void	nag_ken_spe_corr_coeff (Integer n, Integer m, const double x[], Integer tdx, const Integer svar[], const Integer sobs[], double corr[], Integer tdc, NagError *fail)

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

4

References

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

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 $n_{s}$ by $n_{s}$ 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 3.7 in How to Use the NAG Library and its Documentation).

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.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

nag_ken_spe_corr_coeff (g02brc) is not threaded in any implementation.

9

Further Comments

None.

10

Example

A program to calculate the Kendall and Spearman rank correlation coefficients from a set of data.

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

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