Hessian Inverse Transformation for Nonlinear Confounding

Abstract

Many model-free dimension reduction methods have been developed for high-dimensional regression data, but have not paid much attention on problems with nonlinear confounding. In this paper, we propose a response transformation based Hessian directions method for nonlinear confounding data to reduce the dimension of predictors without requiring a prespeci_ed parametric model. The bene_t of using geometrical information from our method is highlighted. A ratio estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. The weighted chi-squared test of dimension for our method is derived. Several simulation examples are reported for illustration and comparisons are made with sliced inverse regression of Li (1997). Illustrative applications to two real data are also presented.