# Multicollinearity: quadratic correlation between two independent variables in polynomial regression

Consider a polynomial regression of the form

$$y = α*x_1^2 + β*x_1 + ​ɣ*x_2$$

My question is: how to deal with multicollinearity between $$x_1^2$$ and $$x2$$? Or in other words, how to control for a quadratic correlation between two independent variables in polynomial regression analyses?

I guess that such correlations must have consequences for model fitting. I know that it is not perfect, but for "normal" multicollinearity in (non-polynomial) regression analyses, scientists in my research field would drop one of the correlated variables in order to avoid multicollinearity issues. Perhaps someone can indicate a sensible approach?

Here I include two example plots, $$x_1$$ vs $$x_2$$ and $$x_1^2$$ vs $$x_2$$. Hmisc::rcorr with type = "Spearman" (inR) yields a correlation coefficient between $$x_1^2$$ vs $$x_2$$ of $$-0.58$$. The variables are scaled variables from a use case.

• 1) Why are $x_1^2$ and $x_2$ necessarily correlated? // What’s wrong with correlation for your modeling task?
– Dave
Feb 24 at 13:17
• It's not a problem. There is no need to worry about it. The purpose here is to model curvature, not to assess the effect of $x$ while holding $x^2$ fixed. Feb 24 at 13:54
• @Dave, this from a use case where I detected correlation for x1² and x2 (I don't know how to write it correctly here... sorry). // I no not know whether this is really wrong for my modeling task. An example would be: in a study, the effect of the variables x1: grazing pressure and x2: diversity of microsites on the dependent variable y: ground beetle diversity is tested. Both variables might have an effect on beetle diversity. The study sites feature highest microsite diversity at intermediate grazing pressure, which results in x1² and x2 to be correlated. I thought, this might be a problem. Feb 24 at 14:27
• @BigBendRegion, I am not sure if this is correct. I do consider more than one variable x (in the example the variables x1 and x2). I would not worry about x1 and x1² being correlated. However, may it not be a problem if x2 and x1² are correlated? Feb 24 at 14:31
• Since you wouldn't worry about correlation between $x_1$ and $x_1^2,$ why do you have any concern about correlation between $x_2$ and $x_1^2$??
– whuber
Feb 24 at 15:01

• Thank you for this promising answer, @Robert Long. If the correlation was higher - is there anything else you'd suggest? And do you think there is some kind of threshold beyond which one should worry about correlations between $x_1^2$ and $x_2$? Mar 3 at 11:12