Calculating VIF for ordinal logistic regression & multicollinearity in R

Question

I am running an ordinal regression model. I have 8 explanatory variables, 4 of them categorical ('0' or '1') , 4 of them continuous. Beforehand I want to be sure there's no multicollinearity, so I use the variance inflation factor (vif function from the car package) :

mod1<-polr(Y ~ X1+X2+X3+X4+X5+X6+X7+X8, Hess = T, data=df)
vif(mod1)

but I get a VIF value of 125 for one of the variables, as well as the following warning :

Warning message: In vif.default(mod1) : No intercept: vifs may not be sensible.

However, when I convert my dependent variable to numeric (instead of a factor), and do the same thing with a linear model :

mod2<-lm(Y ~ X1+X2+X3+X4+X5+X6+X7+X8, data=df)
vif(mod2)

This time all the VIF values are below 3, suggesting that there's no multicollinearity.

I am confused about the vif function. How can it return VIFs > 100 for one model and low VIFs for another ? Should I stick with the second result and still do an ordinal model anyway ?

Answer 1

The vif() function uses determinants of the correlation matrix of the parameters (and subsets thereof) to calculate the VIF. In the linear model, this includes just the regression coefficients (excluding the intercept). The vif() function wasn't intended to be used with ordered logit models. So, when it finds the variance-covariance matrix of the parameters, it includes the threshold parameters (i.e., intercepts), which would normally be excluded by the function in a linear model. This is why you get the warning you get - it doesn't know to look for threshold parameters and remove them. Since the VIF is really a function of inter-correlations in the design matrix (which doesn't depend on the dependent variable or the non-linear mapping from the linear predictor into the space of the response variable [i.e., the link function in a glm]), you should get the right answer with your second solution above, using lm() with a numeric version of your dependent variable.