# Should I use log transformed pharmacokinetic data or use GLM gamma regression with log link?

I was taught, that when we deal with data of multiplicative nature, following the log-normal distribution, like in pharmacokinetic analyses, we should log the data first to enable classic parametric tests. The bonus was that the back-transformed tests of means (of log-transformed data) was about the ratio of geometric means (of raw data). We like the geometric means and standard deviations in the pharmaceutical industry, especially because it is equal to medians. Actually, even the regulatory guidelines advise to use the log transformation. This is just an industry standard.

But recently I had a talk with a statistician, who recommended the gamma regression with log link. He plained me why the outcomes of the two are not equivalent. It was not only about the predicted outcome, but also the fact, that GLM handles the mean-variance relationship, which is not handles by log (which changes both mean and variance of the data), does not bias the outcomes (as back-transformed confidence intervals on log data) and still gives nice interpretation in terms of multiplicative change of the response for unit change in predictors.

But, since the models are not equivalent, which one would you advise me to work with log-normal data? Do you know, if the pharmaceutical regulators allow the gamma regression for that? Are the predicted LS-means the geometric means, as when I just run OLS on log-transformed data? As the two models are different, I bet it is not. So what is this?

This answer is a good place to start. The relevant bits (emphasis my own)...

The gamma has a property shared by the lognormal; namely that when the shape parameter is held constant while the scale parameter is varied (as is usually done when using either for models), the variance is proportional to mean-squared (constant coefficient of variation).

Something approximate to this occurs fairly often with financial data, or indeed, with many other kinds of data.

As a result it's often suitable for data that are continuous, positive, right-skew and where variance is near-constant on the log-scale, though there are a number of other well-known (and often fairly readily available) choices with those properties.

So, from this alone, it seems like gamma regression may be appropriate.

So far as regulation is concerned, it is my understanding that regulators want what they want. So if they insist on log transformed linear regressions then I don't think there is much you can do about that.

I have some PK data I'm playing with right now. Let me compare some models quickly.

• Thank you a lot for your helpful comment. I'm just learning this topic and you guided me well. Kindly please, do the comparison, if possible, but I worry a lot what if the results will be different? And will be, likely, as these are different models? Which one will be "better" to report? Or both, maybe? But what about the interpretation? OLS on log data can be interpreted in terms of ratio of medians or geometric means on the raw scale. But what will be the meaning of means (or LS-means) in the gamma regression? I mean the result of the joint test. Feb 14, 2020 at 0:43
• I can't do a very worthwhile comparison. Looks like some of R's mixed effect tools can't compute CIs for models with non-fixed scale parameters. The models will be slightly different due to different likelihoods. Whether or not they are practically different is really an issue for regulators or pharma to decide. I would report the log-linear model if it is the standard. Feb 14, 2020 at 0:48
• Thank you a lot. I know in R there are numerous packages for fitting it: glmmTMB is quite powered version of the lme4. Mabe the GEE would help you somehow? There is also the mixed beta regression in R, but that's different model. I mention it just by the way. Maybe also this: zeligproject.org and this www2.uaem.mx/r-mirror/web/packages/ZeligMultilevel/vignettes/… Feb 14, 2020 at 0:53
• glmmTMB supports: response distributions: Poisson, binomial, negative binomial (NB1 andNB2 parameterizations), Gamma, Beta, Gaussian; truncated Poissonand negative binomial;Studentt; Tweedie Feb 14, 2020 at 0:55
• I have my own work at the moment, but the comparison on a single data set should be straight forward. Feb 14, 2020 at 0:56