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(*x _{i}* –

*x*) × Sign(

_{j}*y*–

_{i}*y*) } }.

_{j}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.

### See also

Procedure: `KTAU`

.

Commands for: Basic and nonparametric statistics.

### Example

CAPTION 'PRKTAU example',!t(\ 'Calculate the Table 6.1 of Sen & Krishnaiah (1984,',\ 'Handbook of Statistics. Volume 4, Chapter 37, p. 953)');\ STYLE=meta,plain VARIATE [VALUES=0.005,0.01,0.025,0.05] PLevel; DECIMALS=3 & [VALUES=4...35] N; DECIMALS=0 & [NVALUES=N] Pr[1,2,3,4] & [NVALUES=N] CN[1,2,3,4] POINTER [NVALUES=NVALUES(PLevel)] Pos FOR [INDEX=i] n = #N PRKTAU n; TAU=0; LPROBABILITIES=lpr CALCULATE clpr = CUMULATE(lpr) & CN[]$[i] = SUM(clpr < #PLevel) - 1 & Pos[] = CN[]$[i] + 1 + (CN[]$[i] < 0) & Pr[]$[i] = clpr$[Pos[]] & Pr[]$[i] = MVINSERT(Pr[]$[i];CN[]$[i] < 0) & CN[]$[i] = MVINSERT(CN[]$[i];CN[]$[i] < 0) DELETE [Redefine=yes] lpr,clpr ENDFOR PRINT [ORIENT=Across] PLevel; FIELD=11 PRINT [MISSING=' ';IPRINT=*;SQUASH=yes] \ CN[1],Pr[1],CN[2],Pr[2],CN[3],Pr[3],CN[4],Pr[4];\ DECIMALS=(0,4)3; FIELD=4,7