Kendall rank correlation coefficient

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In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities. A τ test is a non-parametric hypothesis test for statistical dependence based on the τ 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]

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 [math]\displaystyle{ \tau }[/math] and Spearman's [math]\displaystyle{ \rho }[/math] can be formulated as special cases of a more general correlation coefficient.

Definition

Let [math]\displaystyle{ (x_1,y_1), ..., (x_n,y_n) }[/math] be a set of observations of the joint random variables X and Y, such that all the values of ([math]\displaystyle{ x_i }[/math]) and ([math]\displaystyle{ y_i }[/math]) are unique (ties are neglected for simplicity). Any pair of observations [math]\displaystyle{ (x_i,y_i) }[/math] and [math]\displaystyle{ (x_j,y_j) }[/math], where [math]\displaystyle{ i \lt j }[/math], are said to be concordant if the sort order of [math]\displaystyle{ (x_i,x_j) }[/math] and [math]\displaystyle{ (y_i,y_j) }[/math] agrees: that is, if either both [math]\displaystyle{ x_i\gt x_j }[/math] and [math]\displaystyle{ y_i\gt y_j }[/math] holds or both [math]\displaystyle{ x_i\lt x_j }[/math] and [math]\displaystyle{ y_i\lt y_j }[/math]; otherwise they are said to be discordant.

The Kendall τ coefficient is defined as:

[math]\displaystyle{ \tau = \frac{(\text{number of concordant pairs}) - (\text{number of discordant pairs})}{ {n \choose 2} } . }[/math][3]

Where [math]\displaystyle{ {n \choose 2} = {n (n-1) \over 2} }[/math] is the binomial coefficient for the number of ways to choose two items from n items.

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.
  • An explicit expression for Kendall's rank coefficient is [math]\displaystyle{ \tau= \frac{2}{n(n-1)}\sum_{i\lt j} \sgn(x_i-x_j)\sgn(y_i-y_j) }[/math].

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 non-parametric, 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

[math]\displaystyle{ \frac{2(2n+5)}{9n (n-1)} }[/math].[4]

Accounting for ties

A pair [math]\displaystyle{ \{(x_i,y_i), (x_j,y_j)\} }[/math] is said to be tied if [math]\displaystyle{ x_i=x_j }[/math] or [math]\displaystyle{ y_i=y_j }[/math]; 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]:

Tau-a

The Tau-a statistic tests the strength of association of the cross tabulations. Both variables have to be ordinal. Tau-a will not make any adjustment for ties. It is defined as:

[math]\displaystyle{ \tau_A = \frac{n_c-n_d}{n_0} }[/math]

where nc, nd and n0 are defined as in the next section.

Tau-b

The Tau-b statistic, unlike Tau-a, makes adjustments for ties.[5] Values of Tau-b 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 Tau-b coefficient is defined as:

[math]\displaystyle{ \tau_B = \frac{n_c-n_d}{\sqrt{(n_0-n_1)(n_0-n_2)}} }[/math]

where

[math]\displaystyle{ \begin{align} n_0 & = n(n-1)/2\\ n_1 & = \sum_i t_i (t_i-1)/2 \\ n_2 & = \sum_j u_j (u_j-1)/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} }[/math]

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.[6]

Tau-c

Tau-c (also called Stuart-Kendall Tau-c)[7] is more suitable than Tau-b for the analysis of data based on non-square (i.e. rectangular) contingency tables.[7][8] So use Tau-b if the underlying scale of both variables has the same number of possible values (before ranking) and Tau-c if they differ. For instance, one variable might be scored on a 5-point scale (very good, good, average, bad, very bad), whereas the other might be based on a finer 10-point scale.

The Kendall Tau-c coefficient is defined as:[8]

[math]\displaystyle{ \tau_C = \frac{2 (n_c-n_d)}{n^2 \frac{(m-1)}{m}} }[/math]

where

[math]\displaystyle{ \begin{align} n_c & = \text{Number of concordant pairs} \\ n_d & = \text{Number of discordant pairs} \\ r & = \text{Number of rows} \\ c & = \text{Number of columns} \\ m & = \min(r, c) \end{align} }[/math]

Significance tests

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

[math]\displaystyle{ z_A = {3 (n_c - n_d) \over \sqrt{n(n-1)(2n+5)/2} } }[/math]

Thus, to test whether two variables are statistically dependent, one computes [math]\displaystyle{ z_A }[/math], and finds the cumulative probability for a standard normal distribution at [math]\displaystyle{ -|z_A| }[/math]. For a 2-tailed test, multiply that number by two to obtain the p-value. If the p-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 [math]\displaystyle{ z_A }[/math] when accounting for ties. The following statistic, [math]\displaystyle{ z_B }[/math], has the same distribution as the [math]\displaystyle{ \tau_B }[/math] distribution, and is again approximately equal to a standard normal distribution when the quantities are statistically independent:

