Advances and Applications in Statistics
Volume 36, Issue 2, Pages 83 - 104
(October 2013)
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ON PINTO DA COSTA AND SOARES’ AND OTHER WEIGHTED RANK CORRELATION MEASURES DERIVING FROM THE SPEARMAN’S RHO
Livia Dancelli, Marica Manisera and Marika Vezzoli
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Abstract: A user who chooses a weighted rank correlation index certainly requires that this gives the right emphasis to the exchanges on the most important ranks and does not return the same value of the related unweighted measure. In 2005, Pinto da Costa and Soares proposed an index introducing linear weights in the simplified formula of the Spearman’s r. We show that it can take the same value at pairs of rankings visibly different on the top and can even return the same value of r. These issues are connected with the concepts of discriminatory power and resistance to change recently applied in the literature to various rank correlation measures. Other existing indices, whichintroducenonlinearweights in the simplified formula of r, seem less resistant. The same is for indices derived by directly introducing weights in the equivalent formula of the Pearson’s product-moment correlation index between ranks.
This contribution may be interesting for users looking for an accurate weighted measurement of the agreement between ranks, wherever the discriminatory power of the rank correlation coefficient ought to be particularly strong. The knowledge of this feature of competing indices can address their choices. For example, in some marketing applications involving a large number of short rankings, we needed a highly responsive weighted index able to clearly distinguish between pairs of rankings. In such cases, the awareness that the Pinto da Costa and Soares’ index has a low discriminatory power and sometimes its weighting system is totally ineffective, has discouraged its use. |
Keywords and phrases: rankings, weighted concordance measures, linear and nonlinear symmetric weights, Blest’s symmetrized index. |
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