g02bn:: Correlation and Regression Analysis (NAG Toolbox)

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 of the

j

th variable.

The quantities calculated are:

(a)

Ranks

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 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, \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 actual observations

x_{i j}

are replaced by the ranks

y_{i j}

(b)

Nonparametric rank correlation coefficients

(i)

Kendall's tau:

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,

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

and

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

, 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

(ii)

Spearman's:

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)

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

References

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

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

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

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

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

NAG Toolbox: nag_correg_coeffs_kspearman_overwrite (g02bn)

▸▿ Contents

Purpose

Syntax