either an estimate or an observed directly computed quantity. For an estimation, several independent samples may be used. Alternanatively, repeat sampling from a specified population may be conducted.
– user10619
May 5, 2018 at 3:29

So, to be precise, let $r_1, r_2, \ldots, r_k$ denote the observed correlation coefficients for the $k$ studies included in our meta-analysis. In each of these studies, there is an underlying true correlation which we can denote with $\rho_1, \rho_2, \ldots, \rho_k$. Imagine we would conduct each study with a very very large sample size; then the observed correlations would be in essence these underlying true correlations.
These $k$ true correlations are assumed to have come from a population of true correlations. One way to imagine this is to think of a population of studies, consisting of studies that were actually conducted, but also containing studies that could have been conducted hypothetically (yes, this part gets a bit abstract).
We typically assume that the true correlations in this population are normally distributed (or at least, approximately so) and hence we write $\rho_i \sim N(\mu_\rho, \tau_\rho^2)$. The random-effects model provides us an estimate of $\mu_\rho$ and $\tau_\rho^2$, which we can denote $\hat{\mu}_\rho$ and $\hat{\tau}_\rho^2$.
Some may use other notation for $\hat{\mu}_\rho$, such as $\hat{\bar{\rho}}$ (so a hat for denoting that this is an estimate and the bar on top of $\rho$ to denote that this is an average). I do not like this notation, because it does not draw a clear distinction between the average in the population of studies (i.e., $\mu_\rho$) and the average of the true correlations of the $k$ studies included in the meta-analysis (i.e., the average of $\rho_1, \rho_2, \ldots, \rho_k$). The random-effects model does not estimate the latter; it estimates the average in the population of studies.
Even worse would be just writing $\hat{\rho}$ for $\hat{\mu}_\rho$. That would be outright incorrect notation as far as I am concerned, because it gives the impression that we are estimating one correlation coefficient.