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# NAG Toolbox: nag_nonpar_concordance_kendall (g08da)

## Purpose

nag_nonpar_concordance_kendall (g08da) calculates Kendall's coefficient of concordance on $k$ independent rankings of $n$ objects or individuals.

## Syntax

[w, p, ifail] = g08da(x, k, 'n', n)
[w, p, ifail] = nag_nonpar_concordance_kendall(x, k, 'n', n)

## 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
 $xi1,xi2,…,xin.$
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 ${r}_{ij}$ 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 $k\left(n+1\right)/2$. 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 $k,2k,\dots ,nk$ (or some permutation thereof). The total squared deviation of these values is ${k}^{2}\left({n}^{3}-n\right)/12$.
Kendall's coefficient of concordance is the ratio
 $W = ∑ j=1 n Rj - 12 kn+1 2 112 k2 n3-n$
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 $T=\sum \left({t}^{3}-t\right)/12$, 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 $n>7,k\left(n-1\right)W$ approximately follows a ${\chi }_{n-1}^{2}$ distribution, so large values of $W$ imply rejection of ${H}_{0}$.) ${H}_{0}$ is rejected by a test of chosen size $\alpha$ if $p<\alpha$. If $n\le 7$, 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:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{n}}\right)$ – double array
ldx, the first dimension of the array, must satisfy the constraint $\mathit{ldx}\ge {\mathbf{k}}$.
${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value ${x}_{\mathit{i}\mathit{j}}$ of object $\mathit{j}$ in comparison $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$ and $\mathit{j}=1,2,\dots ,n$.
2:     $\mathrm{k}$int64int32nag_int scalar
$k$, the number of comparisons.
Constraint: ${\mathbf{k}}\ge 2$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array x.
$n$, the number of objects.
Constraint: ${\mathbf{n}}\ge 2$.

### Output Parameters

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

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<2$.
${\mathbf{ifail}}=2$
 On entry, $\mathit{ldx}<{\mathbf{k}}$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{k}}\le 1$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
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 $nk$.

## 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 $\left(p=0.008<0.01\right)$, indicating that the null hypothesis of there being no agreement between the three rankings x, y, z may be rejected with reasonably high confidence.
```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
```

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