g02bsf computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting cases with missing values from only those calculations involving the variables for which the values are missing; the data array is preserved, and the ranks of the observations are not available on exit from the routine.

2 Specification

Fortran Interface

Subroutine g02bsf (

n, m, x, ldx, miss, xmiss, itype, rr, ldrr, ncases, cnt, ldcnt, kworka, kworkb, kworkc, kworkd, work1, work2, ifail)

Integer, Intent (In)	::	n, m, ldx, miss(m), itype, ldrr, ldcnt
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ncases, kworka(n), kworkb(n), kworkc(n), kworkd(n)
Real (Kind=nag_wp), Intent (In)	::	x(ldx,m), xmiss(m)
Real (Kind=nag_wp), Intent (Inout)	::	rr(ldrr,m), cnt(ldcnt,m)
Real (Kind=nag_wp), Intent (Out)	::	work1(n), work2(n)

C Header Interface

#include <nag.h>

void

g02bsf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer miss[], const double xmiss[], const Integer *itype, double rr[], const Integer *ldrr, Integer *ncases, double cnt[], const Integer *ldcnt, Integer kworka[], Integer kworkb[], Integer kworkc[], Integer kworkd[], double work1[], double work2[], Integer *ifail)

The routine may be called by the names g02bsf or nagf_correg_coeffs_kspearman_miss_pair.

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) ​ and ​ 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 j} = 0

if the

i

th observation for the

j

th variable is a missing value, i.e., if a missing value,

{xm}_{j}

, has been declared for the

j

th variable, and

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

(see also Section 7); and

w_{i j} = 1

otherwise, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

The observations are first ranked, a pair of variables at a time as follows:

For a given pair of variables,

j

and

l

say, each of the observations

x_{i j}

for which the product

w_{i j} w_{i l} = 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 magnitude of the other observations on variable

j

for which

w_{i j} w_{i l} = 1

The smallest of these valid observations for variable

j

is assigned to rank

1

, the second smallest valid observation for variable

j

the rank

2

, the third smallest rank

3

, and so on until the largest such observation is given the rank

n_{j l}

, where

n_{j l} = \sum_{i = 1}^{n} w_{i j} w_{i l} .

If a number of cases all have the same value for the 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 j} w_{i l} = 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 variable

l

is then ranked in a similar way. The process is then repeated for all pairs of variables

j

and

l

, for

j = 1, 2, \dots, m

and

l = j, \dots, m

. Let

y_{i j (l)}

be the rank assigned to the observation

x_{i j}

when the

j

th and

l

th variables are being ranked, and

y_{i l (j)}

be the rank assigned to the observation

x_{i l}

during the same process, for

i = 1, 2, \dots, n

j = 1, 2, \dots, m

and

l = j, \dots, m

The quantities calculated are:

(a)Kendall's tau rank correlation coefficients:

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

where $n_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k}$

and $sign u = 1$ if $u > 0$

$sign u = 0$ if $u = 0$

$sign u = −1$ if $u < 0$

and $T_{j (k)} = \sum t_{j} (t_{j} - 1)$ where $t_{j}$ is the number of ties of a particular value of variable $j$ when the $j$ th and $k$ th variables are being ranked, and the summation is over all tied values of variable $j$ .
(b)Spearman's rank correlation coefficients:

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

where $n_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k}$
and $T_{j (k)}^{*} = \sum t_{j} (t_{j}^{2} - 1)$ , where $t_{j}$ is the number of ties of a particular value of variable $j$ when the $j$ th and $k$ th variables are being ranked, 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 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of observations or cases.

Constraint:

n \geq 2

2: $m$ – Integer Input

On entry:

m

, the number of variables.

Constraint:

m \geq 2

3: $x (ldx, m)$ – Real (Kind=nag_wp) array Input

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

4: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which g02bsf is called.

Constraint:

ldx \geq n

5: $miss (m)$ – Integer array Input

On entry:

miss (j)

must be set equal 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.

6: $xmiss (m)$ – Real (Kind=nag_wp) array Input

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).

7: $itype$ – Integer Input

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

8: $rr (ldrr, m)$ – Real (Kind=nag_wp) array Output

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.)

9: $ldrr$ – Integer Input

On entry: the first dimension of the array rr as declared in the (sub)program from which g02bsf is called.

Constraint:

ldrr \geq m

10: $ncases$ – Integer Output

On exit: the minimum number of cases used in the calculation of any of the correlation coefficients (when cases involving missing values have been eliminated).

11: $cnt (ldcnt, m)$ – Real (Kind=nag_wp) array Output

On exit: the number of cases,

n_{j k}

, actually used in the calculation of the rank correlation coefficient for the

j

th and

k

th variables, for

j = 1, 2, \dots, m

and

k = 1, 2, \dots, m

12: $ldcnt$ – Integer Input

On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bsf is called.

Constraint:

ldcnt \geq m

13: $kworka (n)$ – Integer array Workspace

14: $kworkb (n)$ – Integer array Workspace

15: $kworkc (n)$ – Integer array Workspace

16: $kworkd (n)$ – Integer array Workspace

17: $work1 (n)$ – Real (Kind=nag_wp) array Workspace

18: $work2 (n)$ – Real (Kind=nag_wp) array Workspace

19: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. 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:

Note: in some cases g02bsf may return useful information.

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 2$ .

$ifail = 3$: On entry, $ldcnt = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $ldcnt \geq m$ .

On entry, $ldrr = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $ldrr \geq m$ .

On entry, $ldx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldx \geq n$ .

$ifail = 4$: On entry, $itype = ⟨ value ⟩$ .
Constraint: $itype = −1$ or $1$ .

$ifail = 5$: After observations with missing values were omitted, fewer than two cases remained for at least one pair of variables. (The pairs of variables involved can be determined by examination of the contents of the array cnt). All means, standard deviations, sums of squares and cross-products, and correlation-like coefficients based on two or more cases are returned by the routine even if $ifail = 5$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

You are warned of the need to exercise extreme care in your selection of missing values. g02bsf 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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02bsf is not threaded in any implementation.

9 Further Comments

The time taken by g02bsf depends on

n

and

m

, and the occurrence of missing values.

10 Example

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

0.99

9.0

and

0.0

are declared for the first, second and third variables respectively. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables, omitting cases with missing values from only those calculations involving the variables for which the values are missing. The program, therefore, eliminates cases

4

5

7

and

9

in calculating and correlation between the first and second variables, cases

5

8

and

9

for the first and third variables, and cases

4

7

and

8

for the second and third variables.

g02bs: FL CL CPP AD PY MB

where	$n_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k}$
and	$sign u = 1$ if $u > 0$
	$sign u = 0$ if $u = 0$
	$sign u = −1$ if $u < 0$

NAG FL Interfaceg02bsf (coeffs_​kspearman_​miss_​pair)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g02bsf (coeffs_kspearman_miss_pair)