Title: Impact of correlation between predictors on variance decomposition and variable selection using CAR scores
Authors: Henri Wallard - Ipsos (France) [presenting]
Abstract: Regression is widely used to identify the most important regressors or to quantify their relative importance. If predictors are strongly correlated OLS is difficult to use and other methods such as regularization or variance decomposition have been considered. A method is said to benefit from grouping property if the computed measures of importance tend to equate when a group of regressors are highly correlated. Grouping property has been initially investigated for ridge regression or elastic net. The quantification of variable importance using CAR scores (Correlation-Adjusted Correlation) has been presented by some authors as a way to provide a canonical ordering of importance and has also been credited with grouping property. This paper demonstrates that CAR scores actually do not benefit from the grouping property. Using theoretical examples and geometric interpretation in the case of two and three predictors we will show in contrary that the difference of CAR scores remains stable or can even grow when the correlation between predictors increases. It will also be shown with a real dataset that grouping property is not achieved with CAR scores. As a consequence the usage of CAR scores should not be recommended because of the risk of inconsistent estimation and other methods will be proposed.