There's a number of posts out there on back-transformations but none that I've been able to use to answer this problem, internet searches are returning surprisingly little on the issue of back-transformations. But it seems this is an issue many people run into, so hopefully this is of use to others too.

I've got a series of regression models, one set are log-log models and the other are linear models but I've had to boxcox transform the dependent variable due to non-normality.

One set of models has other GHGs as the dependent, the other CO2 (both per capita). Each model has 8 - 10 independent variables, some use poly terms. The boxcox transformed dependent uses lambda 0.136.

The independent variables are the same for each model. It's important for me to be able to compare the betas from both models. Therefore I want to back-transform the coefficients. However, I'm confused about exactly how this should be done. I gather it's not as simple as doing a straight transformation using the exponent, and bias needs to be accounted for.

Could someone explain how to do this? And if at all possible, share R code to perform the transformations.

  • $\begingroup$ What's a GHG? Do the models use the same explanatory variables? Would they have common values of those variables? Why do you need to back-transform the betas to make a comparison? $\endgroup$ – whuber Jul 29 '20 at 20:54
  • $\begingroup$ GHG is greenhouse gases, yeah all explanatory variables are the same. At the moment the CO2 model has a boxcox transformed dependent variable and explanatory variables all non-transformed. While the other greenhouse gases model is log-log. So comparing at the betas at moment doesn't mean much right? $\endgroup$ – Orla Jul 31 '20 at 6:08
  • $\begingroup$ It's difficult to think how one could compare two such betas, because they have intrinsically different meanings. It would be like comparing a height to a mass in a simpler physical application: it just makes no sense. $\endgroup$ – whuber Jul 31 '20 at 15:50

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