log-transforming standard deviations

[ThisIsLorenzo]

https://github.com/ThisIsLorenzo/PFAS_Trophic_Magnification/blob/a8aeec0edf5d7e5ece631c192ab868798dfe510d/R/code.Rmd#L58

Some studies do not report the slope and its SE. So I am extracting raw data from linear regressions. Doing so, I am collecting means and standard deviations of each observation in the regression. However, PFAS concentrations are not in the log scale. For this reasons, I need to log transform the means (PFAS concentrations) and their standard deviations before running a linear regression analysis. This is to make slopes and errors consistent. For the means, I can simply log transform them. But what about the standard deviations? What formula can I use to convert SD into the log scale? @itchyshin

[ThisIsLorenzo]

I had a meeting with Shinichi, and this is what we concluded:

• First of all, to answer the issues's question, we can apply the Deltha method (Taylor's expansion) to log transforming SD values as we do for SE values. This conclusion was reached following this discussion:
• SE and SD are both measures of variation
• SE is the standard deviation of estimates. SE can be deviation of slope, effect size, whatever.
• SE is a function of SD and n.
• SD is the standard deviation of samples. SD is a property of the population, it is NOT linked to the sample size.
• Shinichi believes that we should not account for heteroscedasticity because most of the studies we are using did not. So rather than being more conservative (because accounting for heteroscedasticity widens the SE) it is better to be more consistent with what we have to make proper comparisons.
• So, if we have SE we can use it as weight.
• If we have SD, we can use the SD^2 as weight (we are taking the assumption that n is the same for each observation! we need to aknowledge it).
• If we do not have SE or SD we just run lm() without assigning weights and without taking into account heteroscedasticity (no robust no coeftest function).
• Originally, I was planning to use the coeftest() function (i.e., coeftest(*lm object*, vcov. = vcovHC)) to taking into account heteroscedasticity but we decided to do not do it because all other studies that provide slope and SE did not do it and we want to keep slope and errors values consistent.