# Variance inflation factor for generalized additive models

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?

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.