In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the association between two measured quantities. A tau test is a nonparametric hypothesis test for statistical dependence based on the tau coefficient.
It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities. It is named after Maurice Kendall, who developed it in 1938,^{[1]} though Gustav Fechner had proposed a similar measure in the context of time series in 1897.^{[2]}
Contents

Definition 1

Hypothesis test 2

Accounting for ties 3

Taua 3.1

Taub 3.2

Tauc 3.3

Significance tests 4

Algorithms 5

See also 6

References 7

Further reading 8

External links 9
Definition
Let (x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n}) be a set of observations of the joint random variables X and Y respectively, such that all the values of (x_{i}) and (y_{i}) are unique. Any pair of observations (x_{i}, y_{i}) and (x_{j}, y_{j}) are said to be concordant if the ranks for both elements agree: that is, if both x_{i} > x_{j} and y_{i} > y_{j} or if both x_{i} < x_{j} and y_{i} < y_{j}. They are said to be discordant, if x_{i} > x_{j} and y_{i} < y_{j} or if x_{i} < x_{j} and y_{i} > y_{j}. If x_{i} = x_{j} or y_{i} = y_{j}, the pair is neither concordant nor discordant.
The Kendall τ coefficient is defined as:

\tau = \frac{(\text{number of concordant pairs})  (\text{number of discordant pairs})}{\frac{1}{2} n (n1) } .^{[3]}
Properties
The denominator 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.

If X and Y are independent, then we would expect the coefficient to be approximately zero.
Hypothesis test
The Kendall rank coefficient is often used as a test statistic in a statistical hypothesis test to establish whether two variables may be regarded as statistically dependent. This test is nonparametric, as it does not rely on any assumptions on the distributions of X or Y or the distribution of (X,Y).
Under the null hypothesis of independence of X and Y, the sampling distribution of τ has an expected 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 the normal distribution, with mean zero and variance

\frac{2(2n+5)}{9n (n1)}.^{[4]}
Accounting for ties
A pair {(x_{i}, y_{i}), (x_{j}, y_{j})} is said to be tied if x_{i} = x_{j} or 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]:
Taua
The Taua statistic tests the strength of association of the cross tabulations. Both variables have to be ordinal. Taua will not make any adjustment for ties. It is defined as:

\tau_A = \frac{n_cn_d}{n_0}
where n_{c}, n_{d} and n_{0} are defined as in the next section.
Taub
The Taub statistic, unlike Taua, makes adjustments for ties.^{[5]} Values of Taub range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
The Kendall Taub coefficient is defined as:

\tau_B = \frac{n_cn_d}{\sqrt{(n_0n_1)(n_0n_2)}}
where

\begin{align} n_0 & = n(n1)/2\\ n_1 & = \sum_i t_i (t_i1)/2 \\ n_2 & = \sum_j u_j (u_j1)/2 \\ n_c & = \text{Number of concordant pairs} \\ n_d & = \text{Number of discordant pairs} \\ t_i & = \text{Number of tied values in the } i^\text{th} \text{ group of ties for the first quantity} \\ u_j & = \text{Number of tied values in the } j^\text{th} \text{ group of ties for the second quantity} \end{align}
Tauc
Tauc differs from Taub as in being more suitable for rectangular tables than for square tables.
Significance tests
When two quantities are statistically independent, the distribution of \tau is not easily characterizable in terms of known distributions. However, for \tau_A the following statistic, z_A, is approximately distributed as a standard normal when the variables are statistically independent:

z_A = {(n_c  n_d) \over \sqrt{2(2n+5)\over 9n(n1)} }
Thus, to test whether two variables are statistically dependent, one computes z_A, and finds the cumulative probability for a standard normal distribution at z_A. For a 2tailed test, multiply that number by two to obtain the pvalue. If the pvalue 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 z_A when accounting for ties. The following statistic, z_B, has the same distribution as the \tau_B distribution, and is again approximately equal to a standard normal distribution when the quantities are statistically independent:

