I have fitted two different binomial logistic regression models A and B. Model A contains only one predictor variable. Model B contains a different set of predictor variables, none of which is the predictor included in model A. There is a notable degree of multicollinearity between the predictors in model B. I want to compare how well the two models can account for the variation in my data.
Usually, when discussing the consequences of multicollinearity in linear regression models, most authors focus on the effect on the predictors. For example Dormann et al (2012) point out that the standard errors of collinear coefficients will be inflated, which leads to "inaccurate tests of significance for the predictors, meaning that important predictors may not be significant, even if they are truly influential" (p. 29).
However, what is less clear to me is the effect of multicollinearity on the overall performance of the model. This question asks whether multicollinearity affects the performance of the model as a classifier, with reference to the Wikipedia article on Multicollinearity which says that "multicollinearity does not reduce the predictive power or reliability of the model as a whole, at least within the sample data set". In his answer to the question, @EdM seems to confirm that multicollinearity does not affect model reliability unless it is used to predict a data set different from the one used to fit it.
My case is somewhat different because I don't want to use the models as classifiers on new data. Instead, I want to compare how well they can explain my data set. So, that answer still leaves me with the following questions:
- Is it valid to say that the explained variance of a model is invariant to the presence of multicollinearity?
- Can I use still measures such as AIC or AUROC to compare the performance of my models A and B even though the predictors of model B are strongly correlated?
- Is there a quotable reference that discusses the effect of multicollinearity on the explained variance of models and on measures such as the AIC or AUROC?