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# PRSPEARMAN procedure

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

### Parameters

`N` = scalars Numbers of pairs of observations Values of the signed rank statistic Cumulative lower probability of `CORRELATION` Cumulative upper probability of `CORRELATION` Probability density of `CORRELATION` Probability densities of `CORRELATION`…1 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 ((Ri-(N+1)/2) × (Si-(N+1)/2)) / (N × (N2-1) / 12

where Ri and Si are the ranks of Xi and Yi respectively.

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.

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,Pr,CN,Pr,CN,Pr,CN,Pr;\
DECIMALS=(0,4)3; FIELD=4,7
ENDFOR
```
Updated on March 6, 2019