Kendall tau metric

Kendall tau

The non-parametric correlation coefficient (or measure of association) known as Kendall's tau was first discussed by G.T. Fechner and others about 1900, and was rediscovered (independently) by M.G. Kendall in 1938 [a3], [a4]. In modern use, the term "correlation" refers to a measure of a linear relationship between variates (such as the Pearson product-moment correlation coefficient), while "measure of association" refers to a measure of a monotone relationship between variates (such as Kendall's tau and the Spearman rho metric). For a historical review of Kendall's tau and related coefficients, see [a5].

Underlying the definition of Kendall's tau is the notion of concordance. If and are two elements of a sample from a bivariate population, one says that and are concordant if and or if and (i.e., if ); and discordant if and or if and (i.e., if ). There are distinct pairs of observations in the sample, and each pair (barring ties) is either concordant or discordant. Denoting by the number of concordant pairs minus the number of discordant pairs, Kendall's tau for the sample is defined as

When ties exist in the data, the following adjusted formula is used:

where for the number of observations that are tied at a given rank, and for the number of observations that are tied at a given rank. For details on the use of in hypotheses testing, and for large-sample theory, see [a2].

Note that is equal to the probability of concordance minus the probability of discordance for a pair of observations and chosen randomly from the sample . The population version of Kendall's tau is defined similarly for random variables and (cf. also Random variable). Let and be independent random vectors with the same distribution as . Then

Since is the Pearson product-moment correlation coefficient of the random variables and , is sometimes called the difference sign correlation coefficient.

When and are continuous,

where is the copula of and . Consequently, is invariant under strictly increasing transformations of and , a property shares with Spearman's rho, but not with the Pearson product-moment correlation coefficient. For a survey of copulas and their relationship with measures of association, see [a6].

Besides Kendall's tau, there are other measures of association based on the notion of concordance, one of which is Blomqvist's coefficient [a1]. Let denote a sample from a continuous bivariate population, and let and denote sample medians (cf. also Median (in statistics)). Divide the -plane into four quadrants with the lines and ; and let be the number of sample points belonging to the first or third quadrants, and the number of points belonging to the second or fourth quadrants. If the sample size is even, the calculation of and is evident. If is odd, then one or two of the sample points fall on the lines and . In the first case one ignores the point; in the second case one assigns one point to the quadrant touched by both points and ignores the other. Then Blomqvist's is defined as

For details on the use of in hypothesis testing, and for large-sample theory, see [a1].

The population parameter estimated by , denoted by , is defined analogously to Kendall's tau (cf. Kendall tau metric). Denoting by and the population medians of and , then

where denotes the joint distribution function of and . Since depends only on the value of at the point whose coordinates are the population medians of and , it is sometimes called the medial correlation coefficient. When and are continuous,

where again denotes the copula of and . Thus , like , is invariant under strictly increasing transformations of and .

References

[a1]	N. Blomqvist, "On a measure of dependence between two random variables" Ann. Math. Stat. , 21 (1950) pp. 503–600
[a2]	J.D. Gibbons, "Nonparametric methods for quantitative analysis" , Holt, Rinehart & Winston (1976)
[a3]	M.G. Kendall, "A new measure of rank correlation" Biometrika , 30 (1938) pp. 81–93
[a4]	M.G. Kendall, "Rank correlation methods" , Charles Griffin (1970) (Edition: Fourth)
[a5]	W.H. Kruskal, "Ordinal measures of association" J. Amer. Statist. Assoc. , 53 (1958) pp. 814–861
[a6]	R.B. Nelsen, "An introduction to copulas" , Springer (1999)