[math]\displaystyle{ z_B = {n_c - n_d \over \sqrt{ v } } }[/math]

where

[math]\displaystyle{ \begin{array}{ccl} v & = & (v_0 - v_t - v_u)/18 + v_1 + v_2 \\ v_0 & = & n (n-1) (2n+5) \\ v_t & = & \sum_i t_i (t_i-1) (2 t_i+5)\\ v_u & = & \sum_j u_j (u_j-1)(2 u_j+5) \\ v_1 & = & \sum_i t_i (t_i-1) \sum_j u_j (u_j-1) / (2n(n-1)) \\ v_2 & = & \sum_i t_i (t_i-1) (t_i-2) \sum_j u_j (u_j-1) (u_j-2) / (9 n (n-1) (n-2)) \end{array} }[/math]

This is sometimes referred to as the Mann-Kendall test.[9]

Algorithms

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

numer := 0
for i := 2..N do
    for 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 [math]\displaystyle{ O(n^2) }[/math] in complexity and becomes very slow on large samples. A more sophisticated algorithm[10] built upon the Merge Sort algorithm can be used to compute the numerator in [math]\displaystyle{ O(n \cdot \log{n}) }[/math] time.

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

[math]\displaystyle{ n_c-n_d = n_0 - n_1 - n_2 + n_3 - 2 S(y), }[/math]

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

A Merge Sort partitions the data to be sorted, [math]\displaystyle{ y }[/math] into two roughly equal halves, [math]\displaystyle{ y_\mathrm{left} }[/math] and [math]\displaystyle{ y_\mathrm{right} }[/math], 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:

[math]\displaystyle{ S(y) = S(y_\mathrm{left}) + S(y_\mathrm{right}) + M(Y_\mathrm{left},Y_\mathrm{right}) }[/math]

where [math]\displaystyle{ Y_\mathrm{left} }[/math] and [math]\displaystyle{ Y_\mathrm{right} }[/math] are the sorted versions of [math]\displaystyle{ y_\mathrm{left} }[/math] and [math]\displaystyle{ y_\mathrm{right} }[/math], and [math]\displaystyle{ M(\cdot,\cdot) }[/math] characterizes the Bubble Sort swap-equivalent for a merge operation. [math]\displaystyle{ M(\cdot,\cdot) }[/math] is computed as depicted in the following pseudo-code:

function M(L[1..n], R[1..m]) is
    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 [math]\displaystyle{ x }[/math] and a sorted version of [math]\displaystyle{ y }[/math]. With these, the factors [math]\displaystyle{ t_i }[/math] and [math]\displaystyle{ u_j }[/math] used to compute [math]\displaystyle{ \tau_B }[/math] are easily obtained in a single linear-time pass through the sorted arrays.

Software Implementations

See also

References

  1. Kendall, M. (1938). "A New Measure of Rank Correlation". Biometrika 30 (1–2): 81–89. doi:10.1093/biomet/30.1-2.81. 
  2. "Ordinal Measures of Association". Journal of the American Statistical Association 53 (284): 814–861. 1958. doi:10.2307/2281954. 
  3. Hazewinkel, Michiel, ed. (2001), "Kendall tau metric", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=K/k130020 
  4. Hazewinkel, Michiel, ed. (2001), "Kendall coefficient of rank correlation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=K/k055200 
  5. Agresti, A. (2010). Analysis of Ordinal Categorical Data (Second ed.). New York: John Wiley & Sons. ISBN 978-0-470-08289-8. 
  6. IBM (2016). IBM SPSS Statistics 24 Algorithms. IBM. p. 168. http://www-01.ibm.com/support/docview.wss?uid=swg27047033#en. Retrieved 31 August 2017. 
  7. 7.0 7.1 Berry, K. J.; Johnston, J. E.; Zahran, S.; Mielke, P. W. (2009). "Stuart's tau measure of effect size for ordinal variables: Some methodological considerations". Behavior Research Methods 41 (4): 1144–1148. doi:10.3758/brm.41.4.1144. PMID 19897822. 
  8. 8.0 8.1 Stuart, A. (1953). "The Estimation and Comparison of Strengths of Association in Contingency Tables". Biometrika 40 (1–2): 105–110. doi:10.2307/2333101. 
  9. Glen_b. "Relationship between Mann-Kendall and Kendall Tau-b". https://stats.stackexchange.com/questions/414038/relationship-between-mann-kendall-and-kendall-tau-b. 
  10. 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. 

Further reading

External links

de:Rangkorrelationskoeffizient#Kendalls Tau