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I'm currently conducting a meta-regression for my moderators from a meta-analysis my team recently conducted. In calculating the fixed effects meta-regressions, my objective is to identify how much heterogeneity is accounted for when the moderators are considered in the model. To do this, I used the rma() function from the metafor package in R:

Reg1 <- rma(yi = LogOdds, sei = SE, data = data, method = 'FE', mods = ~ mod1 + mod2)

The output provides out $I^2$ and $H^2$ values as well as the significance of our moderators, however, it does not provide an $R^2$ value for the amount of heterogeneity accounted for.

I did notice, however, that running an random effects meta regression dos provide the $R^2$.

My questions are: Does a meta-regression have to be random effects in order to garner the amount of heterogeneity accounted for? If I ran a fixed effects meta-analysis, must I also run a fixed effects meta-regression? And if not, how can I estimate the amount of heterogeneity accounted for in a fixed effects meta-regression?

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The standard definition of the 'pseudo' $R^2$ statistic in meta-analysis is based on the proportional reduction in the estimate of $\tau^2$ (the amount of (residual) heterogeneity), that is: $$R^2 = \frac{\hat{\tau}_0^2 - \hat{\tau}_1^2}{\hat{\tau}_0^2},$$ where $\hat{\tau}_0^2$ is the estimate from a standard random-effects model (and hence this estimates the total amount of heterogeneity) and $\hat{\tau}_1^2$ is the estimate from the mixed-effects meta-regression model (and hence this estimates the residual amount of heterogeneity, that is, the amount not accounted for by the moderators included in the model).

For fixed-effects models, we do not estimate $\tau^2$ and hence this definition is no longer applicable. However, one could make the argument that an analogous version in this context is based on the standard definition of $R^2$ in regression models (although using weighted least squares estimation since we typically use inverse-variance weights in meta-analysis). I would suggest to compute the adjusted $R^2$ though, since $k$ is typically small in meta-analyses, so the adjustment is often relevant then.

Since you are using rma() from the metafor package: I have decided to implement this in the development version, so if you follow these instructions, you will now get the (adjusted) $R^2$ also for fixed-effects models.

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