My colleagues and I were wondering if is OK to add a constant to the means in the log transformed ratio of means (known also as log response ratio).
The log response ratio and its variance are:
\begin{align} \newcommand{\var}{{\rm Var}} \ln R &= \ln(\bar{X}_{1})- \ln(\bar{X}_{2}) \\[8pt] \var(\ln R) &= \frac{(SD_{1})^2}{n_{1}\bar{X}_{1}^2}+\frac{(SD_{2})^2}{n_{2}\bar{X}_{2}^2} \end{align}
Hedges, L. V., Gurevitch, J., & Curtis, P. S. (1999). The meta-analysis of response ratios in experimental ecology. Ecology, 80(4), 1150-1156.
Adding a constant to the means I presume would results in the following formulae:
\begin{align} \ln Rk &= \ln(\bar{X}_{1}+k)- \ln(\bar{X}_{2}+k) \\[8pt] \var(\ln Rk) &= \frac{(SD_{1})^2}{n_{1}(\bar{X}_{1}+k)^2}+\frac{(SD_{2})^2}{n_{2}(\bar{X}_{2}+k)^2} \end{align}
It is also unclear if this constant should be added to all cases or only to cases with zero means.
So far we didn’t come yet across literature suggesting this practice. Also when trying to compute the ES and its variance using metaphor::escalc()
function, even though there is an option to add a constant, there is no change taking place. Below is an example in R, but I couldn't find a way to make metaphor::escalc()
add the constant.
library(metafor)
m1i <- 15.6
m2i <- 12.2
n1i <- 15
n2i <- 20
sd1i <- 3.82
sd2i <- 3.22
log(m1i) - log(m2i)
# [1] 0.245835
log(m1i + .5) - log(m2i + .5)
# [1] 0.2372173
sd1i^2/(n1i * m1i^2) + sd2i^2/(n2i * m2i^2)
# [1] 0.007480549
sd1i^2/(n1i * (m1i + .5)^2) + sd2i^2/(n2i * (m2i + .5)^2)
# [1] 0.006967255
lnR <- escalc(measure="ROM", m1i=m1i, m2i=m2i, sd1i=sd1i, sd2i=sd2i, n1i=n1i, n2i=n2i)
lnRk <- escalc(measure="ROM", add=0.5, to="all", m1i=m1i, m2i=m2i, sd1i=sd1i, sd2i=sd2i, n1i=n1i, n2i=n2i)
lnR; lnRk
# yi vi
# 1 0.2458 0.0075
# yi vi
# 1 0.2458 0.0075
all.equal(lnR, lnRk)
# [1] TRUE
We came across literature against this practice though: Koricheva, Handbook of Meta-analysis:
"One should also not use a ratio as an effect size measure when either the numerator or denominator would be equal to zero; the transform is undefined and trying to adjust the values by adding a tiny fraction to the numerator and denominator usually results in abnormally large estimates of effect size"
If adding a constant to the means is a bad idea, then advice on other options dealing with zero means would be greatly appreciated.
We understand that Hedges’d (or g) would be maybe a better solution, but then we confront with some cases yielding extreme effect sizes. Unfortunately those cases are particularly interesting for our study and we would not want to drop them (e.g. dropping extreme Hedges’d was practiced in Golivets, M., & Wallin, K. F. (2018). Neighbour tolerance, not suppression, provides competitive advantage to non‐native plants. Ecology letters.)