## An Inferential Method for Determining Which of Two Independent Variables Is Most Important When There Is Curvature

## Main Article Content

## Abstract

Consider three random variables *Y*, *X*_{1} and *X*_{2}, where the typical value of *Y*, given *X*_{1} and *X*_{2}, is given by some unknown function *m*(*X*_{1}, *X*_{2}). A goal is to determine which of the two independent variables is most important when both variables are included in the model. Let *τ*_{1} denote the strength of the association associated with *Y* and *X*_{1}, when *X*_{2} is included in the model, and let *τ*_{2} be defined in an analogous manner. If it is assumed that *m*(*X*_{1}, *X*_{2}) is given by *Y* = *β*_{0} + *β*_{1}*X*_{1} + *β*_{2}*X*_{2} for some unknown parameters *β*_{0}, *β*_{1} and *β*_{2}, a robust method for testing H_{0} : *τ*_{1} = *τ*_{2} is now available. However, the usual linear model might not provide an adequate approximation of the regression surface. Many smoothers (nonparametric regression estimators) were proposed for estimating the regression surface in a more flexible manner. A robust method is proposed for assessing the strength of the empirical evidence that a decision can be made about which independent variable is most important when using a smoother. The focus is on LOESS, but it is readily extended to any nonparametric regression estimator of interest.