Calculates variance-inflation and generalized variance-inflation factors for linear and generalized linear models. It's a measure describing how much the variance of an estimated coefficient is increased because of collinearity.

`VIF(mod)`

A vector of vifs, or a matrix containing one row for each term in the model, and columns for the GVIF, df, and \(GVIF^{1/(2\times df)}\).

- mod
an object that responds to

`coef`

,`vcov`

, and`model.matrix`

, such as an`lm`

or`glm`

object.

Henric Nilsson and John Fox jfox@mcmaster.ca

If all terms in an unweighted linear model have 1 df, then the usual variance-inflation factors are calculated.

The vif are defined as $$vif_j=\frac{1}{1-R_j^2}$$

where \(R_j^2\) equals the coefficient of determination for regressing the explanatory variable `j`

in question on the other terms in the model. This is one of the well-known collinearity diagnostics.

If any terms in an unweighted linear model have more than 1 df, then generalized variance-inflation factors (Fox and Monette, 1992) are calculated. These are interpretable as the inflation in size of the confidence ellipse or ellipsoid for the coefficients of the term in comparison with what would be obtained for orthogonal data.

The generalized vifs are invariant with respect to the coding of the terms in the model (as long as the subspace of the columns of the model matrix pertaining to each term is invariant). To adjust for the dimension of the confidence ellipsoid, the function also prints \(GVIF^{1/(2\times df)}\) where \(df\) is the degrees of freedom associated with the term.

Through a further generalization, the implementation here is applicable as well to other sorts of models, in particular weighted linear models and generalized linear models.

Values of vif up to 5 are usually interpreted as uncritical, values above 5 denote a considerable multicollinearity.

Fox, J. and Monette, G. (1992)
Generalized collinearity diagnostics.
*JASA*, **87**, 178--183.

Fox, J. (2008)
*Applied Regression Analysis and Generalized Linear Models*,
Second Edition. Sage.

Fox, J. and Weisberg, S. (2011)
*An R Companion to Applied Regression*, Second Edition, Sage.

```
VIF(lm(Fertility ~ Agriculture + Education, data=swiss))
VIF(lm(Fertility ~ ., data=swiss))
```

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