[R-meta] Effect size and variance from synthetic control studies

Dario Schulz d@r|o@@chu|z @end|ng |rom ||r@un|-bonn@de
Fri Jan 22 17:01:43 CET 2021

Hello there,

I have two questions, the context is explained below.

First, I'd like to know whether Borenstein's 2009 formula for the variance of standard mean differences is appropriate if there is just one single treatment observation. Related to that, is Hedges small sample correction also applicable in such circumstances?

And second, how should the pooled standard deviation be calculated if the controls have unequal weights?

A number of primary studies in my meta-analysis used the Synthetic Control method (see for example  <https://doi.org/10.1073/pnas.2004334117> https://doi.org/10.1073/pnas.2004334117). This method is used when few (e.g. just one) treatment units, but many potential controls are available. The basic idea is to compare the observed treatment outcome with a synthetic control, which is a weighted combination of several units from a "donor pool". Should I therefore calculate the SD in the control group using a weighting method such as Hmisc::wtd.var() in R?

In my context, there are typically multiple reported differences, i.e. one per year, so an approach that I thought of would be to calculate the treatment SD based on all observations (one for each year). A 2020 working paper by Hollingsworth and Coady (doi.org/10.31235/osf.io/fc9xt) calculates a type of Cohens d by using the SD in pre-treatment periods. But if there is a temporal trend in both treated an control units that has nothing to do with the treatment, this would, if I understand it correctly, inflate the pooled SD and deflate the effect estimate. I therefore consider an approach that uses only the observations from one given year more useful. These can either be aggregated to an overall mean, or analyzed individually as dependent effect sizes.

Looking forward to hearing your feedback and ideas!

Kind regards

Dario Schulz


Doctoral Student

PhenoRob (Cluster of Excellence)

Institute for Food and Resource Economics (ILR)

University of Bonn

Nu�allee 21, D-53115 Bonn

Email: dario.schulz using ilr.uni-bonn.de <mailto:dario.schulz using ilr.uni-bonn.de>

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