• Bernoulli
  • Volume 20, Number 2 (2014), 1006-1028.

A consistent test of independence based on a sign covariance related to Kendall’s tau

Wicher Bergsma and Angelos Dassios

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The most popular ways to test for independence of two ordinal random variables are by means of Kendall’s tau and Spearman’s rho. However, such tests are not consistent, only having power for alternatives with “monotonic” association. In this paper, we introduce a natural extension of Kendall’s tau, called $\tau^{*}$, which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measures of dependence for ordinal random variables is given, and it is shown that, in a well-defined sense, $\tau^{*}$ is the simplest, similarly to Kendall’s tau being the simplest of ordinal measures of monotone association. Simulation studies show our test compares well with the alternatives in terms of average $p$-values.

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Bernoulli, Volume 20, Number 2 (2014), 1006-1028.

First available in Project Euclid: 28 February 2014

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concordance copula discordance measure of association ordinal data permutation test sign test


Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based on a sign covariance related to Kendall’s tau. Bernoulli 20 (2014), no. 2, 1006--1028. doi:10.3150/13-BEJ514.

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  • [1] Agresti, A. (2010). Analysis of Ordinal Categorical Data, 2nd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley.
  • [2] Bergsma, W.P. (2006). A new correlation coefficient, its orthogonal decomposition, and associated tests of independence. Available at arXiv:math/0604627v1 [math.ST].
  • [3] Bergsma, W. and Dassios, A. (2014). Supplement to “A consistent test of independence based on a sign covariance related to Kendall’s tau.” DOI:10.3150/13-BEJ514SUPP.
  • [4] Blum, J.R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function. Ann. Math. Statist. 32 485–498.
  • [5] de Wet, T. (1980). Cramér–von Mises tests for independence. J. Multivariate Anal. 10 38–50.
  • [6] Deheuvels, P. (1981). An asymptotic decomposition for multivariate distribution-free tests of independence. J. Multivariate Anal. 11 102–113.
  • [7] Escoufier, Y. (1973). Le traitement des variables vectorielles. Biometrics 29 751–760.
  • [8] Feuerverger, A. (1993). A consistent test for bivariate dependence. International Statistical Review/Revue Internationale de Statistique 61 419–433.
  • [9] Gretton, A., Bousquet, O., Smola, A. and Schölkopf, B. (2005). Measuring statistical dependence with Hilbert–Schmidt norms. In Algorithmic Learning Theory. Lecture Notes in Computer Science 3734 63–77. Berlin: Springer.
  • [10] Heller, R., Heller, Y. and Gorfine, M. (2012). A consistent multivariate test of association based on ranks of distances. Available at arXiv:1201.3522.
  • [11] Hoeffding, W. (1948). A non-parametric test of independence. Ann. Math. Statistics 19 546–557.
  • [12] Hollander, M. and Wolfe, D.A. (1999). Nonparametric Statistical Methods, 2nd ed. Wiley Series in Probability and Statistics: Texts and References Section. New York: Wiley.
  • [13] Kendall, M. and Gibbons, J.D. (1990). Rank Correlation Methods, 5th ed. A Charles Griffin Title. London: Edward Arnold.
  • [14] Kendall, M.G. (1938). A new measure of rank correlation. Biometrika 30 81–93.
  • [15] Kimeldorf, G. and Sampson, A.R. (1978). Monotone dependence. Ann. Statist. 6 895–903.
  • [16] Kruskal, W.H. (1958). Ordinal measures of association. J. Amer. Statist. Assoc. 53 814–861.
  • [17] Lyons, R. (2013). Distance covariance in metric spaces. Ann. Probab. 41 3284–3305.
  • [18] Nelsen, R.B. (2006). An Introduction to Copulas, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [19] Rényi, A. (1959). On measures of dependence. Acta Math. Acad. Sci. Hungar. 10 441–451.
  • [20] Robert, P. and Escoufier, Y. (1976). A unifying tool for linear multivariate statistical methods: The $RV$-coefficient. J. R. Stat. Soc. Ser. C. Appl. Stat. 25 257–265.
  • [21] Schweizer, B. and Wolff, E.F. (1981). On nonparametric measures of dependence for random variables. Ann. Statist. 9 879–885.
  • [22] Sejdinovic, D., Gretton, A., Sriperumbudur, B. and Fukumizu, K. (2012). Hypothesis testing using pairwise distances and associated kernels. In Proc. International Conference on Machine Learning. Edinburgh, UK: ICML.
  • [23] Sejdinovic, D., Sriperumbudur, B., Gretton, A. and Fukumizu, K. (2012). Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Available at arXiv:1207.6076.
  • [24] Sheskin, D.J. (2007). Handbook of Parametric and Nonparametric Statistical Procedures, 4th ed. Boca Raton, FL: Chapman & Hall/CRC.
  • [25] Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology 15 72–101.
  • [26] Székely, G.J., Rizzo, M.L. and Bakirov, N.K. (2007). Measuring and testing dependence by correlation of distances. Ann. Statist. 35 2769–2794.
  • [27] Wilding, G.E. and Mudholkar, G.S. (2008). Empirical approximations for Hoeffding’s test of bivariate independence using two Weibull extensions. Stat. Methodol. 5 160–170.

Supplemental materials

  • Supplementary material: A shorter proof of the main theorem for the continuous case and some miscellaneous further results. The supplement contains the following results: (i) a shorter proof of the main theorem, but only for the continuous case, (ii) the Cramér von Mises test as a special case, (iii) a shorter proof of main theorem for the case that one of the variables is binary, and (iv) a result for an extension to the case of variables in metric spaces.