NAG Library Routine Document

G02BQF

+− 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

1 Purpose

G02BQF 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 routine.

2 Specification

SUBROUTINE G02BQF (

N, M, X, LDX, ITYPE, RR, LDRR, KWORKA, KWORKB, WORK1, WORK2, IFAIL)

INTEGER	N, M, LDX, ITYPE, LDRR, KWORKA(N), KWORKB(N), IFAIL
REAL (KIND=nag_wp)	X(LDX,M), RR(LDRR,M), WORK1(N), WORK2(N)

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 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

4 References

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

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

On entry:

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

4: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G02BQF 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 G02BQF is called.

Constraint:

LDRR \geq M

8: KWORKA(N) – INTEGER arrayWorkspace

9: KWORKB(N) – INTEGER arrayWorkspace

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

11: WORK2(N) – 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).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

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	$ITYPE > 1$ .

7 Accuracy

The method used is believed to be stable.

8 Further Comments

The time taken by G02BQF depends on

n

and

m

9 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.

NAG Library Routine DocumentG02BQF

+− 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

G02BQF