PRKTAU procedure

Calculates probabilities for Kendall's rank correlation coefficient τ (D.B. Baird).


No options


Parameters

N = scalars
Sizes of the first groups of observations

TAU = scalars
Values of Kendall's τ statistic

CLPROBABILITY = scalars
Cumulative lower probability of TAU

CUPROBABILITY = scalars
Cumulative upper probability of TAU

PROBABILITY = scalars
Probability density of TAU

LPROBABILITIES = variates
Probability densities of -1...TAU

LTAU = variates
Values of Tau at corresponding values of LPROBABILITIES


Description

PRKTAU calculates various probabilities for the Kendall's rank correlation coefficient, τ (tau). The τ statistic arises from Kendall's rank correlation test, which can be used to give a nonparametric assessment as to whether paired samples are correlated. τ is calculated as

T / NCOMBINATIONS(N; 2)

where T is

∑i = 1...N { ∑j = i...N { Sign(xi - xj) × Sign(yi - yj) } }.

   The number of sample pairs of observations is specified by the N parameter, and the TAU parameter specifies the value of the Kendall rank correlation coefficient for which the probabilities are required. The CLPROBABILITY and CUPROBABILITY parameters can specify scalars to save the cumulative lower and upper probabilities, pr(s ≤= τ) and pr(s > τ) respectively. PROBABILITY can save the probability density at τ, pr(s = τ), and LPROBABILITIES can save a variate containing the densities for -1...τ, and LTAU can save the values of τ for the elements in LPROBABILITIES.

 

Options: none.

Parameters: N, TAU, CLPROBABILITY, CUPROBABILITY, PROBABILITY, LPROBABILITIES, LTAU.


Method

The procedure calculates the coefficents of the generating function for the Kendall rank correlation coefficient under the null hypothesis using recurrence functions (See van de Weil et al. 1999). The central limit theorem is used when N exceeds 35, and a Normal approximation of the cumulative density function is returned.


Reference

van de Wiel, M.A. Di Bucchianico, A. & van de Laan, P. (1999). Symbolic computation and exact distributions of nonparametric test statistics. The Statistician, 48, 507-516.