nag_ken_spe_corr_coeff (g02brc) : NAG Library, Mark 24

3 Description

nag_ken_spe_corr_coeff (g02brc) calculates both the Kendall rank correlation coefficients and the Spearman rank correlation coefficients.

The data consists of

n

observations for each of

m

variables:

x_{i j}, i = 1, 2, \dots, n ​ and  ​ j = 1, 2, \dots, m (m, n \geq 2)

where

x_{i j}

is the

i

th observation on the

j

th variable. The function eliminates any variable

x_{i j}

, for

i = 1, 2, \dots, n

, where the argument

svar [j - 1] = 0

, and any observation

x_{i j}

, for

j = 1, 2, \dots, m

, where the argument

sobs [i - 1] = 0

The observations are first ranked as follows:

For a given variable,

j

say, each of the observations

x_{i j}

for which

sobs [i - 1] > 0

, 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

sobs [i - 1] > 0

The smallest of these valid observations for variable

j

is assigned the rank 1, the second smallest observation for variable

j

the rank 2, and so on until the largest such observation is given the rank

n_{s}

, where

n_{s}

is the number of observations for which

sobs [i - 1] > 0

If a number of cases all have the same value for a 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

sobs [i - 1] > 0

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 for which

svar [j - 1] > 0

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

sobs [i - 1] = 0

y_{i j} = 0

, for

j = 1, 2, \dots, m

For variables

j, k

the following are computed:

(a)

Kendall's tau 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_{s} (n_{s} - 1) - T_{j}] [n_{s} (n_{s} - 1) - T_{k}]}} j, k = 1, 2, \dots, m;

where	$n_{s}$ is the number of observations for which $sobs [i - 1] > 0$ ,
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

(b)

Spearman's rank correlation coefficients:

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

where

n_{s}

is the number of observations for which

sobs [i - 1] > 0

, 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

5 Arguments

1: n – IntegerInput

On entry: the number of observations in the dataset.

Constraint:

n \geq 2

2: m – IntegerInput

On entry: the number of variables.

Constraint:

m \geq 2

3: x[

n \times tdx

] – const doubleInput

On entry:

x [i - 1 \times tdx + j - 1]

must contain the

i

th observation on the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

4: tdx – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

5: svar[m] – const IntegerInput

On entry:

svar [j - 1]

indicates which variables are to be included, for the

j

th variable to be included,

svar [j - 1] > 0

. If all variables are to be included then a NULL pointer (Integer *)0 may be supplied.

Constraint:

svar [j - 1] \geq 0

, and there is at least one positive element, for

j = 1, 2, \dots, m

6: sobs[n] – const IntegerInput

On entry:

sobs [i - 1]

indicates which observations are to be included, for the

i

th observation to be included,

sobs [i - 1] > 0

. If all observations are to be included then a NULL pointer (Integer *)0 may be supplied.

Constraint:

sobs [i - 1] \geq 0

, and there are at least two positive elements, for

i = 1, 2, \dots, n

7: corr[

m \times tdc

] – doubleOutput

On exit: the upper

n_{s}

n_{s}

part of corr contains the correlation coefficients, the upper triangle 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

corr [j - 1 \times tdc + k - 1]

contains the Spearman rank correlation coefficient, and

corr [k - 1 \times tdc + j - 1]

contains Kendall's tau, for

j, k = 1, 2, \dots, n_{s}

. The diagonal will be set to 1.

8: tdc – IntegerInput

On entry: the stride separating matrix column elements in the array corr.

Constraint:

tdc \geq m

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT

On entry,

tdc = ⟨value⟩

while

m = ⟨value⟩

. These arguments must satisfy

tdc \geq m

On entry,

tdx = ⟨value⟩

while

m = ⟨value⟩

. These arguments must satisfy

tdx \geq m

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_INT_ARG_LT

On entry,

m = ⟨value⟩

.
Constraint:

m \geq 2

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 2

NE_INT_ARRAY_1

Value

⟨value⟩

given to

sobs [⟨value⟩]

not valid. Correct range for elements of sobs is

sobs [i] \geq 0

Value

⟨value⟩

given to

svar [⟨value⟩]

not valid. Correct range for elements of svar is

svar [i] \geq 0

NE_INTERNAL_ERROR

An initial error has occurred in this function. Check the function call and any array sizes.

NE_SOBS_LOW

On entry, sobs must contain at least 2 positive elements.

Too few observations have been selected.

NE_SVAR_LOW

No variables have been selected.

On entry, svar must contain at least 1 positive element.

NAG Library Function Document

nag_ken_spe_corr_coeff (g02brc)

+− 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 Library Function Documentnag_ken_spe_corr_coeff (g02brc)

+− 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 Library Function Document

nag_ken_spe_corr_coeff (g02brc)