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I would like to compare two models, one of which has a transformation on the response variable, that is:

First model: $y \sim x+z$

Second model: $y^{1/k} \sim x+z$

Since AIC can be used only for models with the same form on the response, how could I potentially solve this problem? I know that if we have a log transformation on normal, we can correct it by adding twice the sum of log response, but how would we deal with a generalised case when y is transformed by raising it to the power of $1/k$, where $k\in\mathbb{N_+}$?

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The most logical approach would seem to be to look at the predictive performance - obviously with predicted transformed values back-transformed. E.g. for the fitted models you would draw lots of predictions (e.g. predict the whole dataset 10000 times) and then look at e.g. root mean squared error of these predictions. You likely need to look at drawing random predictions for each observations covariates - rather than looking at the mean prediction on the respective scales -, because the error distributions are also transformed.

I also suspect you should take the uncertainty around the regression coefficients by treating estimate $\pm$ SE as a quasi-posterior - i.e. for each predicted dataset draw the coefficients from N(estimate, SE$^2$). That’s again, because the error goes into a non-linear transformation.

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