NAG Library Routine Document

G08DAF

, 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 Parameters

1: X(LDX,N) – REAL (KIND=nag_wp) arrayInput: 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 – INTEGERInput: 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 – INTEGERInput: On entry: $k$ , the number of comparisons.
Constraint: $K \geq 2$ .
4: N – INTEGERInput: On entry: $n$ , the number of objects.
Constraint: $N \geq 2$ .
5: RNK(LDX,N) – REAL (KIND=nag_wp) arrayWorkspace
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 – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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 < 2

$IFAIL = 2$

On entry,

LDX < K

$IFAIL = 3$

On entry,

K \leq 1

7 Accuracy

All computations are believed to be stable. The statistic

W

should be accurate enough for all practical uses.

8 Further Comments

The time taken by G08DAF is approximately proportional to the product

n k

9 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.

NAG Library Routine DocumentG08DAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G08DAF