[R-meta] Calculation of variance of ratio of means from summary statistics

[Mick Girdwood]

Thank you for this message board, it’s always so helpful, and sorry for the slightly long question. I am planning a meta-analysis where we are using the ratio of means (RoM) effect measure for comparing outcomes from two sides of the body (i.e. the affected side and unaffected side). E.g. we are looking at a particular muscle strength outcome, this is reported for each side. 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 = the affected / non-affected side. These are then summarised and reported for the group as the mean(LSI). In these cases we cannot compute the RoM effect measure, as we don’t have the summaries for each side. My issue is about how to incorporate the limb symmetry index (LSI) data into the meta-analysis.
I am aware that the RoM is essentially mean[affected] / mean[non-affected], which is slightly different to LSI - mean(affected/nonaffected). The LSI is a good almost perfect approximation of the RoM (I have checked this with data where studies reported both). So this is easy to just use log(LSI) to get an estimate for yi.
My issue is around how to determine a precision parameter (vi sei etc) 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 a large number of studies where only the LSI mean and SD are provide (and so cannot calculate the RoM from the data for each side). I have done an extensive amount of reading and haven't been able to find anything that works well.
I first tried converting the SD to confidence intervals, then log transforming these to use, however they overestimate the precision by about a factor of ~2, a problem with using inverse variance methods. I also tried methods from this paper by Quan 2003 (equation 5) https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.1525 however this majorly underestimates the precision.
The best solution I've found was by accident - I was looking at using the formula for sample variance SD^2 / (n-1) and accidentally forgot to square the SD in my code, so in effect my formula became sd / (n-1). This was very good/comparable when comparing to the CIs generated by RoM calculations. However this is an obvious accident so I don’t have any justification to proceed with this
My questions are How can I use the LSI data and convert these (specifically the variance) for use in analyses In the case of my accidental approach SD/(n-1) - 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. Thank you for your help!
Mick Girdwood
La Trobe University | Australia

n.b. I have posted a similar question here - https://stats.stackexchange.com/questions/590611/use-of-ratio-of-means-effect-measure-and-estimation-of-variance-from-mean-of-rat



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