Use of ratio of means effect measure and estimation of variance from mean of ratios I am running a meta-analysis where we are using the ratio of means (RoM) effect measure as the data we are using compare the two sides of the body (i.e. left and right). All RoM calculations are completed by escalc function in metafor r package, based on Friedrich 2008.
Most studies we are concerned with will report a mean and sd for each side of the body. We can then use these for the RoM calculations. In some cases authors will report only a 'limb symmetry index ' - e.g. for each participant the $LSI = \frac{left}{right}$. These are then summarised and reported for the group as a $mean_{LSI}$ and $SD_{LSI}$.
My issue is about how to incorporate the limb symmetry index (LSI) data into the meta-analysis. I am aware that the RoM is in effect $\frac{mean_{left}}{mean_{right}}$, where as the LSI is
$ mean (\frac{left}{right})$
The LSI is a good approximation of the RoM (have checked this with data where studies report both). My issue is around how to determine a precision parameter i.e. how can I estimate the variance from the LSI standard deviation provided. I am struggling with this especially as RoM meta-analysis is performed with log transformation, so am not sure how to use the arithmetic SD in this situation. I have done an extensive amount of reading and haven't been able to find anything of any help. This paper I thought would be useful, but after using their recommended formula for $σ^2$, looking at the confidence intervals comparing those calculated from ROM method to the ones calculated from $σ^2$ are very different.
The best solution I've found was by accident - I was looking at $\frac{SD^2}{n}$ (sample variance) and accidentally forgot to square the SD in my code, so in effect my formula became: $\frac{SD}{n}$ - this was close to perfect when looking at the length of CIs compared to those calculated from RoM effects. However I am not sure why and obviously would need justification to pursue with this method for a paper.
My questions are

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*What is the best approach for incorporating the LSI data into the RoM analyses?

*In the case of my accidental approach $\frac{SD}{n}$ - why does this work well? Is there any justification I could call on to use this? Or is it just a major co-incidence that it provides a good estimation? I have a dataset of n=40 on which I tested this on and all were very reasonable approximations.


I can answer my own question thanks to the great clear help from Wolfgang Viechtbauer over on the r meta-analysis forum where I also asked this question.
Answer in-depth here: https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2022-October/004236.html
The issue came down to two things, transposed from Wolfgang's answer above:

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*My data is paired and so the RoM needs to account for the correlated nature of the data. Use of ROMC as the effect measure solves this, and requires a correlation parameter ri. Detailed here - Lajeunesse 2011 as referenced on escalc page


*Because the effect calculation now accounts for the correlation, the precision is much greater, i.e. the confidence intervals are much tighter. This now means my issues with variance calculation being too precise are no longer an issue. Wolfgang provided the answer there too, with one I had also tried but discarded thinking it wasn't right. Turns out it was:
effect = yi = log(Mean of Ratios) = $log(mean_{LSI})$
variance = vi = $\frac{SD_{LSI}}{(n \times mean_{LSI}^2)}$
Thanks to Wolfgang!
