, 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.

nag_nonpar_concordance_kendall (g08da) 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)).

References

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

Parameters

Compulsory Input Parameters

1: $x (ldx, n)$ – double array: ldx, the first dimension of the array, must satisfy the constraint $ldx \geq k$ .
$x (i, j)$ 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: $k$ – int64int32nag_int scalar: $k$ , the number of comparisons.

Constraint: $k \geq 2$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the array x.
$n$ , the number of objects.

Constraint: $n \geq 2$ .

Output Parameters

1: $w$ – double scalar: The value of Kendall's coefficient of concordance, $W$ .
2: $p$ – double scalar: The approximate significance, $p$ , of $W$ .
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

n < 2

$ifail = 2$

On entry,

ldx < k

$ifail = 3$

On entry,

k \leq 1

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

All computations are believed to be stable. The statistic

W

should be accurate enough for all practical uses.

Further Comments

The time taken by nag_nonpar_concordance_kendall (g08da) is approximately proportional to the product

n k

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.

Open in the MATLAB editor: g08da_example

function g08da_example


fprintf('g08da example results\n\n');

x = [1,   4.5, 2,   4.5, 3,   7.5, 6,  9,    7.5, 10;
     2.5, 1,   2.5, 4.5, 4.5, 8,   9,  6.5, 10,    6.5;
     2,   1,   4.5, 4.5, 4.5, 4.5, 8,  8,    8,   10  ];

fprintf('Kendall''s coefficient of concordance\n\n');
% Table Labels
labrow = 'Character';
rlabs  = {'Comparison 1 scores';
          'Comparison 2 scores';
          'Comparison 3 scores'};
labcol = 'None';
clabs  = {'       '};
ncols  = int64(80);
indent = int64(0);

[ifail] =  x04cb( ...
                 'General', ' ', x, 'F5.1', 'Data values', labrow, ...
                 rlabs, labcol, clabs, ncols, indent);

k = int64(3);
[w, p, ifail] = g08da(x, k);

fprintf('\nKendall''s coefficient = %8.3f\n', w);
fprintf('         Significance = %8.3f\n', p);

g08da example results

Kendall's coefficient of concordance

 Data values
 Comparison 1 scores   1.0  4.5  2.0  4.5  3.0  7.5  6.0  9.0  7.5 10.0
 Comparison 2 scores   2.5  1.0  2.5  4.5  4.5  8.0  9.0  6.5 10.0  6.5
 Comparison 3 scores   2.0  1.0  4.5  4.5  4.5  4.5  8.0  8.0  8.0 10.0

Kendall's coefficient =    0.828
         Significance =    0.008

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox