PRSPEARMAN procedure

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

No options

Parameters

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 × (N²-1) / 12

where R_i and S_i are the ranks of X_i and Y_i 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.

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