Back-transformation of regression coefficients

Question

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:

$$Y=\sqrt{50-Y_{orig}}$$

where $Y_{orig}$ is the dependent variable on the original scale.

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

$$Y=\sqrt{50-Y_{orig}}=\alpha+\beta \cdot X$$

and by fixing $X=0$, we have

$$\alpha=\sqrt{50-Y_{orig}}=\sqrt{50-\alpha_{orig}}$$

And finally,

$$\alpha_{orig}=50-\alpha^2$$

Using the same logic, I found

$$\beta_{orig}=\alpha\space(\alpha-2\beta)+\beta^2+\alpha_{orig}-50$$

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

$$Y=\alpha+X_1\beta_{X_1}+X_2\beta_{X_2}+X_1X_2\beta_{X_1X_2}$$

Then the back-transformation for the $\beta$s 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 $\beta$ (for the interaction term). Before going into crazy algebra, I thought I'd ask for advice...

Answer 1

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.

Answer 2

I salute your efforts here, but you're barking up the wrong tree. You don't back transform betas. Your model holds in the transformed data world. If you want to make a prediction, for example, you back transform $\hat{y}_i$, but that's it. Of course, you can also get a prediction interval by computing the high and low limit values, and then back transform them as well, but in no case do you back transform the betas.