Department of Psychology, University of Southern California, Los Angeles, California 90089, USA.
Br J Math Stat Psychol. 2010 May;63(Pt 2):379-93. doi: 10.1348/000711009X467618. Epub 2009 Dec 14.
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 eta(2). When the regression surface is a plane, an estimate of eta(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 eta(2) is approximately equal to zero, estimation using smoothers has relatively high bias. A variation of eta(2) is suggested for dealing with this problem. Methods for testing H(0): eta(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.
本文考虑了估计关联总体强度的问题，包括存在曲率的情况。一般策略是拟合稳健回归线或允许曲率的某种平滑器，然后使用解释能力的稳健类似物，例如 eta(2)。当回归曲面为平面时，通过 Theil-Sen 估计量得到的 eta(2)估计值在均方误差和偏差方面相对于其他一些稳健回归估计量表现良好。当存在曲率时，Fan 导出的核估计量的一种推广形式表现相对较好，但两种替代平滑器具有某些实际优势。当 eta(2)近似为零时，使用平滑器进行估计会有相对较高的偏差。提出了一种改进 eta(2)的方法来解决这个问题。基于平滑器，本文还研究了检验 H(0)：eta(2)=0 的方法。在模拟中发现了两种方法，它们可以合理地控制第一类错误概率。提供了用于应用更成功方法的软件。
