SuppDists (version 1.1-9.8)

Kendall: The distribution of Kendall's tau

Description

Density, distribution function, quantile function, random generator and summary function for Kendall's tau.

Usage

dKendall(x, N, log=FALSE)
pKendall(q, N, lower.tail=TRUE, log.p=FALSE)
qKendall(p, N, lower.tail=TRUE, log.p=FALSE)
rKendall(n, N)
sKendall(N)

Value

The output values conform to the output from other such functions in R. dKendall() gives the density, pKendall() the distribution function and qKendall() its inverse. rKendall() generates random numbers. sKendall() produces a list containing parameters corresponding to the arguments -- mean, median, mode, variance, sd, third cental moment, fourth central moment, Pearson's skewness, skewness, and kurtosis.

Arguments

x,q

vector of non-negative quantities

p

vector of probabilities

n

number of values to generate. If n is a vector, length(n) values will be generated

N

vector number of treatments

log, log.p

logical vector; if TRUE, probabilities p are given as log(p)

lower.tail

logical vector; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\)

Author

Bob Wheeler

Details

There are two categories with N treatments each. The treatments are ranked for each category, and then sorted according to the ranks for the first category. This produces a 2 by N array in which the numbers in the first row are increasing from 1 to N. The array is scanned, and every time two adjacent ranks in the second row are not in order, they are exchanged. The scanning is repeated until the second row is in increasing order. Let s denote the number of exchanges, then Kendall's tau is given by

$$\tau=1-\frac{4s}{N(N-1)}$$

This too is a product-moment correlation coefficient. See Kendall (1975), Chapter 2. Other methods for calculating the statistic are also discussed there.

The calculated values are exact for \(N < 13\), thereafter an Edgeworth expansion is used.

References

Kendall, M. (1975). Rank Correlation Methods. Griffin, London.

Examples

pKendall(0, N=10)
pKendall(c(-.42,0.02,.42), N=10) ## approximately 5% 50% and 95% 
qKendall(.95,N=c(10,20))
sKendall(N=10)
plot(function(x)dKendall(x, N=10),-0.5,0.5)