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

[Viechtbauer, Wolfgang (NP)]

Dear Mick,

So for some studies you have means and SDs for each side and can compute the (log-transformed) ratio of means (RoM) and corresponding sampling variances with:

yi = log(m1/m2i)
vi = sd1i^2/(ni*m1i^2) + sd2i^2/(ni*m2i^2) - 2*ri*sd1i*sd2i/(m1i*m2i*ni)

where ni refers to the number of individuals. Note that you have paired data, so the equation for computing the sampling variance must incorporate the correlation between the measurements for the two sides (ri). This is measure="ROMC" in escalc(). If you do not know this correlation (because presumably it is not reported), then you could try to make an educated guestimate.

But for other studies, you have the mean of the ratios (MoR), mean(x1i/x2i), and I assume the corresponding standard deviation of the ratios (SDoR), sd(x1i/x2i)).

To what extent MoR approximates RoM depends on how close the true means are to 0. To demonstrate:

x1 <- rnorm(1000000, 10, 1)
x2 <- rnorm(1000000,  5, 1)

mean(x1/x2)
mean(x1)/mean(x2)

x1 <- rnorm(1000000, 100, 1)
x2 <- rnorm(1000000,  50, 1)

mean(x1/x2)
mean(x1)/mean(x2)

Assuming you are in a situation where the means are far enough away from 0 so that this doesn't matter, then how can we incorporate MoR studies into the meta-analysis? It would be:

yi = log(mean(x1i/x2i))
vi = sd(x1i/x2i))^2 / (ni * mean(x1i/x2i)^2)

or using the abbreviations above

yi = log(MoR)
vi = SDoR^2 / (ni * MoR^2)

where ni is again the sample size. The vi equation comes from a simple application of the delta method (https://en.wikipedia.org/wiki/Delta_method).

I would recommend to code for each yi value whether it is a (log transformed) RoM or MoR value and examine via a moderator analysis whether there is a systematic difference between the two sets.

P.S.: Thanks for pointing out that this was also posted on Cross Validated.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>Behalf Of Mick Girdwood
>Sent: Tuesday, 04 October, 2022 3:01
>To: r-sig-meta-analysis using r-project.org
>Subject: [R-meta] Calculation of variance of ratio of means from summary
>statistics
>
>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