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I am running a general linear model in which I have two predictors: X and X-squared.

I entered both these predictors in my analysis because I think X might explain the variance in the outcome measure partially linearly and partially in a quadratic fashion.

Obviously, there is multicollinearity in this example. However, I was wondering if there are maybe some reasons why it is not a good idea to put both these predictors in my model.

Vincent
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  • No problem in principle. Plot the data and the fitted curve as well as looking at the numeric results to see how well it works. – Nick Cox Oct 15 '13 at 22:23
  • Well, as you already note, you can get multicollinearity (though this can be avoided with the use of orthogonal polynomials). – Glen_b Oct 15 '13 at 22:42
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    An alternative to the suggestion of @Glen_b is to first center X, then create a squared version of the centered X variable. Include them both and multicollinearity will be substantially reduced. As an added bonus, model interpretation is often nicer with centered IVs. – ndoogan Oct 16 '13 at 00:12
  • In this case, what most makes the model interpretable is, I suggest, to plot data and model fit. Centering X does no harm, but the most interesting level for X is often that at which the quadratic has a turning point, whenever that occurs within the range of the data. (It may lie way outside that range.) Numerical stability given correlated predictors is much less of a difficulty with modern statistical software than it was a few decades ago when regression programs did not always use good algorithms. But an X and its square can't be interpreted as having separate effects, regardless. – Nick Cox Oct 16 '13 at 00:35
  • @Glen_b How do you orthogonalize these predictors? – Vincent Oct 16 '13 at 19:58
  • Outline discussion [here](http://stats.stackexchange.com/questions/72626/how-to-include-a-linear-and-quadratic-term-when-also-including-interaction-with/73042#73042) – Glen_b Oct 17 '13 at 06:08
  • I used 'orthog' in Stata to orthogonalize X and X^2. I have a couple of questions about the orthogonalized variables: 1) How do you decide if you want to orthogonalize X with respect to X^2, or vice versa? 2) The centered variables have a different effect in my model than the orthogonalized variables. How do I decide which one to chose? 3) After orthogonalization, is there any interpretation possible of the beta values of these variables? – Vincent Oct 19 '13 at 13:38

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