Kendall tau rank correlation coefficient
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau (τ) coefficient, 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.
Specifically, it is a measure of rank correlation, i.e., 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
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:
 ^{3}
Properties
The denominator is the total number 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
 .^{4}
Accounting for ties
This article needs additional citations for verification. (June 2010) 
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:
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:
where
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 is not easily characterizable in terms of known distributions. However, for the following statistic, , is approximately distributed as a standard normal when the variables are statistically independent:
Thus, to test whether two variables are statistically dependent, one computes , and finds the cumulative probability for a standard normal distribution at . 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 when accounting for ties. The following statistic, , has the same distribution as the distribution, and is again approximately equal to a standard normal distribution when the quantities are statistically independent:
where
Algorithms
The direct computation of the numerator , 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 in complexity and becomes very slow on large samples. A more sophisticated algorithm^{6} built upon the Merge Sort algorithm can be used to compute the numerator in time.
Begin by ordering your data points sorting by the first quantity, , and secondarily (among ties in ) by the second quantity, . With this initial ordering, is not sorted, and the core of the algorithm consists of computing how many steps a Bubble Sort would take to sort this initial . An enhanced Merge Sort algorithm, with complexity, can be applied to compute the number of swaps, , that would be required by a Bubble Sort to sort . Then the numerator for is computed as:
 ,
where is computed like and , but with respect to the joint ties in and .
A Merge Sort partitions the data to be sorted, into two roughly equal halves, and , 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:
where and are the sorted versions of and , and characterizes the Bubble Sort swapequivalent for a merge operation. 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 and a sorted version of . With these, the factors and used to compute are easily obtained in a single lineartime pass through the sorted arrays.
A second algorithm with time complexity, based on AVL trees, was devised by David Christensen.^{7}
See also
 Correlation
 Kendall tau distance
 Kendall's W
 Spearman's rank correlation coefficient
 Goodman and Kruskal's gamma
 Theil–Sen estimator
References
 ^ Kendall, M. (1938). "A New Measure of Rank Correlation". Biometrika 30 (1–2): 81–89. doi:10.1093/biomet/30.12.81. JSTOR 2332226.
 ^ Kruskal, W.H. (1958). "Ordinal Measures of Association". Journal of the American Statistical Association 53 (284): 814–861. doi:10.2307/2281954. JSTOR 2281954. MR 100941.
 ^ Nelsen, R.B. (2001), "Kendall tau metric", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 ^ Prokhorov, A.V. (2001), "Kendall coefficient of rank correlation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 ^ Agresti, A. (2010). Analysis of Ordinal Categorical Data, Second Edition. New York, John Wiley & Sons.
 ^ Knight, W. (1966). "A Computer Method for Calculating Kendall's Tau with Ungrouped Data". Journal of the American Statistical Association 61 (314): 436–439. doi:10.2307/2282833. JSTOR 2282833.
 ^ Christensen, David (2005). "Fast algorithms for the calculation of Kendall's τ". Computational Statistics 20 (1): 51–62. doi:10.1007/BF02736122.
 Abdi, H. (2007). "Kendall rank correlation". In Salkind, N.J. Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.
 Kendall, M. (1948) Rank Correlation Methods, Charles Griffin & Company Limited
External links
 Tied rank calculation
 Why Kendall tau?
 Software for computing Kendall's tau on very large datasets
 Online software: computes Kendall's tau rank correlation
 The CORR Procedure: Statistical Computations

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