Only transform the left hand side. The rhs variables are fixed so they are less likely to need a transformation. Example is attached. Be sure to plot the data before you do the regression. Best, David Booth

The first question is of you want to adjust for the expected (mean) value or by the geometric mean predicted by 𝐶𝑅𝑃𝑜𝑏𝑠−𝐶𝑅𝑃𝑟𝑒𝑓 and 𝐴𝐺𝑃𝑜𝑏𝑠−𝐴𝐺𝑃𝑟𝑒𝑓?

If it is important to correct for the expected value, you will need a generalized linear model that fits the expectation directly. A suitable distribution family for concentrations of biological compounds is usually the gamma-distribution, and usually the log-link function makes sense. How you can do that practically depends on the software you use. It would not be a great deal in R, for instance.

If you log-transform your response (zinc) and use a (general) linear model, you estimate the expectation on the log-scale. This is very often a simple and nice way if the aim is to model systematic differences / effects depending on some experimental factors on such concentration variables. The expected value (=mean) on the log-scale corresponds to the geometric mean on the original scale (the expected value on the original scale is higher than that and depends on the variance as well).

If you should transform the predictors (CRP, AG) depends on their relationship with the response. This relationship should be approximately linear. Transformations can help to linearize these relationships. I doubt that taking the logs of the differences makes sense (particularily if the differences may be zero or negative). You may consider to use the log ratios instead (e.g. log(𝐶𝑅𝑃𝑜𝑏𝑠/𝐶𝑅𝑃𝑟𝑒𝑓) instead of log(𝐶𝑅𝑃𝑜𝑏𝑠-𝐶𝑅𝑃𝑟𝑒𝑓)).

And as David suggested: plot, plot, plot

In models a & b the right part is systematic, left is random

a. log Zinc~ logCRP + logAGP => log Zinc~ log(CRP *AGP)

that equation corresponds two fitting

log(y) ~beta*X +eps for linear model

log(y+eps ) ~beta*X for GLM with link function

in both cases eps distibuted ~ N(0, sigma^2)

the results are different in general cases E[ln(Y|x)] not equal ln ([E(Y|x])

b. Zinc~ log CRP + log AGP=> Zinc~ log(CRP *AGP)

that equation corresponds

y ~beta*X +eps for linear model, eps distibuted ~ N(0, sigma^2)

Disrtibutions of observed variables affect Your model choice.

Thanks all for your responses, I can't say I understood everything that was written(statistics is not my strongest forte), but I appreciate the assistance. I dug more into the literature and some papers which use this method suggest log transforming all the variables (Zn, CRP, AGP), so it would be model a. Furthermore, in the equation, I log transformed all the variables, that way the value I get in the right side is also log transformed. as below

ln(𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑠𝑒𝑟𝑢𝑚 𝑧𝑖𝑛𝑐)

= ln(𝑢𝑛𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑠𝑒𝑟𝑢𝑚 𝑧𝑖𝑛𝑐) − 𝛽1*(ln(𝐶𝑅𝑃𝑜𝑏𝑠) − ln(𝐶𝑅𝑃𝑟𝑒𝑓)) − 𝛽2*(ln(𝐴𝐺𝑃𝑜𝑏𝑠) − ln(𝐴𝐺𝑃𝑟𝑒𝑓))

This way I can just back transform the adjusted values to get the values with the unit.

Jochen Wilhelm I understood most of what you said. On the bit about using ratios instead of differences, that is part of the method itself (I am comparing several methodologies, where one uses geometric mean ratios and the other one uses differences ). As such if I alter this one to work with the mean ratios instead, it becomes a different method and not the one suggested by BRINDA. I do my analyses in R and will look into the gamma distribution.

Thanks to everyone for your assistance.

Blessings Hendrina Likoswe, note that the (arithmetic) mean of logs is the log of the geometric mean.

Say you values are x1, x2, x3, ..., xn

The logs are ln(x1), ln(x2), ln(x3), ..., ln(xn)

The mean log is h = 1/n * (ln(x1)+ln(x2)+ln(x3)+...+ln(xn))

Then exp(h) is the geometric mean of x1, x2, ..., xn

(that is, h = log(geoMean(x1,x2, ..., xn))