Calculates probabilities for Spearman’s rank correlation statistic (D.B. Baird).

### No options

### Parameters

`N` = scalars |
Numbers of pairs of observations |
---|---|

`CORRELATION` = scalars |
Values of the signed rank statistic |

`CLPROBABILITY` = scalars |
Cumulative lower probability of `CORRELATION` |

`CUPROBABILITY` = scalars |
Cumulative upper probability of `CORRELATION` |

`PROBABILITY` = scalars |
Probability density of `CORRELATION` |

`UPROBABILITIES` = variates |
Probability densities of `CORRELATION` …1 |

`UCORRELATION` = variates |
Values of `CORRELATION` at corresponding elements of `UPROBABILITIES` |

### Description

`PRSPEARMAN`

calculates various probabilities for Spearman’s rank correlation coefficient (see procedure `SPEARMAN`

). These can be used to give a nonparametric assessment of whether paired samples are correlated.

correlation = ∑_{i=1…N} ((*R _{i}*-(

*N*+1)/2) × (

*S*-(

_{i}*N*+1)/2)) / (

*N*× (

*N*

^{2}-1) / 12

where *R _{i}* and

*S*are the ranks of

_{i}*X*and

_{i}*Y*respectively.

_{i}The number of sample pairs of observations is specified by the `N`

parameter, and the `CORRELATION`

parameter specifies the value of the rank correlation for which the probabilities are required. The `CLPROBABILITY`

and `CUPROBABILITY`

parameters can specify scalars to save the cumulative lower and upper probabilities,

Pr.(*s* ≤ `CORRELATION`

)

and

Pr.(*s* > `CORRELATION`

)

respectively. `PROBABILITY`

can save the probability density at `CORRELATION`

,

Pr.(*s* == `CORRELATION`

),

`UPROBABILITIES`

can save a variate containing the densities for `CORRELATION`

…1, and `UCORRELATION`

can save the values of `CORRELATION`

for the elements in `UPROBABILITIES`

.

Options: none.

Parameters: `N`

, `CORRELATION`

, `CLPROBABILITY`

, `CUPROBABILITY`

, `PROBABILITY`

, `UPROBABILITIES`

, `UCORRELATION`

.

### Method

The procedure uses `PASS`

to call an external program which calculates the coefficients of the generating function for the Spearman rank correlation coefficient under the null hypothesis using recurrence functions (see van de Weil *et al*. 1999). A t approximation is used when N exceeds 20.

### Action with `RESTRICT`

Restrictions are not applicable to any of the parameters.

### 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: `SPEARMAN`

.

Commands for: Basic and nonparametric statistics.

### Example

CAPTION 'PRSPEARMAN example',\ !t('Calculate the Table 6.2 of Sen & Krishnaiah (1984,',\ 'Handbook of Statistics. Volume 4, Chapter 37, p.954)',\ 'Note: Table 6.2 has mistakenly printed 2*s rather than s.');\ STYLE=meta,plain VARIATE [VALUES=0.005,0.01,0.025,0.05] PLevel; DECIMALS=3 & [VALUES=4...16] 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 PRSPEARMAN n; CORRELATION=0; UPROBABILITIES=upr CALCULATE cupr = CUMULATE(upr) & CN[]$[i] = SUM(cupr < #PLevel) - 1 & Pos[] = CN[]$[i] + 1 + (CN[]$[i] < 0) & Pr[]$[i] = cupr$[Pos[]] & Pr[]$[i] = MVINSERT(Pr[]$[i];CN[]$[i] < 0) & CN[]$[i] = MVINSERT(CN[]$[i];CN[]$[i] < 0) DELETE [Redefine=yes] upr,cupr ENDFOR FOR [NTIMES=1] 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 ENDFOR