# Testing for publication bias in meta-analysis when effect is raw mean

I am conducting a multi-level meta-analysis using rma.mv in the metafor package in R. The effect size measure I am working with is the raw mean (measure="MN" in metafor). The sampling variance (vi) for each datum is calculated as sd^2/n.

I would like to test for publication bias using something akin to Egger's linear regression method, but I am not sure which predictor variable to use, given my effect size measure. I am aware of several predictor variable options (e.g., n, 1/vi, sqrt(vi), etc.). However, I'd like my choice of predictor to be appropriate for my ES measure and defensible (to reviewers).

The "standard" Egger regression test uses $\sqrt{v_i} = SE_i$ (i.e., the standard error of the outcomes) as the predictor. This is sometimes not advisable, especially when the standard error is a function of the outcome measure itself. For example, suppose you are not meta-analyzing means, but raw (Pearson product-moment) correlation coefficients. The large-sample variance of a correlation coefficient $r_i$ can be estimated with $v_i = (1-r_i^2)^2 / (n_i-1)$. Note that $v_i$ (and hence the SE) is a function of $r_i$, so by construction, there is a relationship between the outcomes and the SEs. In that case, it may be better to use $n_i$ or some other function of $n_i$ (e.g., $1/\sqrt{n_i}$) as the predictor.
For means, there is no inherent relationship between $v_i = SD^2/n_i$ and $\bar{x}$, so this issue does not apply. Therefore, I think it would be fine to stick to the default approach and use $\sqrt{v_i} = SE_i$.
• Actually, Macaskill et al. (2001) suggested using $n_i$, while Peters et al. (2006) suggested $1/n_i$. – Wolfgang May 15 '16 at 10:48