G02BPF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

G02BPF computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting completely any cases with a missing observation for any variable; the data array is overwritten with the ranks of the observations.

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 Section 7); 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

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

On entry:

n

, the number of observations or cases.

Constraint:

N \geq 2

On entry:

m

, the number of variables.

Constraint:

M \geq 2

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

. (For those observations containing missing values, and therefore excluded from the calculation,

y_{i j} = 0

, for

j = 1, 2, \dots, m

On entry: the first dimension of the array X as declared in the (sub)program from which G02BPF is called.

Constraint:

LDX \geq N

On entry:

MISS (j)

must be set to

1

if a missing value,

x m_{j}

, is to be specified for the

j

th variable in the array X, or set equal to

0

otherwise. Values of MISS must be given for all

m

variables in the array X.

On exit: the array MISS is overwritten by the routine, and the information it contained on entry is lost.

On entry:

XMISS (j)

must be set to the missing value,

x m_{j}

, to be associated with the

j

th variable in the array X, for those variables for which missing values are specified by means of the array MISS (see Section 7).

On exit: the array XMISS is overwritten by the routine, and the information it contained on entry is lost.

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

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.)

On entry: the first dimension of the array RR as declared in the (sub)program from which G02BPF is called.

Constraint:

LDRR \geq M

On exit: the number of cases,

n_{c}

, actually used in the calculations (when cases involving missing values have been eliminated).

On exit:

INCASE (i)

holds the value

1

if the

i

th case was included in the calculations, and the value

0

if the

i

th case contained a missing value for at least one variable. That is,

INCASE (i) = w_{i}

(see Section 3), for

i = 1, 2, \dots, n

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).

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:

After observations with missing values were omitted, fewer than

2

cases remained.

You are warned of the need to exercise extreme care in your selection of missing values. G02BPF treats all values in the inclusive range

(1 \pm {0.1}^{(X02BEF - 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.

The time taken by G02BPF depends on

n

and

m

, and the occurrence of missing values.

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.

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

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

NAG Library Routine Document

G02BPF

+− 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 DocumentG02BPF

+− 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

G02BPF