Nonlinear confounding in high-dimensional regression

Abstract

It is not uncommon to find nonlinear patterns in the scatterplots of regressor variables. But how such findings affect standard regression analysis remains largely unexplored. This article offers a theory on nonlinear confounding, a term for describing the situation where a certain nonlinear relationship in regressors leads to difficulties in modeling and related analysis of the data. The theory begins with a measure of nonlinearity between two regressor variables. It is then used to assess nonlinearity between any two projections from the high-dimensional regressor and a method of finding most nonlinear projections is given. Nonlinear confounding is addressed by taking a fresh new look at fundamental issues such as the validity of prediction and inference, diagnostics, regression surface approximation, model uncertainty and Fisher information loss.