Multicollinearity between ln(x) and ln(x)^2 I am running a negative binomial model and one of my predictor variables is a count variable. Since this variable was heavily skewed, I decided to log-transform it.
However, the effect of this variable is hypothesized to be non-linear. However, as soon as I include the squared term in my model, I obtain VIFs of these two variables that are >20, while all other predictors remain stable at VIFs between 1 and 5.
To my current understanding, the relationship should not be linear and hence multicollineairy should not arise.
Can anyone explain the cause of the multi-collinearty and give possible solutions to this problem?
 A: Except for very small counts, $\log(x)^2$ is essentially a linear function of $\log(x)$:

The colored lines are least squares fits to $\log(x)^2$ vs $\log(x)$ for various ranges of counts $x$.  They are extremely good once $x$ exceeds $10$ (and still awfully good even when $x\gt 4$ or so).
Introducing the square of a variable sometimes is used to test goodness of fit, but (in my experience) is rarely a good choice as an explanatory variable.  To account for a nonlinear response, consider these options:


*

*Study the nature of the nonlinearity.  Select appropriate variables and/or transformation to capture it.

*Keep the count itself in the model.  There will still be collinearity for larger counts, so consider creating a pair of orthogonal variables from $x$ and $\log(x)$ in order to achieve a numerically stable fit.

*Use splines of $x$ (and/or $\log(x)$) to model the nonlinearity.

*Ignore the problem altogether.  If you have enough data, a large VIF may be inconsequential.  Unless your purpose is to obtain precise coefficient estimates (which your willingness to transform suggests is not the case), then collinearity scarcely matters anyway.
A: The source of the collinearity is that $f(x) = x^2$. One way to reduce the correlation between $x$ and $x^2$ is to center $x$. Let $z=x-E(x)$ and compute $z^2$. Because the low end of the scale now has large absolute values, its square becomes large, making the relationship between $z$ and $z^2$ less linear than that between $x$ and $x^2$. This advice comes from The Analysis Factor: http://www.theanalysisfactor.com/centering-for-multicollinearity-between-main-effects-and-interaction-terms/
Note: When interpreting the effects, please remember that you scaled the covariate. Also, some researchers may caution against scaling because then the results of your model are data-dependent. Here is some perspective from Andrew Gelman on that issue: http://andrewgelman.com/2009/07/11/when_to_standar/
