Measuring and detecting associations: Methods based on robust regression estimators or smoothers that allow curvature

This paper considers the problem of estimating the overall strength of an association, including situations where there is curvature. The general strategy is to fit a robust regression line, or some type of smoother that allows curvature, and then use a robust analogue of explanatory power, say η 2 . When the regression surface is a plane, an estimate of η 2 via the Theil–Sen estimator is found to perform well, relative to some other robust regression estimators, in terms of mean squared error and bias. When there is curvature, a generalization of a kernel estimator derived by Fan performs relatively well, but two alternative smoothers have certain practical advantages. When η 2 is approximately equal to zero, estimation using smoothers has relatively high bias. A variation of η 2 is suggested for dealing with this problem. Methods for testing H 0 : η 2 =0 are examined that are based in part on smoothers. Two methods are found that control Type I error probabilities reasonably well in simulations. Software for applying the more successful methods is provided.