Weighting by Tying: A New Approach to Weighted Rank Correlation

TL;DR

This paper introduces scaled gamma, a flexible and formally sound weighted rank correlation measure based on fuzzy order relations, allowing emphasis on specific ranking positions.

Contribution

It proposes a new weighted rank correlation measure using fuzzy order relations, enhancing flexibility and formal properties over existing methods.

Findings

The scaled gamma measure is related to Goodman and Kruskal's gamma.

It is parametrized by a fuzzy equivalence relation.

The approach allows flexible weighting of rank positions.

Abstract

Measures of rank correlation are commonly used in statistics to capture the degree of concordance between two orderings of the same set of items. Standard measures like Kendall's tau and Spearman's rho coefficient put equal emphasis on each position of a ranking. Yet, motivated by applications in which some of the positions (typically those on the top) are more important than others, a few weighted variants of these measures have been proposed. Most of these generalizations fail to meet desirable formal properties, however. Besides, they are often quite inflexible in the sense of committing to a fixed weighing scheme. In this paper, we propose a weighted rank correlation measure on the basis of fuzzy order relations. Our measure, called scaled gamma, is related to Goodman and Kruskal's gamma rank correlation. It is parametrized by a fuzzy equivalence relation on the rank positions, which in turn is specified conveniently by a so-called scaling function. This approach combines soundness with flexibility: it has a sound formal foundation and allows for weighing rank positions in a flexible way.