Centering to reduce collinearity of x^2?

[wvguy8258]

Hi,

I would like to exponentiate the values of independent variables in a regression model, possibly using splines. I know that collinearity between X and X^2 is to be expected and the standard remedy is to center by taking X-average(X) prior to exponetiating. This seems odd to me because it will change the shape of the relationship between X and X^2 to one that is U-shaped, not monotonic. For example, if after centering you have

-2
-1
0
1
2

then the formerly lowest value will now be equal to the highest when squared. This seems to disrupt the idea that lower values have a lesser effect, let's say. What am I not getting that makes this make sense to do? Thanks. Seth

 

[EnumaElish]

Since X and X^2 are monotonically related, they have a high correlation. When their relationship is U-shaped, they are not, and that's the tradeoff.

 

[wvguy8258]

Hi,

A trade-off implies that it is a bit of a pain in the butt, I suppose. I hadn't played around with this much before posting. I made a small sample data set with and without variable centering. The model fit and significance levels were identical but the parameter estimates were very different for the linear term X and intercept. The noncentered version returned the parameter estimates I used to create the sample data. The centered data set returned the same parameter estimates for the squared term but not the linear term, the intercept was much different as well. I've since learned that the parameter estimates after centering reflect the linear trend when X = 0 (at the mean). Working now on a formula to transform parameter estimates between equation systems so that I understand this a bit more. I can see this should be possible when only X and X^2 are in the right side of the equation. My feeling is that understanding the variables in terms of their untransformed state will not be possible if several variables are transformed and so are all adding to changing the intercept away from what it would if the data were not centered. Anyone know a link that shows the equation I'll be looking for per the last few sentences? Thanks. -seth