Weighting by Tying: A New Approach to Weighted Rank Correlation
This provides a more adaptable solution for applications where certain ranking positions are more important, though it is incremental as it builds on existing rank correlation methods.
The paper tackles the problem of standard rank correlation measures lacking flexibility and desirable properties for weighted variants, proposing a new weighted measure called scaled gamma based on fuzzy order relations that is both formally sound and flexible.
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.