Back-transformation of regression coefficients

I’m doing a linear regression with a transformed dependent variable. The following transformation was done so that the assumption of normality of residuals would hold. The untransformed dependant variable was negatively skewed, and the following transform made it close to normal:

Y=50Yorig

where Yorig is the dependent variable on the original scale.

I think it makes sense to use some transformation on the β coefficients to work our way back to the original scale. Using the following regression equation,

Y=50Yorig=α+βX

and by fixing X=0, we have

α=50Yorig=50αorig

And finally,

αorig=50α2

Using the same logic, I found

βorig=α (α2β)+β2+αorig50

Now things work very well for a model with 1 or 2 predictors; the back-transformed coefficients resemble the original ones, only now I can trust the standard errors. The problem comes when including an interaction term, such as

Y=α+X1βX1+X2βX2+X1X2βX1X2

Then the back-transformation for the βs are not so close to the ones from the original scale, and I’m not sure why that happens. I’m also unsure if the formula found for back-transforming a beta coefficient is usable as is for the 3rd β (for the interaction term). Before going into crazy algebra, I thought I’d ask for advice…

Answer

One problem is that you’ve written

Y=α+β⋅X

That is a simple deterministic (i.e. non-random) model. In that case, you could back transform the coefficients on the original scale, since it’s just a matter of some simple algebra. But, in usual regression you only have E(Y|X)=α+β⋅X ; you’ve left the error term out of your model. If transformation from Y back to Y_{orig} is non-linear, you may have a problem since E\big(f(X)\big)≠f\big(E(X)\big), in general. I think that may have to do with the discrepancy you’re seeing.

Edit: Note that if the transformation is linear, you can back transform to get estimates of the coefficients on the original scale, since expectation is linear.

Attribution
Source : Link , Question Author : Dominic Comtois , Answer Author : Macro

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