[R-meta] Multivariate multi-level meta-analysis: adjusting control variables when modeling publication bias?

[Daniel Foster]

Hello Wolfgang and all,

I am carrying out multivariate multilevel meta-analysis using the
rma.mv function in the metafor package, and I have come across an
issue that has been giving me a lot of trouble. At this point I am at
a stand still and any insight would be greatly appreciated!!

When testing for publication bias using the PET approach, Doucouliagos
& Stanley (2009), suggest using the following model in a weighted
least squares formula:

ES= B1(1/SE) + SIGMA ak(Zj/SEi )+ e

Where SE is the standard error of the effect estimate,  Z is a vector
of meta-independent variables reflecting differences across studies
for the jth study in literature, ak is the meta-regression coefficient
which reflects the effect of particular study characteristics.

My confusion lies in the fact that they are suggesting that the
control variables (Z) included need to be divided by the standard
error of the effect estimate. My questions are this:

Should I be dividing my control variables by the standard error of
the effect estimate when using the rma.mv function?  I have found some
multivariate multilevel meta-analyses that follow this method
(Klomp, 2009), but then others that don't (at least explicitly;
Akgunduz, 2018)

If I do need to do this, it is not clear to me how a binary control
variable can be incorporated in my mra.mv model (i.e., 1, 0). To my
mind, it seems strange to divide a dichotomous variable by a
continuous variable. What steps do I need to execute to include these
variables in my mra.mv model?

Thank you so much in advance for your insights!

Daniel

Akgunduz, Y. E., & Plantenga, J. (2018). doi: 10.1111/joes.12192
Doucouliagos, H., & Stanley, T. D. (2009). doi: 10.1111/j.1467-8543.2009.00723.x
Klomp, J., & De Haan, J. (2010). doi: 10.1111/j.1467-6419.2009.00597.x