Asymmetric properties of the Pearson correlation coefficient: Correlation as the negative association between linear regression residuals

ABSTRACT

An interpretation of the Pearson correlation coefficient as the negative association between linear regression residuals is used to develop asymmetric formulas, which allow researchers to decide upon directional dependence. Model selection based on residuals extends direction dependence methodology (originally proposed for non normal variables) to normally distributed predictors. Simulation results on the robustness of the methods and an empirical example are presented. We discuss potential advantages of a change in perspective in which non normality is not treated as a source of bias, but as a valuable characteristic of variables, which can be used to gain further insights into bi- and multivariate relations.

KEYWORDS:

• Correlation
• Direction of dependence
• Linear regression
• Skewness

Acknowledgment

The authors are indebted to Joseph L. Rodgers and the anonymous reviewers for constructive comments which helped improve the manuscript.