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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_correg_coeffs_kspearman_overwrite (g02bn)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_correg_coeffs_kspearman_overwrite (g02bn) 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.

Syntax

[x, rr, ifail] = g02bn(x, itype, 'n', n, 'm', m)
[x, rr, ifail] = nag_correg_coeffs_kspearman_overwrite(x, itype, 'n', n, 'm', m)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 22: n was made optional

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,,mm2,  
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=1ni=1nsignyhj-yijsignyhk-yik nn-1-Tjnn-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=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*=nn2-1-6i=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.

References

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

Parameters

Compulsory Input Parameters

1:     xldxm – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn.
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.
2:     itype int64int32nag_int scalar
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.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array x.
n, the number of observations or cases.
Constraint: n2.
2:     m int64int32nag_int scalar
Default: the second dimension of the array x.
m, the number of variables.
Constraint: m2.

Output Parameters

1:     xldxm – double array
xij contains the rank yij of the observation xij, for i=1,2,,n and j=1,2,,m.
2:     rrldrrm – double array
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.)
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,n<2.
   ifail=2
On entry,m<2.
   ifail=3
On entry,ldx<n,
orldrr<m.
   ifail=4
On entry,itype<-1,
oritype>1.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

The method used is believed to be stable.

Further Comments

The time taken by nag_correg_coeffs_kspearman_overwrite (g02bn) depends on n and m.

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.
function g02bn_example


fprintf('g02bn example results\n\n');

x = [1.7,  1, 0.5;
     2.8,  4, 3.0;
     0.6,  6, 2.5;
     1.8,  9, 6.0;
     0.99, 4, 2.5;
     1.4,  2, 5.5;
     1.8,  9, 7.5;
     2.5,  7, 0.0;
     0.99, 5, 3.0];
[n,m] = size(x);
fprintf('Number of variables (columns) = %d\n', m);
fprintf('Number of cases     (rows)    = %d\n\n', n);
disp('Data matrix is:-');
disp(x);

itype = int64(0);
[x, rr, ifail] = g02bn(x, itype);

fprintf('\nMatrix of ranks:-\n');
disp(x);
fprintf('Matrix of rank correlation coefficients:\n');
fprintf('Upper triangle -- Spearman''s\n');
fprintf('Lower triangle -- Kendall''s tau\n\n');
disp(rr);


g02bn example results

Number of variables (columns) = 3
Number of cases     (rows)    = 9

Data matrix is:-
    1.7000    1.0000    0.5000
    2.8000    4.0000    3.0000
    0.6000    6.0000    2.5000
    1.8000    9.0000    6.0000
    0.9900    4.0000    2.5000
    1.4000    2.0000    5.5000
    1.8000    9.0000    7.5000
    2.5000    7.0000         0
    0.9900    5.0000    3.0000


Matrix of ranks:-
    5.0000    1.0000    2.0000
    9.0000    3.5000    5.5000
    1.0000    6.0000    3.5000
    6.5000    8.5000    8.0000
    2.5000    3.5000    3.5000
    4.0000    2.0000    7.0000
    6.5000    8.5000    9.0000
    8.0000    7.0000    1.0000
    2.5000    5.0000    5.5000

Matrix of rank correlation coefficients:
Upper triangle -- Spearman's
Lower triangle -- Kendall's tau

    1.0000    0.2246    0.1186
    0.0294    1.0000    0.3814
    0.1176    0.2353    1.0000


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Chapter Contents
Chapter Introduction
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