Integer, Intent (In)	::	ldx, k, n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(ldx,n)
Real (Kind=nag_wp), Intent (Inout)	::	rnk(ldx,n)
Real (Kind=nag_wp), Intent (Out)	::	w, p

C Header Interface

#include <nag.h>

void	g08daf_ (const double x[], const Integer ldx, const Integer k, const Integer n, double rnk[], double w, double p, Integer ifail)

The routine may be called by the names g08daf or nagf_nonpar_concordance_kendall.

3 Description

Kendall's coefficient of concordance measures the degree of agreement between

k

comparisons of

n

objects, the scores in the

i

th comparison being denoted by

x_{i 1}, x_{i 2}, \dots, x_{i n} .

The hypothesis under test,

H_{0}

, often called the null hypothesis, is that there is no agreement between the comparisons, and this is to be tested against the alternative hypothesis,

H_{1}

, that there is some agreement.

The

n

scores for each comparison are ranked, the rank

r_{i j}

denoting the rank of object

j

in comparison

i

, and all ranks lying between

1

and

n

. Average ranks are assigned to tied scores.

For each of the

n

objects, the

k

ranks are totalled, giving rank sums

R_{j}

, for

j = 1, 2, \dots, n

. Under

H_{0}

, all the

R_{j}

would be approximately equal to the average rank sum

k (n + 1) / 2

. The total squared deviation of the

R_{j}

from this average value is, therefore, a measure of the departure from

H_{0}

exhibited by the data. If there were complete agreement between the comparisons, the rank sums

R_{j}

would have the values

k, 2 k, \dots, n k

(or some permutation thereof). The total squared deviation of these values is

k^{2} (n^{3} - n) / 12

Kendall's coefficient of concordance is the ratio

W = \frac{\sum_{j = 1}^{n} {(R_{j} - \frac{1}{2} k (n + 1))}^{2}}{\frac{1}{12} k^{2} (n^{3} - n)}

and lies between

0

and

1

, the value

0

indicating complete disagreement, and

1

indicating complete agreement.

If there are tied rankings within comparisons,

W

is corrected by subtracting

k \sum T

from the denominator, where

T = \sum (t^{3} - t) / 12

, each

t

being the number of occurrences of each tied rank within a comparison, and the summation of

T

being over all comparisons containing ties.

g08daf returns the value of

W

, and also an approximation,

p

, of the significance of the observed

W

. (For

n > 7, k (n - 1) W

approximately follows a

χ_{n - 1}^{2}

distribution, so large values of

W

imply rejection of

H_{0}

H_{0}

is rejected by a test of chosen size

α

p < α

. If

n \leq 7

, tables should be used to establish the significance of

W

(e.g., Table R of Siegel (1956)).

4 References

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

5 Arguments

1: $x (ldx, n)$ – Real (Kind=nag_wp) array Input: On entry: $x (i, j)$ must be set to the value $x_{i j}$ of object $j$ in comparison $i$ , for $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, n$ .
2: $ldx$ – Integer Input: On entry: the first dimension of the arrays x and rnk as declared in the (sub)program from which g08daf is called.

Constraint: $ldx \geq k$ .
3: $k$ – Integer Input: On entry: $k$ , the number of comparisons.

Constraint: $k \geq 2$ .
4: $n$ – Integer Input: On entry: $n$ , the number of objects.

Constraint: $n \geq 2$ .
5: $rnk (ldx, n)$ – Real (Kind=nag_wp) array Workspace
6: $w$ – Real (Kind=nag_wp) Output: On exit: the value of Kendall's coefficient of concordance, $W$ .
7: $p$ – Real (Kind=nag_wp) Output: On exit: the approximate significance, $p$ , of $W$ .
8: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ 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 $−1$ or $1$ 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).

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:

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

$ifail = 2$: On entry, $ldx = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $ldx \geq k$ .

$ifail = 3$: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 2$ .

$ifail = - 99$: 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

All computations are believed to be stable. The statistic

W

should be accurate enough for all practical uses.

8 Parallelism and Performance

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

g08daf is not threaded in any implementation.

9 Further Comments

The time taken by g08daf is approximately proportional to the product

n k

10 Example

This example is taken from page 234 of Siegel (1956). The data consists of

10

objects ranked on three different variables: X, Y and Z. The computed values of Kendall's coefficient is significant at the

1 %

level of significance

(p = 0.008 < 0.01)

, indicating that the null hypothesis of there being no agreement between the three rankings X, Y, Z may be rejected with reasonably high confidence.

g08da: FL CL CPP AD PY MB

NAG FL Interfaceg08daf (concordance_​kendall)

▸▿ 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
g08daf (concordance_kendall)