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[R-meta] Egger-type test for multi-level meta-analysis

Just me thinking out loud here for the moment:

While the extension of the 'Egger regression test' (and the PET/PEESE methods) to multilevel models is straightforward, a bit of thinking is required as to what we are really trying to capture by adding something like sqrt(vi) (or just vi or any other transformation thereof) to the model as a predictor. Selective reporting within studies of the larger / significant effects? Or selective reporting of studies in general? In the latter case, selection may depends on whether at least one effect is significant / large, the focal one (in case there is a defined primary endpoint), or something else. One could argue that cor(sqrt(vi), yi) (in essence what we are examining with the regression test) might capture a bit of all of this and maybe that's the best we can generally do.

Best,
Wolfgang