g02bnf (coeffs_kspearman_overwrite) : NAG Library, Mark 27

g02bnf computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is overwritten with the ranks of the observations.

The routine may be called by the names g02bnf or nagf_correg_coeffs_kspearman_overwrite.

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
1. (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 $j$ .
2. (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 $j$ .

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

On entry: the first dimension of the array x as declared in the (sub)program from which g02bnf is called.

Constraint:

ldx \geq n

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 g02bnf is called.

Constraint:

ldrr \geq m

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

0

is recommended. 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:

On entry,

n = ⟨ value ⟩

.
Constraint:

n \geq 2

On entry,

m = ⟨ value ⟩

.
Constraint:

m \geq 2

On entry,

ldrr = ⟨ value ⟩

and

m = ⟨ value ⟩

.
Constraint:

ldrr \geq m

On entry,

ldx = ⟨ value ⟩

and

n = ⟨ value ⟩

.
Constraint:

ldx \geq n

On entry,

itype = ⟨ value ⟩

.
Constraint:

itype = −1

1

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.

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.

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

The method used is believed to be stable.

g02bnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by g02bnf depends on

n

and

m

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

NAG FL Interface
g02bnf (coeffs_kspearman_overwrite)

▸▿ 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

where	$sign u = 1$ if $u > 0$ ,
	$sign u = 0$ if $u = 0$ ,
	$sign u = −1$ if $u < 0$ ,

NAG FL Interfaceg02bnf (coeffs_​kspearman_​overwrite)

▸▿ 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
g02bnf (coeffs_kspearman_overwrite)