Skip to content
Prev 20114 / 20628 Next

Collinearity diagnostics for (mixed) multinomial models

Dear Phillip (and Juho),

You raise a reasonable point but, unfortunately, one that isn't really 
relevant to the problem at hand.

Applied to a linear model, which is the context in which generalized 
variance inflation was originally defined in the paper by me and Georges 
Monette cited in ?car::vif, the GVIF *is* invariant with respect to 
inessential changes to the model such as centering regressors or any 
change in the bases for the regressor subspaces of terms in the model. 
The GVIF compares the size of the joint confidence region for the set of 
coefficients for a term in the model to its size in a utopian situation 
in which the subspace for the term is orthogonal to the subspaces of the 
other terms, and reduces to the usual VIF when the term in one-dimensional.

Generalized variance inflation has subsequently been extended to some 
other regression models, such as generalized linear models, and it 
retains these essential invariances (although interpretation isn't as 
straightforward).

In response to Juho's original question, I conjectured an extension to 
multinomial logit models, tested some of its invariance properties, but 
unfortunately didn't test sufficiently extensively. (I did suggest 
additional tests that I didn't perform.) It's clear from Juho's example 
that my conjecture was wrong.

The reason that I hadn't yet responded to Juho's recent question is that 
Georges and I are still trying to understand why my proposed definition 
fails for multinomial logit models. It appears to work, for example, for 
multivariate linear models. Neither of us, at this point, has a solution 
to Juho's problem, and it's possible that there isn't one. We're 
continuing to discuss the problem, and one of us will post an update to 
the list if we come up with either a solution or a clear explanation of 
why my proposal failed.

Thank you for prompting me to reply, if only in a preliminary manner.

Best,
  John