z_B = {n_c  n_d \over \sqrt{ v } }
where

\begin{array}{ccl} v & = & (v_0  v_t  v_u)/18 + v_1 + v_2 \\ v_0 & = & n (n1) (2n+5) \\ v_t & = & \sum_i t_i (t_i1) (2 t_i+5)\\ v_u & = & \sum_j u_j (u_j1)(2 u_j+5) \\ v_1 & = & \sum_i t_i (t_i1) \sum_j u_j (u_j1) / (2n(n1)) \\ v_2 & = & \sum_i t_i (t_i1) (t_i2) \sum_j u_j (u_j1) (u_j2) / (9 n (n1) (n2)) \end{array}
pvrank^{[6]} is a very recent R package that computes rank correlations and their pvalues with various options for tied ranks. It is possible to compute exact Kendall coefficient test pvalues for n ≤ 60.
Algorithms
The direct computation of the numerator n_c  n_d, involves two nested iterations, as characterized by the following pseudocode:
numer := 0
for i:=2..N do
for j:=1..(i1) do
numer := numer + sign(x[i]  x[j]) * sign(y[i]  y[j])
return numer
Although quick to implement, this algorithm is O(n^2) in complexity and becomes very slow on large samples. A more sophisticated algorithm^{[7]} built upon the Merge Sort algorithm can be used to compute the numerator in O(n \cdot \log{n}) time.
Begin by ordering your data points sorting by the first quantity, x, and secondarily (among ties in x) by the second quantity, y. With this initial ordering, y is not sorted, and the core of the algorithm consists of computing how many steps a Bubble Sort would take to sort this initial y. An enhanced Merge Sort algorithm, with O(n \log n) complexity, can be applied to compute the number of swaps, S(y), that would be required by a Bubble Sort to sort y_i. Then the numerator for \tau is computed as:

n_cn_d = n_0  n_1  n_2 + n_3  2 S(y),
where n_3 is computed like n_1 and n_2, but with respect to the joint ties in x and y.
A Merge Sort partitions the data to be sorted, y into two roughly equal halves, y_\mathrm{left} and y_\mathrm{right}, then sorts each half recursive, and then merges the two sorted halves into a fully sorted vector. The number of Bubble Sort swaps is equal to:

S(y) = S(y_\mathrm{left}) + S(y_\mathrm{right}) + M(Y_\mathrm{left},Y_\mathrm{right})
where Y_\mathrm{left} and Y_\mathrm{right} are the sorted versions of y_\mathrm{left} and y_\mathrm{right}, and M(\cdot,\cdot) characterizes the Bubble Sort swapequivalent for a merge operation. M(\cdot,\cdot) is computed as depicted in the following pseudocode:
function M(L[1..n], R[1..m])
i := 1
j := 1
nSwaps := 0
while i <= n and j <= m do
if R[j] < L[i] then
nSwaps := nSwaps + n  i + 1
j := j + 1
else
i := i + 1
return nSwaps
A side effect of the above steps is that you end up with both a sorted version of x and a sorted version of y. With these, the factors t_i and u_j used to compute \tau_B are easily obtained in a single lineartime pass through the sorted arrays.
A second algorithm with O(n \cdot \log{n}) time complexity, based on AVL trees, was devised by David Christensen.^{[8]} Yet another algorithm for O(n \cdot \log{n}) time complexity was proposed more recently.^{[9]}
See also
References

^ Kendall, M. (1938). "A New Measure of Rank Correlation".

^

^ Nelsen, R.B. (2001), "Kendall tau metric", in Hazewinkel, Michiel,

^ Prokhorov, A.V. (2001), "Kendall coefficient of rank correlation", in Hazewinkel, Michiel,

^ Agresti, A. (2010). Analysis of Ordinal Categorical Data (Second ed.). New York: John Wiley & Sons.

^ Amerise, I.L.; Marozzi, M.; Tarsitano, A. "R package pvrank".

^ Knight, W. (1966). "A Computer Method for Calculating Kendall's Tau with Ungrouped Data".

^ Christensen, David (2005). "Fast algorithms for the calculation of Kendall's τ".

^ Campello, R.J.G.B.; Hruschka, E.R. (29 March 2009). "On comparing two sequences of numbers and its applications to clustering analysis". Information Sciences 179 (8): 1025–1039.
Further reading

Abdi, H. (2007). "Kendall rank correlation" (PDF). In Salkind, N.J. Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.

Daniel, Wayne W. (1990). "Kendall's tau". Applied Nonparametric Statistics (2nd ed.). Boston: PWSKent. pp. 365–377.

Kendall, M. (1948) Rank Correlation Methods, Charles Griffin & Company Limited

Bonett, DG & Wright, TA (2000) Sample size requirements for Pearson, Kendall, and Spearman correlations, Psychometrika, 65, 23–28.
External links

Tied rank calculation

Software for computing Kendall's tau on very large datasets

Online software: computes Kendall's tau rank correlation

The CORR Procedure: Statistical Computations – McDonough School of Business
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