[R-meta] multilevel models and bias assessment

Dear Catia,

one could conduct a modified Egger's regression test by accounting for 
the dependency (e.g. StudyID/EffectID and respective variance-covariance 
matrix) and using a precision estimate as moderator variable, e.g.

model <- rma.mv(Effects, Vmatrix, mods = ~ Precision,random = ~ 1 | 
StudyID/EffectID, data = data)

Please note that the choice of the precision estimate depends on the 
effect size (e.g., r, d): "A complication with Egger?s regression is 
that for certain effect size metrics, the standard error is naturally
correlated with the effect size estimate even in the ab-
sence of selective reporting or other sources of asym-
metry. Different variants of Egger?s regression have
been developed to reduce the correlation by using alter-
native measures of precision, specifically for log odds
ratios (Macaskill, Walter, & Irwig, 2001; Moreno et
al., 2009; Peters et al., 2006), raw proportions (Hunter
et al., 2014), hazard ratios (Debray, Moons, & Riley,
2018), and standardized mean differences (Pustejovsky
& Rodgers, 2019)."

If you are using correlation coefficients then see this reply from Wolfgang:
https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2020-May/002086.html

For recommendations about other effect sizes you can use the references 
in the cited article:
Rodgers, M. A., & Pustejovsky, J. E. (2021). Evaluating meta-analytic 
methods to detect selective reporting in the presence of dependent 
effect sizes. Psychological Methods, 26(2), 141?160. 
https://doi.org/10.1037/met0000300

Best wishes,
Lukasz