Resampling Techniques to Determine Direction of Effects in Linear Regression Models

by Wolfgang Wiedermann, Michael Hagmann, Michael Kossmeier, and Alexander von Eye.

Abstract: Previous studies have shown that, in the context of linear regression analysis, the cube of the Pearson correlation coefficient can be expressed by the ratio of the third moment of the response variable to the third moment of the explanatory variable (Dodge & Rousson, 2001). This relation implies that the skewness of the response variable is always smaller than the skewness of the explanatory variable, and directional dependency can be determined based on the third moments of variables. The current study extends the concept of directional dependency and focuses on distributional properties of the residuals of two competing linear regression models. It is shown that the residual skewness of the mis-specified regression model is larger than the residual skewness of the true regression model. Based on this result, three significance tests are developed that can be used to determine the direction of dependence in non-normally distributed samples. A Monte-Carlo simulation experiment is performed to analyze robustness and power properties of the proposed tests under various degrees of correlations, sample sizes, and population distributions. Additionally, an empirical example is provided which underlines important assumptions of the proposed resampling procedures. Recommendations are given for making decisions concerning the direction of effects based on the three significance tests.

Key Words: Direction of Effects, Directional Dependence, Permutation, Bootstrap, Significance Test

Authors:
Wolfgang Wiedermann, wolfgang.wiedermann@univie.ac.at
Michael Hagmann, michael.hagmann@univie.ac.at
Michael Kossmeiser, voneye@msu.edu

Editor: Richard G. Graf, rgraf@sunstroke.sdsu.edu

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