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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_correg_coeffs_kspearman (g02bq)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_correg_coeffs_kspearman (g02bq) computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is preserved, and the ranks of the observations are not available on exit from the function.


[rr, ifail] = g02bq(x, itype, 'n', n, 'm', m)
[rr, ifail] = nag_correg_coeffs_kspearman(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


The input data consists of n observations for each of m variables, given as an array
xij,  i=1,2,,nn2,j=1,2,,mm2,  
where xij is the ith observation on the jth variable.
The observations are first ranked, as follows.
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 magnitude 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
all k observations would be assigned the rank
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 quantities calculated are:
(a) Kendall's tau rank correlation coefficients:
Rjk=h=1ni=1nsignyhj-yijsignyhk-yik nn-1-Tjnn-1-Tk ,  j,k=1,2,,m,  
and signu=1 if u>0
signu=0 if u=0
signu=-1 if u<0
and Tj=tjtj-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
(b) Spearman's rank correlation coefficients:
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 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.


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


Compulsory Input Parameters

1:     xldxm – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn.
xij must be set to data value 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.
Only Kendall's tau coefficients are calculated.
Both Kendall's tau and Spearman's coefficients are calculated.
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:     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.)
2:     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:
On entry,n<2.
On entry,m<2.
On entry,ldx<n,
On entry,itype<-1,
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The method used is believed to be stable.

Further Comments

The time taken by nag_correg_coeffs_kspearman (g02bq) depends on n and m.


This example reads in a set of data consisting of nine observations on each of three variables. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables.
function g02bq_example

fprintf('g02bq 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:-');
itype = int64(0);

[rr, ifail] = g02bq(x, itype);

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

g02bq 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 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 Introduction
NAG Toolbox

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