Standardization of Weighted Ranking Correlation Coefficients

Abstract: A relevant problem in statistics is defining the correlation of two rankings of a list of items. Kendall's tau and Spearman's rho are two well established correlation coefficients, characterized by a symmetric form that ensures zero expected value between two pairs of rankings randomly chosen with uniform probability. However, in recent years, several weighted versions of the original Spearman and Kendall coefficients have emerged that take into account the greater importance of top ranks compared to low ranks, which is common in many contexts. The weighting schemes break the symmetry, causing a non-zero expected value between two random rankings. This issue is very relevant, as it undermines the concept of uncorrelation between rankings. In this paper, we address this problem by proposing a standardization function $g(x)$ that maps a correlation ranking coefficient $\Gamma$ in a standard form $g(\Gamma)$ that has zero expected value, while maintaining the relevant statistical properties of $\Gamma$.