Standardization of Weighted Ranking Correlation Coefficients

TL;DR

This paper tackles the problem that weighted ranking correlation coefficients lose the symmetry and zero-mean property of standard coefficients, complicating interpretation for uncorrelated rankings. It introduces a standardization function $g(x)$ that maps any ranking coefficient $\Gamma$ to a zero-mean form $g(\Gamma)$ while preserving domain, monotonicity, and boundary constraints, using estimates of $\bar{\Gamma}$, $V$, and $V^{\ell}$. The estimation framework combines exact calculations for small $n$, numerical sampling, and polynomial regression to model these distribution parameters, and defines a piecewise quadratic construction of $g(x)$ conditioned on variance-ratio behavior. The method is demonstrated for Spearman and Kendall variants with additive and multiplicative weighting, providing practical recipes and numerical results that enable meaningful cross-context comparison of weighted ranking correlations.

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 $Γ$ in a standard form $g(Γ)$ that has zero expected value, while maintaining the relevant statistical properties of $Γ$.

Figures (5)

• Figure 1: Numerical estimate of $\bar{\Gamma}$ as a function of $x$ defined in Eq. \ref{['eq:n_to_x']} (left) and Eq. \ref{['eq:n_to_x_log']} (right) for Spearman coefficient with additive (top) and multiplicative (bottom) weight $w_i=1/(i+n_0)^2$.
• Figure 2: MSE as a function of the polynomial degree $D_{\gamma}$ for the Spearman and Kendall coefficients with the different weighting schemes described in Section \ref{['sec:w_coeff']}.
• Figure 3: Results of the $\bar{\Gamma}$ estimation procedure described in Section \ref{['sec:gamma_bar_estimate']} for Kendall with multiplicative weight and Spearman with additive weight, in both cases with weighting function $f(i)=1/(i+n_0)^2$.
• Figure 4: Standardization function $g(x)$ for the Spearman $\rho$ (Kendall $\tau$) coefficient with additive and multiplicative weight, with $f(i)=1/i^2$ ($f(i)=1/i$) and ranking length $n=500$ ($n=30$).
• Figure 5: Estimate of the distribution function $p(\gamma)$ before and after the standardization of the coefficients for the Spearman $\rho$ (Kendall $\tau$) coefficient with additive and multiplicative weight, with $f(i)=1/i^2$ ($f(i)=1/i$) and ranking length $n=500$ ($n=30$).