Instatistics, theKendall rank correlation coefficient, commonly referred to asKendall's τ coefficient (after the Greek letterτ, tau), is astatistic used to measure theordinal association between two measured quantities. Aτ test is anon-parametrichypothesis test for statistical dependence based on the τ coefficient. It is a measure ofrank correlation: the similarity of the orderings of the data whenranked by each of the quantities. It is named afterMaurice Kendall, who developed it in 1938,^[1] thoughGustav Fechner had proposed a similar measure in the context oftime series in 1897.^[2]

Intuitively, the Kendall correlation between two variables will be high when observations have a similar (or identical for a correlation of 1)rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully different for a correlation of −1) rank between the two variables.

Both Kendall's${\displaystyle \tau }$ andSpearman's${\displaystyle \rho }$ can be formulated as special cases of a moregeneral correlation coefficient. Its notions ofconcordance and discordance also appear in other areas of statistics, like theRand index incluster analysis.

Let${\displaystyle (x_1,y_1),...,(x_n,y_n)}$ be a set of observations of the joint random variablesX andY, such that all the values of (${\displaystyle x_i}$) and (${\displaystyle y_i}$) are unique. (See the section#Accounting for ties for ways of handling non-unique values.) Any pair of observations${\displaystyle (x_i,y_i)}$ and${\displaystyle (x_j,y_j)}$, where, are said to beconcordant if the sort order of${\displaystyle (x_i,x_j)}$ and${\displaystyle (y_i,y_j)}$ agrees: that is, if either both${\displaystyle x_i>x_j}$ and${\displaystyle y_i>y_j}$ holds or both and; otherwise they are said to bediscordant.

In the absence of ties, the Kendall τ coefficient is defined as:

${\displaystyle \tau =\frac{(\text{number of concordant pairs})-(\text{number of discordant pairs})}{(\text{number of pairs})}=1-\frac{2(\text{number of discordant pairs})}{(\genfrac{}{}{0ex}{}{n}{2})}.}$^[3]

for where${\displaystyle (\genfrac{}{}{0ex}{}{n}{2})=\frac{n(n-1)}{2}}$ is thebinomial coefficient for the number of ways to choose two items from n items.

The number of discordant pairs is equal to theinversion number that permutes the y-sequence into the same order as the x-sequence.

Thedenominator is the total number of pair combinations, so the coefficient must be in the range −1 ≤ τ ≤ 1.

• If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value 1.
• If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value −1.
• IfX andY areindependent random variables and not constant, then the expectation of the coefficient is zero.
• An explicit expression for Kendall's rank coefficient is.

The Kendall rank coefficient is often used as atest statistic in astatistical hypothesis test to establish whether two variables may be regarded as statistically dependent. This test isnon-parametric, as it does not rely on any assumptions on the distributions ofX orY or the distribution of (X,Y).

Under thenull hypothesis of independence ofX andY, thesampling distribution ofτ has anexpected value of zero. The precise distribution cannot be characterized in terms of common distributions, but may be calculated exactly for small samples; for larger samples, it is common to use an approximation to thenormal distribution, with mean zero and variance$2(2n+5)/9n(n-1)$.^[4]

Theorem. If the samples are independent, then the variance of${\tau }_A$ is given by$Var[{\tau }_A]=2(2n+5)/9n(n-1)$.

If$(x_1,y_1),(x_2,y_2),...,(x_n,y_n)$ are IID samples from the same jointly normal distribution with a knownPearson correlation coefficient$r$, then the expectation of Kendall rank correlation has a closed-form formula.^[8]

The name is credited to Richard Greiner (1909)^[9] byP. A. P. Moran.^[10]

A pair${\displaystyle \{(x_i,y_i),(x_j,y_j)\}}$ is said to betied if and only if${\displaystyle x_i=x_j}$ or${\displaystyle y_i=y_j}$; a tied pair is neither concordant nor discordant. When tied pairs arise in the data, the coefficient may be modified in a number of ways to keep it in the range [−1, 1]:

The Tau statistic defined by Kendall in 1938^[1] was retrospectively renamed Tau-a. It represents the strength of positive or negative association of two quantitative orordinal variables without any adjustment for ties. It is defined as:

${\displaystyle {\tau }_A=\frac{n_c-n_d}{n_0}}$

wheren_c,n_d andn₀ are defined as in the next section.

When ties are present, and, the coefficient can never be equal to +1 or -1. Even a perfect equality of the two variables (X=Y) leads to a Tau-a < 1.

The Tau-b statistic, unlike Tau-a, makes adjustments for ties.This Tau-b was first described by Kendall in 1945 under the name Tau-w^[12] as an extension of the original Tau statistic supporting ties.Values of Tau-b range from −1 (100% negative association, or perfect disagreement) to +1 (100% positive association, or perfect agreement). In case of the absence of association, Tau-b is equal to zero.

