G02BNF : NAG Library, Mark 24

3 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

5 Parameters

1: N – INTEGERInput

On entry:

n

, the number of observations or cases.

Constraint:

N \geq 2

2: M – INTEGERInput

On entry:

m

, the number of variables.

Constraint:

M \geq 2

3: X(LDX,M) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

X (i, j)

must be set to

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

On exit:

X (i, j)

contains the rank

y_{i j}

of the observation

x_{i j}

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

4: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02BNF is called.

Constraint:

LDX \geq N

5: ITYPE – INTEGERInput

On entry: 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

6: RR(LDRR,M) – REAL (KIND=nag_wp) arrayOutput

On exit: 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.)

7: LDRR – INTEGERInput

On entry: the first dimension of the array RR as declared in the (sub)program from which G02BNF is called.

Constraint:

LDRR \geq M

8: KWORKA(N) – INTEGER arrayWorkspace

9: KWORKB(N) – INTEGER arrayWorkspace

10: WORK1(M) – REAL (KIND=nag_wp) arrayWorkspace

11: WORK2(M) – REAL (KIND=nag_wp) arrayWorkspace

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

On entry,	$LDX < N$ ,
or	$LDRR < M$ .

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

NAG Library Routine Document

G02BNF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG02BNF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G02BNF