Consider three random variables Y, X1 and X2, where the typical value of Y, given X1 and X2, is given by some unknown function m(X1, X2). A goal is to determine which of the two independent variables is most important when both variables are included in the model. Let τ1 denote the strength of the association associated with Y and X1, when X2 is included in the model, and let τ2 be defined in an analogous manner. If it is assumed that m(X1, X2) is given by Y = β0 + β1X1 + β2X2 for some unknown parameters β0, β1 and β2, a robust method for testing H0 : τ1 = τ2 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.
Wilcox, R. (2018). An Inferential Method for Determining Which of Two Independent Variables Is Most Important When There Is Curvature. Journal of Modern Applied Statistical Methods, 17(1), eP2588. doi: 10.22237/jmasm/1525132920