The Kendall Tau-b coefficient is defined as :

${\displaystyle {\tau }_B=\frac{n_c-n_d}{\sqrt{(n_0-n_1)(n_0-n_2)}}}$

where

A simple algorithm developed in BASIC computes Tau-b coefficient using an alternative formula.^[13]

Be aware that some statistical packages, e.g. SPSS, use alternative formulas for computational efficiency, with double the 'usual' number of concordant and discordant pairs.^[14]

Tau-c (also called Stuart-Kendall Tau-c)^[15] was first defined by Stuart in 1953.^[16]Contrary to Tau-b, Tau-c can be equal to +1 or -1 for non-square (i.e. rectangular)contingency tables,^[15][16] i.e. when the underlying scale of both variables have different number of possible values. For instance, if the variable X has a continuous uniform distribution between 0 and 100 and Y is a dichotomous variable equal to 1 if X ≥ 50 and 0 if X < 50, the Tau-c statistic of X and Y is equal to 1 while Tau-b is equal to 0.707. A Tau-C equal to 1 can be interpreted as the best possible positive correlation conditional to marginal distributions while a Tau-B equal to 1 can be interpreted as the perfect positive monotonic correlation where the distribution of X conditional to Y has zero variance and the distribution of Y conditional to X has zero variance so that a bijective function f with f(X)=Y exists.

The Stuart-Kendall Tau-c coefficient is defined as:^[16]

${\displaystyle {\tau }_C=\frac{2(n_c-n_d)}{n^2\frac{(m-1)}{m}}={\tau }_A\frac{n-1}{n}\frac{m}{m-1}}$

where

When two quantities are statistically dependent, the distribution of${\displaystyle \tau }$ is not easily characterizable in terms of known distributions. However, for${\displaystyle {\tau }_A}$ the following statistic,${\displaystyle z_A}$, is approximately distributed as a standard normal when the variables are statistically independent:

${\displaystyle z_A=\frac{n_c-n_d}{\sqrt{\frac{1}{18}{v}_0}}}$

where${\displaystyle v_0=n(n-1)(2n+5)}$.

Thus, to test whether two variables are statistically dependent, one computes${\displaystyle z_A}$, and finds the cumulative probability for a standard normal distribution at${\displaystyle -|z_A|}$. For a 2-tailed test, multiply that number by two to obtain thep-value. If thep-value is below a given significance level, one rejects the null hypothesis (at that significance level) that the quantities are statistically independent.

Numerous adjustments should be added to${\displaystyle z_A}$ when accounting for ties. The following statistic,${\displaystyle z_B}$, has the same distribution as the${\displaystyle {\tau }_B}$ distribution, and is again approximately equal to a standard normal distribution when the quantities are statistically independent:

${\displaystyle z_B=\frac{n_c-n_d}{\sqrt{v}}}$

where

This is sometimes referred to as the Mann-Kendall test.^[17]

The direct computation of the numerator${\displaystyle n_c-n_d}$, involves two nested iterations, as characterized by the following pseudocode:

numer := 0for i := 2..Ndofor j := 1..(i − 1)do        numer := numer + sign(x[i] − x[j]) × sign(y[i] − y[j])return numer

Although quick to implement, this algorithm is${\displaystyle O(n^2)}$ in complexity and becomes very slow on large samples. A more sophisticated algorithm^[18] built upon theMerge Sort algorithm can be used to compute the numerator in${\displaystyle O(n\cdot \log n)}$ time.

Begin by ordering your data points sorting by the first quantity,${\displaystyle x}$, and secondarily (among ties in${\displaystyle x}$) by the second quantity,${\displaystyle y}$. With this initial ordering,${\displaystyle y}$ is not sorted, and the core of the algorithm consists of computing how many steps aBubble Sort would take to sort this initial${\displaystyle y}$. An enhancedMerge Sort algorithm, with${\displaystyle O(n\log n)}$ complexity, can be applied to compute the number of swaps,${\displaystyle S(y)}$, that would be required by aBubble Sort to sort${\displaystyle y_i}$. Then the numerator for${\displaystyle \tau }$ is computed as:

${\displaystyle n_c-n_d=n_0-n_1-n_2+n_3-2S(y),}$

where${\displaystyle n_3}$ is computed like${\displaystyle n_1}$ and${\displaystyle n_2}$, but with respect to the joint ties in${\displaystyle x}$ and${\displaystyle y}$.

