Comparing the strength of association of two predictors via smoothers or robust regression estimators

Abstract

Consider three random variables, Y, X1 and X2, having some unknown trivariate distribution and let η2j (j = 1, 2) be some measure of the strength of association between Y and Xj. When η2j is taken to be Pearsons correlation numerous methods for testing H0 η21 = η22 have been proposed. However, Pearsons correlation is not robust and the methods for testing H are not level robust in general. This article examines methods for testing H0 based on a robust fit. The first approach assumes a linear model and the second approach uses a nonparametric regression estimator that provides a flexible way of dealing with curvature. The focus is on the Theil-Sen estimator and Clevelands LOESS smoother. It is found that a basic percentile bootstrap method avoids Type I errors that exceed the nominal level. However, situations are identified where this approach results in Type I error probabilities well below the nominal level. Adjustments are suggested for dealing with this problem. © 2011 JMASM, Inc.