g02bp:: Correlation and Regression Analysis (NAG Toolbox)

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. In addition, each of the

m

variables may optionally have associated with it a value which is to be considered as representing a missing observation for that variable; the missing value for the

j

th variable is denoted by

{xm}_{j}

. Missing values need not be specified for all variables.

Let

w_{i} = 0

if observation

i

contains a missing value for any of those variables for which missing values have been declared; i.e., if

x_{i j} = {xm}_{j}

for any

j

for which an

{xm}_{j}

has been assigned (see also Accuracy); and

w_{i} = 1

otherwise, for

i = 1, 2, \dots, n

The quantities calculated are:

(a)

Ranks

For a given variable,

j

say, each of the observations

x_{i j}

for which

w_{i} = 1

, for

i = 1, 2, \dots, n

, 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 observations on that same variable for which

w_{i} = 1

The smallest of these valid observations 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 such observation is given the rank

n_{c}

, where

n_{c} = \sum_{i = 1}^{n} w_{i}

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 for which

w_{i} = 1

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. For those observations,

i

, for which

w_{i} = 0

y_{i j} = 0

, for

j = 1, 2, \dots, m

The actual observations

x_{i j}

are replaced by the ranks

y_{i j}

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

(b)

Nonparametric rank correlation coefficients

(i)

Kendall's tau:

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

where	$n_{c} = \sum_{i = 1}^{n} w_{i}$
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)

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

where

n_{c} = \sum_{i = 1}^{n} w_{i}

and

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

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

You are warned of the need to exercise extreme care in your selection of missing values. nag_correg_coeffs_kspearman_miss_case_overwrite (g02bp) treats all values in the inclusive range

(1 \pm {0.1}^{(x02be - 2)}) \times {x m}_{j}

, where

{xm}_{j}

is the missing value for variable

j

specified in xmiss.

You must therefore ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.

Further Comments

The time taken by nag_correg_coeffs_kspearman_miss_case_overwrite (g02bp) depends on

n

and

m

, and the occurrence of missing values.

Example

This example reads in a set of data consisting of nine observations on each of three variables. Missing values of

0.99

and

0.0

are declared for the first and third variables respectively; no missing value is specified for the second variable. 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, omitting completely all cases containing missing values; cases

5

8

and

9

are therefore eliminated, leaving only six cases in the calculations.

function g02bp_example


fprintf('g02bp 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);

miss  = [int64(1); 0; 1];
xmiss = [0.99;       0; 0];
itype = int64(0);

[x, rr, ncases, incase, ifail] = ...
  g02bp( ...
         x, miss, xmiss, itype);

fprintf('\nMatrix of ranks (zero rows were omitted from calculations):-\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);
fprintf('Number of cases actually used  = %d\n', ncases);

g02bp 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 (zero rows were omitted from calculations):-
    3.0000    1.0000    1.0000
    6.0000    3.0000    3.0000
    1.0000    4.0000    2.0000
    4.5000    5.5000    5.0000
         0         0         0
    2.0000    2.0000    4.0000
    4.5000    5.5000    6.0000
         0         0         0
         0         0         0

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

    1.0000    0.2941    0.4058
    0.1429    1.0000    0.7537
    0.2760    0.5521    1.0000

Number of cases actually used  = 6

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

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

NAG Toolbox: nag_correg_coeffs_kspearman_miss_case_overwrite (g02bp)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example