Abstract
Weighted Rank Correlation indices are useful for measuring the agreement of two rankings when the top ranks are considered more important than the lower ones. This paper investigates, from a descriptive perspective, the behaviour of (i) five existing indices that introduce suitable weights in the simplified formula of the Spearman’s ρ and (ii) an additional five indices we derive using the same weights in the Pearson’s product-moment correlation index between ranks. For their evaluation, we consider that a good Weighted Rank Correlation index should (1) differ from ρ, if computed on the same pair of rankings and (2) assume a broad variety of values in the range \([-1,+1]\), in order to better discriminate amongst different reorderings of the ranks. Results suggest that linear weights should be avoided and show that indices (ii) do not have equalities with ρ and are more sensitive.
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Notes
- 1.
Computations were obtained by the statistical software R 2.13.2 with full precision.
- 2.
An inverse permutation B inv of B is obtained by substituting each number with the number of the place it occupies. A WRC index with symmetric weights must give the same result when computed between A and B and between B inv and A, because the pairs \((a_{i},b_{i})\) and \((b_{i}^{\mathit{inv}},a_{i})\) are the same. Then, one of the two rankings, B or B inv, must be excluded from the n! permutations, unless B and B inv coincide. For example, let A : 1, 2, 3, 4, 5, 6 and B : 2, 3, 1, 5, 4, 6. The inverse permutation of B is B inv : 3, 1, 2, 5, 4, 6. It is evident that the pairs are the same if we rewrite B inv in the natural order (B inv′ : 1, 2, 3, 4, 5, 6) and, consequently, rearrange A, obtaining A′ : 2, 3, 1, 5, 4, 6.
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We wish to thank the anonymous referee for his/her comments that greatly improved the quality of the paper.
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Dancelli, L., Manisera, M., Vezzoli, M. (2013). On Two Classes of Weighted Rank Correlation Measures Deriving from the Spearman’s ρ . In: Giudici, P., Ingrassia, S., Vichi, M. (eds) Statistical Models for Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00032-9_13
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DOI: https://doi.org/10.1007/978-3-319-00032-9_13
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