-->(BOX-COX) Transformation breaks down for large (positive) coefficient values <--

#1
Hi,

I have done a box-cox transformation of my response variable, using the following formula: (Y^lambda - 1)/lambda
Previously, I have got some excellent help in understanding the way interpretation works for different levels of Y (q1,median,q3). My formula for inverse transformation is:

x=(lambda*z + 1)^(1/lamda)

In my analysis, lambda= -1

My interpretation of this formula is that z=y+x (the response times the specific coefficient value). Or just z=y which would return the original values if the response variable.
In short, this works very well. When z=y the back-transformation produces almost the same values as the original.

But a problem arises with one of the beta-coefficients. It is to large, so that z>1 (or, y+x>1 in the transformed scale).
That returns negative values for the back-transformed value of y with respect to x. Does anyone knows how to deal with this problem?

As an example:
Back-transformed values of Y (very close to the real data):
Q1: Q2: Q3:
353,04077 403,8761 496,2258
After the effect of X1, holding the other variables constant (producing plausible results):
Q1: Q2: Q3:
331,2585 373,374179 450,9129
After the effect of X2 (inplausible results):
Q1: Q2: Q3:
-428,079 -373,61786 -318,767

Another question I am searching an answer to, is how to correctly describe the change process?
If the beta-value is -0,00019481 (in transformed scale), and its effect to y varies between 6-9% with different values of y (quantile 1-3 in back-transformed, original scale), how do I describe the change process with respect to a change in x?
Is it correct to describe the change in Y as an interval of 6-9% depending on the value of Y, with respect to a unit change in X? It is this last part of the sentence that I is still not certain about..

Any input is much appreciated! I have searched the webb for any material to read about interpreting back-transformed boc-cox transformation, but although there is plenty to read about the method of transforming, interpretation and inverse transformation is rarely mentioned.

Best regards,
Hank
 
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