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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.

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    $\begingroup$ 1) Why are $x_1^2$ and $x_2$ necessarily correlated? // What’s wrong with correlation for your modeling task? $\endgroup$
    – Dave
    Feb 24 at 13:17
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    $\begingroup$ 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. $\endgroup$ Feb 24 at 13:54
  • $\begingroup$ @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. $\endgroup$
    – yenats
    Feb 24 at 14:27
  • $\begingroup$ @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? $\endgroup$
    – yenats
    Feb 24 at 14:31
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    $\begingroup$ 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 $x_1^2$?? $\endgroup$
    – whuber
    Feb 24 at 15:01
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It is to be expected that independent variables in a multivariable regression model will be correlated. Only in an experimental setting would you expect them to be orthogonal. A correlation coefficient of 0.58 is not especially large and I would not be worried about this.

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  • $\begingroup$ 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 $x_2$? $\endgroup$
    – yenats
    Mar 3 at 11:12
  • $\begingroup$ You're welcome. There is no threshold but when you are dealing with high correlations, say >0.9 and you have a lot of variables then some kind of penalised regression may be appropriate. $\endgroup$ Mar 3 at 11:29

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