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In general I standardize my features before regression by subtracting the mean and dividing by unit variance: $$ \hat{X} = \frac{X - \bar{X}}{Var(X)}$$ With this basic standardization, interpreting the regression coefficients is quite straight-forward.

Now I've got some not-really-normal-dist data, for which a power transform is a good choice. Since some features contain negative values, I chose to use a Yeo-Johnson transformation. (I could also transform the features containing neg. values to all-positive features if Box-Cox helps the cause...)
After this transform, the data is standardized additionally, again by stubtracting the mean and dividing by variance.

Question:

Is there an easy way to interprete the regression coefficients after a power transform (may it be Yeo-Johnson or Box-Cox...)?
A similar question is asked here and the answer helps with the fact that feature scaling changes the coefficient interpretation in general, but it is specific to a transformation with the 4th root.

beyond the question

After the power transformation, I apply polynomial transformation of degree 2 or 3 with interaction terms and then perform a feature extraction with a PCA to reduce the dimensionality of the problem. Finally the design matrix is passed to a regression algorithm.

I guess this further reduces the intepretability for both only standardizing and/or power transform? Any other way to interprete the coefficients by using words? (A graphical interpretation using ICE-plots and the like is imho still fairly easy.)

JE_Muc
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  • My answer that you reference is explicitly *not* specific to the fourth root: at the end it points out the easy generalization where the power $1/4$ equals any nonzero number $p.$ That answers your question. BTW, a polynomial transformation is an exceedingly rare and tricky beast because (a) it is difficult to interpret and (b) unless its coefficients are carefully chosen, it's not even a transformation: it will not be one-to-one. – whuber Jul 30 '20 at 15:21
  • Yeah, maybe I expressed it a bit vague, but I was looking for a *specific answer considering the differences in coeff. interpretation between standardization and power transform*, thus my question imho requires a more answer. What do you mean by "one-to-one"? Considering standardized/power-transformed features? – JE_Muc Jul 30 '20 at 15:54
  • I can't tell what you intend your question to be, but maybe after you edit your post it will become clear. For the meaning of "one-to-one" please [choose your favorite reference](https://www.google.com/search?client=firefox-b-1-d&q=one-to-one+transformation). – whuber Jul 30 '20 at 16:15

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