Effectiveness of rank correlations in curvilinear relationships

Abstract

In cases where only linear relationships are suspected, Pearson’s correlation is generally used to measure the strength of the association between variables. It is well-known, however, that when a non-linear or non-linearizable connection exists, the use of Pearson’s coefficient on original values can be deceiving. On the other hand, rank correlations should perform satisfactorily because of their properties and versatility. There are many coefficients of rank correlation, from simple ones to complicated definitions invoking one or more special transformations. Each of these methods is sensitive to a different feature of dependence between variables The purpose of this article is to find a coefficient, if one exists, that tends to be different from zero at least in a meaningful way more often than others when the relationship between two rankings is of a non linear type. In this regard, we analyze the behavior of a few well-known rank correlation coefficients by focusing on some frequently encountered non-linear patterns. We conclude that a reasonably robust answer to the special needs arising from non-linear relationships could be given by a variant of the Fisher–Yates coefficient, which has a more marked tendency to reject the hypothesis of independence between pairs of rankings connected by several forms of non-linear interaction.