NAG FL Interface
g02bnf (coeffs_​kspearman_​overwrite)

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1 Purpose

g02bnf computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is overwritten with the ranks of the observations.

2 Specification

Fortran Interface
Subroutine g02bnf ( n, m, x, ldx, itype, rr, ldrr, kworka, kworkb, work1, work2, ifail)
Integer, Intent (In) :: n, m, ldx, itype, ldrr
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: kworka(n), kworkb(n)
Real (Kind=nag_wp), Intent (Inout) :: x(ldx,m), rr(ldrr,m)
Real (Kind=nag_wp), Intent (Out) :: work1(m), work2(m)
C Header Interface
#include <nag.h>
void  g02bnf_ (const Integer *n, const Integer *m, double x[], const Integer *ldx, const Integer *itype, double rr[], const Integer *ldrr, Integer kworka[], Integer kworkb[], double work1[], double work2[], Integer *ifail)
The routine may be called by the names g02bnf or nagf_correg_coeffs_kspearman_overwrite.

3 Description

The input data consists of n observations for each of m variables, given as an array
[xij],  i=1,2,,n (n2),j=1,2,,m(m2),  
where xij is the ith observation of the jth variable.
The quantities calculated are:
  1. (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.
  2. (b)Nonparametric rank correlation coefficients
    1. (i)Kendall's tau:
      Rjk=h=1ni=1nsign(yhj-yij)sign(yhk-yik) [n(n-1)-Tj][n(n-1)-Tk] ,  j,k=1,2,,m,  
      where signu=1 if u>0,
      signu=0 if u=0,
      signu=−1 if u<0,
      and Tj=tj(tj-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.
    2. (ii)Spearman's:
      Rjk*=n(n2-1)-6i=1n (yij-yik) 2-12(Tj*+Tk*) [n(n2-1)-Tj*][n(n2-1)-Tk*] ,  j,k=1,2,,m,  
      where Tj*=tj(tj2-1), tj being the number of ties of a particular value of variable j, and the summation being over all tied values of variable j.

4 References

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

5 Arguments

1: n Integer Input
On entry: n, the number of observations or cases.
Constraint: n2.
2: m Integer Input
On entry: m, the number of variables.
Constraint: m2.
3: x(ldx,m) Real (Kind=nag_wp) array Input/Output
On entry: x(i,j) 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: x(i,j) contains the rank yij of the observation xij, for i=1,2,,n and j=1,2,,m.
4: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02bnf is called.
Constraint: ldxn.
5: itype Integer Input
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) array Output
On exit: the requested correlation coefficients.
If only Kendall's tau coefficients are requested (itype=−1), rr(j,k) contains Kendall's tau for the jth and kth variables.
If only Spearman's coefficients are requested (itype=1), rr(j,k) 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, rr(j,k) contains the Spearman rank correlation coefficient, and rr(k,j) contains Kendall's tau, for j=1,2,,m and k=1,2,,m.
(Diagonal terms, rr(j,j), are unity for all three values of itype.)
7: ldrr Integer Input
On entry: the first dimension of the array rr as declared in the (sub)program from which g02bnf is called.
Constraint: ldrrm.
8: kworka(n) Integer array Workspace
9: kworkb(n) Integer array Workspace
10: work1(m) Real (Kind=nag_wp) array Workspace
11: work2(m) Real (Kind=nag_wp) array Workspace
12: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, n=value.
Constraint: n2.
ifail=2
On entry, m=value.
Constraint: m2.
ifail=3
On entry, ldrr=value and m=value.
Constraint: ldrrm.
On entry, ldx=value and n=value.
Constraint: ldxn.
ifail=4
On entry, itype=value.
Constraint: itype=−1 or 1.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The method used is believed to be stable.

8 Parallelism and Performance

g02bnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by g02bnf depends on n and m.

10 Example

This example reads in a set of data consisting of nine observations on each of three variables. The program then calculates and prints the rank of each observation, and both Kendall's tau and Spearman's rank correlation coefficients for all three variables.

10.1 Program Text

Program Text (g02bnfe.f90)

10.2 Program Data

Program Data (g02bnfe.d)

10.3 Program Results

Program Results (g02bnfe.r)