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In the usual VIF calculation for a linear regression, each independent/explanatory variable $X_j$ is treated as the dependent variable in an ordinary least squares regression. i.e.

$$ X_j = \beta_0 + \sum_{i=1, i \neq j}^n \beta_i X_i $$

The $R^2$ values are stored for each of the $n$ regressions and VIF is determined by

$$ VIF_j = \frac{1}{1-R^2_j} $$

for a particular explanatory variable.

Suppose my generalize additive model takes the form, $$ Y=\beta_0+ \sum_{i=1}^n \beta_iX_i + \sum_{j=1}^m s_j(X_i) . $$

Is there an equivalent VIF calculation for this type of model? Is there a way I can control for the smooth terms $s_j$ to test for multicollinearity?

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There is a function in r corvif() which can be found in AED package. For examples and references see Zuur et al. 2009. Mixed Effects Models and Extensions in Ecology with R pp. 386-387. The code for the package is available from the book website http://www.highstat.com/book2.htm.

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    $\begingroup$ You'll get my vote if you spruce up your answer with some of the math. :) $\endgroup$ – Alexis Mar 16 '15 at 19:50
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    $\begingroup$ @Alexis is right. This is helpful, but note that the question isn't asking for R code. Can you explain the application of the VIF to GAMs conceptually? $\endgroup$ – gung Mar 16 '15 at 19:55

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