Kendall {Kendall}

Kendall rank correlation

Description

Computes the Kendall rank correlation and its p-value on a two-sided test of H0: x and y are independent. If there are no ties, the test is exact and in this case it should agree with the base function cor(x,y,method="kendall") and cor.test(x,y,method="kendall"). When there are ties, the normal approximation given in Kendall is used as discussed below. In the case of ties, both Kendall and cor produce the same result but cor.test produces a p-value which is not as accurate

Usage

Kendall(x, y)

Arguments

x

first variable, a vector

y

second variable, a vector the same length as x

Details

In many applications x and y may be ranks or even ordered categorical variables. In our function x and y should be numeric vectors or factors. Any observations correspondings to NA in either x or y are removed.

Kendall's rank correlation measures the strength of monotonic association between the vectors x and y. In the case of no ties in the x and y variables, Kendall's rank correlation coefficient, tau, may be expressed as \tau = S/D where

S=\sum_{i<j} (sign(x[j]-x[i])*sign(y[j]-y[i]))

and D=n(n-1)/2. S is called the score and D, the denominator, is the maximum possible value of S. When there are ties, the formula for D is more complicated (Kendall, 1974, Ch. 3) and this general forumla for ties in both reankings is implemented in our function.

The p-value of tau under the null hypothesis of no association is computed by in the case of no ties using an exact algorithm given by Best and Gipps (1974).

When ties are present, a normal approximation with continuity correction is used by taking S as normally distributed with mean zero and variance var(S), where var(S) is given by Kendall (1976, eqn 4.4, p.55). Unless ties are very extensive and/or the data is very short, this approximation is adequate. If extensive ties are present then the bootstrap provides an expedient solution (Davis and Hinkley, 1997). Alternatively an exact method based on exhaustive enumeration is also available (Valz and Thompson, 1994) but this is not implemented in this package.

An advantage of the Kendall rank correlation over the Spearman rank correlation is that the score function S nearly normally distributed for small n and the distribution of S is easier to work with.

It may also be noted that usual Pearson correlation is fairly robust and it usually agrees well in terms of statistical significance with results obtained using Kendall's rank correlation.

An error is returned if length(x) is less than 3.

Value

A list with class Kendall is returned with the following components:

tau

Kendall's tau statistic

sl

two-sided p-value

S

Kendall Score

D

denominator, tau=S/D

varS

variance of S

Note

Generic functions print.Kendall and summary.Kendall are provided.

If you want to use the output from Kendall, save the result as in out<-Kendall(x,y) and then select from the list out the value(s) needed.

Author(s)

A.I. McLeod, [e-mail]

References

Best, D.J. and Gipps, P.G. (1974), Algorithm AS 71: The Upper Tail Probabilities of Kendall's Tau Applied Statistics, Vol. 23, No. 1. (1974), pp. 98-100.

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Kendall, M.G. (1976). Rank Correlation Methods. 4th Ed. Griffin.

Hill, I.D. (1973), Algorithm AS 66: The Normal Integral Applied Statistics, Vol. 22, No. 3. (1973), pp. 424-427.

Valz, P. (1990). Developments in Rank Correlation Procedures with Applications to Trend Assessment in Water Resources Research, Ph.D. Thesis, Department of Statistical and Actuarial Sciences, The University of Western Ontario.

Valz, P.D. and Thompson, M.E. (1994), Exact inference for Kendall's S and Spearman's rho. Journal of Computational and Graphical Statistics, 3, 459–472.

See Also

cor, print.Kendall, summary.Kendall, MannKendall, SeasonalMannKendall

Examples

#First Example
#From Kendall (1976, p.42-43, Example 3.4)
A<-c(2.5,2.5,2.5,2.5,5,6.5,6.5,10,10,10,10,10,14,14,14,16,17)
B<-c(1,1,1,1,2,1,1,2,1,1,1,1,1,1,2,2,2)
summary(Kendall(A,B))
#Kendall obtains S=34, D=sqrt(116*60), tau=0.41

#Second Example
#From Kendall (1976, p.55, Example 4.3)
x<-c(1.5,1.5,3,4,6,6,6,8,9.5,9.5,11,12)
y<-c(2.5,2.5,7,4.5,1,4.5,6,11.5,11.5,8.5,8.5,10)
summary(Kendall(x,y))
#Kendall obtains S=34 and Var(S)=203.30