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Concordance between two linear orders: The Spearman and Kendall coefficients revisited

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Abstract

This paper discusses the two classic measures of concordance between two linear orders L and L′, the Kendall tau and the Spearman rho, equivalently, the Kendall and Spearman distances between such orders. We give an expression for ρ(L,L′)−τ(L,L′) as a function of the parameters of the partial order L∪L′, which allows the determination of extremal values for this difference and an investigation of when tau and rho are equal. This expression for ρ(L,L′)−τ(L,L′) is derived from a relation between the Kendall and Spearman distances between linear orders that is equivalent to both the Guilbaud (1980) formula linking rho, tau, and a third coefficient sigma, and Daniels’s (1950) inequality. We also prove an apparently new monotonicity property of rho. In the conclusion we point out possible extensions and add general historical comments.

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Authors and Affiliations

  1. CERMSEM, Université Paris I Panthéon Sorbonne, 90 rue de Tolbiac, 75634, Paris Cédex 13

    B. Monjardet

  2. CAMS, EHESS, 54 bd. Raspail, 75270, Paris Cédex 06

    B. Monjardet

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  1. B. Monjardet
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Correspondence to B. Monjardet.

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The author wrote this paper during a sabbatical leave spent at the GERAD and at the Département de Mathématiques et de Statistique de l’Université de Montréal. He is glad to thank them for their kind hospitality and support. He thanks also the referees, P. Hansen and P. Arabie for their useful remarks and important editorial help on the first versions of this paper.

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Monjardet, B. Concordance between two linear orders: The Spearman and Kendall coefficients revisited. Journal of Classification 14, 269–295 (1997). https://doi.org/10.1007/s003579900013

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