Myths About Linear and Monotonic Associations: Pearson’s r, Spearman’s ρ, and Kendall’s τ

Abstract

Pearson’s correlation coefficient is considered a measure of linear association between bivariate random variables X and Y. It is recommended not to use it for other forms of associations. Indeed, for nonlinear monotonic associations alternative measures like Spearman’s rank and Kendall’s tau correlation coefficients are considered more appropriate. These views or opinions on the estimation of association are strongly rooted in the statistical and other empirical sciences. After defining linear and monotonic associations, we will demonstrate that these opinions are incorrect. Pearson’s correlation coefficient should not be ruled out a priori for measuring nonlinear monotonic associations. We will provide examples of practically relevant families of bivariate distribution functions with nonlinear monotonic associations for which Pearson’s correlation is preferred over Spearman’s rank and Kendall’s tau correlation in testing the dependency between X and Y. Alternatively, we will provide a family of bivariate distributions with a linear association between X and Y for which Spearman’s rank and Kendall’s tau are preferred over Pearson’s correlation. Our examples show that existing views on linear and monotonic associations are myths.