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.

Syntax

[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

Description

The input data consists of

n

observations for each of

m

variables, given as an array

[x_{i j}], i = 1, 2, \dots, n (n \geq 2), j = 1, 2, \dots, m (m \geq 2),

where

x_{i j}

is the

i

th observation on the

j

th variable.

The observations are first ranked, as follows.

For a given variable,

j

say, each of the

n

observations,

x_{1 j}, x_{2 j}, \dots, x_{n j}

, 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

h + 1, h + 2, \dots, h + k,

all

k

observations would be assigned the rank

\frac{2 h + k + 1}{2}

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

y_{i j}

be the rank assigned to the observation

x_{i j}

when the

j

th variable is being ranked.

The quantities calculated are:

(a)

Kendall's tau rank correlation coefficients:

R_{j k} = \frac{\sum_{h = 1}^{n} \sum_{i = 1}^{n} sign (y_{h j} - y_{i j}) sign (y_{h k} - y_{i k})}{\sqrt{[n (n - 1) - T_{j}] [n (n - 1) - T_{k}]}}, j, k = 1, 2, \dots, m,

and	$sign u = 1$ if $u > 0$
	$sign u = 0$ if $u = 0$
	$sign u = - 1$ if $u < 0$

and

T_{j} = \sum t_{j} (t_{j} - 1)

t_{j}

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:

R_{j k}^{*} = \frac{n (n^{2} - 1) - 6 \sum_{i = 1}^{n} {(y_{i j} - y_{i k})}^{2} - \frac{1}{2} (T_{j}^{*} + T_{k}^{*})}{\sqrt{[n (n^{2} - 1) - T_{j}^{*}] [n (n^{2} - 1) - T_{k}^{*}]}}, j, k = 1, 2, \dots, m,

where

T_{j}^{*} = \sum t_{j} (t_{j}^{2} - 1)

where

t_{j}

is the number of ties of a particular value of variable

j

, and the summation is 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: $x (ldx, m)$ – double array

ldx, the first dimension of the array, must satisfy the constraint

ldx \geq n

x (i, j)

must be set to data value

x_{i j}

, the value of the

i

th observation on the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, 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

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: $n \geq 2$ .
2: $m$ – int64int32nag_int scalar: Default: the second dimension of the array x.
$m$ , the number of variables.

Constraint: $m \geq 2$ .

Output Parameters

1: $rr (ldrr, m)$ – double array: The requested correlation coefficients.
If only Kendall's tau coefficients are requested ( $itype = - 1$ ), $rr (j, k)$ contains Kendall's tau for the $j$ th and $k$ th variables.

If only Spearman's coefficients are requested ( $itype = 1$ ), $rr (j, k)$ contains Spearman's rank correlation coefficient for the $j$ th and $k$ th 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 $j$ th and $k$ th 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 $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .

(Diagonal terms, $rr (j, j)$ , 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:

$ifail = 1$

On entry,

n < 2

$ifail = 2$

On entry,

m < 2

$ifail = 3$

On entry,	$ldx < n$ ,
or	$ldrr < m$ .

$ifail = 4$

On entry,	$itype < - 1$ ,
or	$itype > 1$ .

$ifail = - 99$: 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 (g02bq) 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 both Kendall's tau and Spearman's rank correlation coefficients for all three variables.

Open in the MATLAB editor: g02bq_example

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:-');
disp(x);
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');
disp(rr);

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox