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

By Wicher P. Bergsma, Angelos Dassios. 2010

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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.

Subfields: Artificial Intelligence, Finance, Statistics and Probability

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Citations in the corpus

Invariant Causal Prediction for Nonlinear Models (2017, background): “There has been some progress towards nonparametric independence tests (Bergsma and Dassios, 2014; Hoeﬀding, 1948; Blum et al., 1961; R´enyi, 1959; Sz´ekely et al., 2007; Zhang et al., 2011).”
Kernel‐based tests for joint independence (2016, background): “, 2005, 2007), and other related approaches for estimating or testing pairwise (in-)dependence including distance correlations (Székely and Rizzo, 2009, 2014), rank-based correlations (Bergsma and Dassios, 2014; Leung and Drton, 2016; Nandy et al., 2016) or also non- and semiparametric copula-based ”
Testing independence in high dimensions with sums of rank correlations (2015, methodology): “…h = ht∗ or h = hD , (3.8) is an effective measure of the overall dependency of X(1), . . . ,X(m), since any pairwise signal θ(pq)h is non-negative and equals zero if and only if X(p) and X(q) are independent under the weak assumptions in the work of Hoeffding (1948a) and Bergsma and Dassios (2014).”
A New Coefficient of Correlation (2019, background): “…and ranks (Hoeffding 1948; Blum, Kiefer, and Rosenblatt 1961; Yanagimoto 1970; Puri and Sen 1971; Rosenblatt 1975; Csörgő 1985; Romano 1988; Bergsma and Dassios 2014; Nandy, Weihs, and Drton 2016; Weihs, Drton, and Leung 2016; Han, Chen, and Liu 2017; Wang, Jiang, and Liu 2017; Drton,…”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, background): “…the bivariate case with p = q = 1, Hoeffding (1940), Hoeffding (1948), Blum, Kiefer, and Rosenblatt (1961), Yanagimoto (1970), Feuerverger (1993), Bergsma and Dassios (2014), among many others, have proposed tests that are consistent against all alternatives from slightly different but rather…”
High-dimensional consistent independence testing with maxima of rank correlations (2018, methodology): “Bergsma and Dassios (2014) introduced a rank correlation statistic as a U-statistic of order 4 with the symmetric kernel hτ∗(z1, . . . ,z4) := 1 16 ∑ (i1,...,i4)∈P4 { 1(zi1,1, zi3,1 zi2,1, zi4,1) + 1(zi2,1, zi4,1 zi1,1, zi3,1) − 1(zi1,1, zi4,1 zi2,1, zi3,1)− 1(zi2,1, zi3,1 zi1,1, zi4,1) } {…”
Symmetric rank covariances: a generalized framework for nonparametric measures of dependence (2017, background): “Bergsma and Dassios (2014) defined τ ∗ only for bivariate distributions so let r = s = 1 for this section. While τ ∗ has a natural definition in terms of concordant and discordant quadruples of points, we will present an alternative definition that will be more useful for our purposes. First for any”
On boosting the power of Chatterjee’s rank correlation (2021, background): “For the sake of testing consistency, Hoeffding (1948), Blum et al. (1961), Yanagimoto (1970), and Bergsma and Dassios (2014) have proposed variants ∗Department of Statistics, University of Washington, Seattle, WA 98195, USA; e-mail: zxlin@uw.edu †Department of Statistics, University of Washington,…”
Nonparametric independence testing via mutual information (2017, methodology): “…R package dHSIC (Pﬁster and Peters, 2017), a test based on the distance covariance implemented in the R package energy (Rizzo and Szekely, 2017) and the improvement of Hoeﬀding’s test described in Bergsma and Dassios (2014) and implemented in the R package TauStar (Weihs, Drton and Leung, 2016).”
Predicting the effect of interventions using invariance principles for nonlinear models (2016, background): “There has been some progress towards nonparametric independence tests [1, 7, 2, 12, 14, 16].”
On the power of Chatterjee’s rank correlation (2020, background): “3(iv) still holds for F c replaced by Fτ ∗ (Theorem 1 in Bergsma and Dassios (2014), Theorem 6.”
Testing mutual independence in high dimensions with sums of squares of rank correlations (2015, background): “(2007) and Bergsma and Dassios (2014), which allow for a refined assessment of dependence.”