G02BNF : NAG Library, Mark 23

The input data consists of n observations for each of m variables, given as an array

xij, i=1,2,…,n n≥2,j=1,2,…,mm≥2,

where xij is the ith observation of the jth variable.

The quantities calculated are:

(a)

Ranks

For a given variable, j say, each of the n observations, x1j,x2j,…,xnj, has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitudes of the other n-1 observations on that same variable.

The smallest observation for variable j is assigned the rank 1, the second smallest observation for variable j the rank 2, the third smallest the rank 3, and so on until the largest observation for variable j is given the rank n.

If a number of cases all have the same value for the given variable, j, then they are each given an ‘average’ rank, e.g., if in attempting to assign the rank h+1, k observations were found to have the same value, then instead of giving them the ranks

h+1,h+2,…,h+k,

all k observations would be assigned the rank

2h+k+12

and the next value in ascending order would be assigned the rank

h+k+ 1.

The process is repeated for each of the m variables.

Let yij be the rank assigned to the observation xij when the jth variable is being ranked. The actual observations xij are replaced by the ranks yij.

(b)

Nonparametric rank correlation coefficients

(i)

Kendall's tau:

Rjk=∑h=1n∑i=1nsignyhj-yijsignyhk-yik nn-1-Tjnn-1-Tk , j,k=1,2,…,m,

where	sign⁡u=1 if u>0,
	sign⁡u=0 if u=0,
	sign⁡u=-1 if u<0,

and Tj=∑tjtj-1, where tj is the number of ties of a particular value of variable j, and the summation is over all tied values of variable j.

(ii)

Spearman's:

Rjk*=nn2-1-6∑i=1n yij-yik 2-12Tj*+Tk* nn2-1-Tj*nn2-1-Tk* , j,k=1,2,…,m,

where Tj*=∑tjtj2-1, tj being the number of ties of a particular value of variable j, and the summation being over all tied values of variable j.

5 Parameters

1: N – INTEGERInput

On entry: n, the number of observations or cases.

Constraint: N≥2.

2: M – INTEGERInput

On entry: m, the number of variables.

Constraint: M≥2.

3: X(LDX,M) – REAL (KIND=nag_wp) arrayInput/Output

On entry: Xij must be set to xij, the value of the ith observation on the jth variable, for i=1,2,…,n and j=1,2,…,m.

On exit: Xij contains the rank yij of the observation xij, for i=1,2,…,n and j=1,2,…,m.

4: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02BNF is called.

Constraint: LDX≥N.

5: ITYPE – INTEGERInput

On entry: the type of correlation coefficients which are to be calculated.

ITYPE=-1: Only Kendall's tau coefficients are calculated.
ITYPE=0: Both Kendall's tau and Spearman's coefficients are calculated.
ITYPE=1: Only Spearman's coefficients are calculated.

Constraint: ITYPE=-1, 0 or 1.

6: RR(LDRR,M) – REAL (KIND=nag_wp) arrayOutput

On exit: the requested correlation coefficients.

If only Kendall's tau coefficients are requested (ITYPE=-1), RRjk contains Kendall's tau for the jth and kth variables.

If only Spearman's coefficients are requested (ITYPE=1), RRjk contains Spearman's rank correlation coefficient for the jth and kth variables.

If both Kendall's tau and Spearman's coefficients are requested (ITYPE=0), the upper triangle of RR contains the Spearman coefficients and the lower triangle the Kendall coefficients. That is, for the jth and kth variables, where j is less than k, RRjk contains the Spearman rank correlation coefficient, and RRkj contains Kendall's tau, for j=1,2,…,m and k=1,2,…,m.

(Diagonal terms, RRjj, are unity for all three values of ITYPE.)

7: LDRR – INTEGERInput

On entry: the first dimension of the array RR as declared in the (sub)program from which G02BNF is called.

Constraint: LDRR≥M.

8: KWORKA(N) – INTEGER arrayWorkspace

9: KWORKB(N) – INTEGER arrayWorkspace

10: WORK1(M) – REAL (KIND=nag_wp) arrayWorkspace

11: WORK2(M) – REAL (KIND=nag_wp) arrayWorkspace

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.

On entry,	LDX<N,
or	LDRR<M.

On entry,	ITYPE<-1,
or	ITYPE>1.

NAG Library Routine Document

G02BNF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG02BNF

+− Contents

NAG Library Routine Document

G02BNF