AMerge Sort partitions the data to be sorted,${\displaystyle y}$ into two roughly equal halves,${\displaystyle y_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}}$ and${\displaystyle y_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}$, then sorts each half recursive, and then merges the two sorted halves into a fully sorted vector. The number ofBubble Sort swaps is equal to:

${\displaystyle S(y)=S(y_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}})+S(y_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}})+M(Y_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}},Y_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}})}$

where${\displaystyle Y_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}}$ and${\displaystyle Y_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}$ are the sorted versions of${\displaystyle y_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}}$ and${\displaystyle y_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}$, and${\displaystyle M(\cdot ,\cdot )}$ characterizes theBubble Sort swap-equivalent for a merge operation.${\displaystyle M(\cdot ,\cdot )}$ is computed as depicted in the following pseudo-code:

function M(L[1..n], R[1..m])is    i := 1    j := 1    nSwaps := 0while i ≤ nand j ≤ mdoif R[j] < L[i]then            nSwaps := nSwaps + n − i + 1            j := j + 1else            i := i + 1return nSwaps

A side effect of the above steps is that you end up with both a sorted version of${\displaystyle x}$ and a sorted version of${\displaystyle y}$. With these, the factors${\displaystyle t_i}$ and${\displaystyle u_j}$ used to compute${\displaystyle {\tau }_B}$ are easily obtained in a single linear-time pass through the sorted arrays.

Efficient algorithms for calculating the Kendall rank correlation coefficient as per the standard estimator have${\displaystyle O(n\cdot \log n)}$ time complexity. However, these algorithms necessitate the availability of all data to determine observation ranks, posing a challenge in sequential data settings where observations are revealed incrementally. Fortunately, algorithms do exist to estimate the Kendall rank correlation coefficient in sequential settings.^[19][20] These algorithms have${\displaystyle O(1)}$ update time and space complexity, scaling efficiently with the number of observations. Consequently, when processing a batch of${\displaystyle n}$ observations, the time complexity becomes${\displaystyle O(n)}$, while space complexity remains a constant${\displaystyle O(1)}$.

The first such algorithm^[19] presents an approximation to the Kendall rank correlation coefficient based on coarsening the joint distribution of the random variables. Non-stationary data is treated via a moving window approach. This algorithm^[19] is simple and is able to handle discrete random variables along with continuous random variables without modification.

The second algorithm^[20] is based on Hermite series estimators and utilizes an alternative estimator for the exact Kendall rank correlation coefficient i.e. for the probability of concordance minus the probability of discordance of pairs of bivariate observations. This alternative estimator also serves as an approximation to the standard estimator. This algorithm^[20] is only applicable to continuous random variables, but it has demonstrated superior accuracy and potential speed gains compared to the first algorithm described,^[19] along with the capability to handle non-stationary data without relying on sliding windows. An efficient implementation of the Hermite series based approach is contained in the R package packagehermiter.^[20]

• R implements the test for${\displaystyle {\tau }_B}$cor.test(x, y, method = "kendall") in its "stats" package (alsocor(x, y, method = "kendall") will work, but the latter does not return the p-value). All three versions of the coefficient are available in the "DescTools" package along with the confidence intervals:KendallTauA(x,y,conf.level=0.95) for${\displaystyle {\tau }_A}$,KendallTauB(x,y,conf.level=0.95) for${\displaystyle {\tau }_B}$,StuartTauC(x,y,conf.level=0.95) for${\displaystyle {\tau }_C}$. Fast batch estimates of the Kendall rank correlation coefficient along with sequential estimates are provided for in the packagehermiter.^[20]
• ForPython, theSciPy library implements the computation of${\displaystyle {\tau }_B}$ inscipy.stats.kendalltau
• InStata is implemeted asktauvarlist.

• Correlation
• Kendall tau distance
• Kendall's W
• Spearman's rank correlation coefficient
• Goodman and Kruskal's gamma
• Theil–Sen estimator
• Mann–Whitney U test - it is equivalent to Kendall's tau correlation coefficient if one of the variables is binary.

• Abdi, H. (2007)."Kendall rank correlation"(PDF). In Salkind, N.J. (ed.).Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.
• Daniel, Wayne W. (1990)."Kendall's tau".Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 365–377.ISBN 978-0-534-91976-4.
• Kendall, Maurice;Gibbons, Jean Dickinson (1990) [First published 1948].Rank Correlation Methods. Charles Griffin Book Series (5th ed.). Oxford: Oxford University Press.ISBN 978-0195208375.
• Bonett, Douglas G.; Wright, Thomas A. (2000). "Sample size requirements for estimating Pearson, Kendall, and Spearman correlations".Psychometrika.65 (1):23–28.doi:10.1007/BF02294183.S2CID 120558581.

• Tied rank calculation
• Software for computing Kendall's tau on very large datasets
• Online software: computes Kendall's tau rank correlation

• Covariance and correlation
• Nonparametric statistics
• Statistical tests
• Independence (probability theory)

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