On multicollinearity and concurvity in some nonlinear multivariate models

Recent developments of multivariate smoothing methods provide a rich collection of feasible models for nonparametric multivariate data analysis. Among the most interpretable are those with smoothed additive terms. Construction of various methods and algorithms for computing the models have been the main concern in literature in this area. Less results are available on the validation of computed fit, instead, and many applications of nonparametric methods end up in computing and comparing the generalized validation error or related indexes. This article reviews the behavior of some of the best known multivariate nonparametric methods, based on subset selection and on projection, when (exact) collinearity or multicollinearity (near collinearity) is present in the input matrix. It shows the possible aliasing effects in computed fits of some selection methods and explores the properties of the projection spaces reached by projection methods in order to help data analysts to select the best model in case of ill conditioned input matrices. Two simulation studies and a real data set application are presented to illustrate further the effects of collinearity or multicollinearity in the fit.