NAG Library Routine Document
g08daf (concordance_kendall)
1
Purpose
g08daf calculates Kendall's coefficient of concordance on $k$ independent rankings of $n$ objects or individuals.
2
Specification
Fortran Interface
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 nagmk26.h
void 
g08daf_ (const double x[], const Integer *ldx, const Integer *k, const Integer *n, double rnk[], double *w, double *p, Integer *ifail) 

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
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
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.
g08daf returns the value of
$W$, and also an approximation,
$p$, of the significance of the observed
$W$. (For
$n>7,k\left(n1\right)W$ approximately follows a
${\chi}_{n1}^{2}$ 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)).
4
References
Siegel S (1956) Nonparametric Statistics for the Behavioral Sciences McGraw–Hill
5
Arguments
 1: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ 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: $\mathbf{ldx}$ – IntegerInput

On entry: the first dimension of the arrays
x and
rnk as declared in the (sub)program from which
g08daf is called.
Constraint:
${\mathbf{ldx}}\ge {\mathbf{k}}$.
 3: $\mathbf{k}$ – IntegerInput

On entry: $k$, the number of comparisons.
Constraint:
${\mathbf{k}}\ge 2$.
 4: $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of objects.
Constraint:
${\mathbf{n}}\ge 2$.
 5: $\mathbf{rnk}\left({\mathbf{ldx}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace

 6: $\mathbf{w}$ – Real (Kind=nag_wp)Output

On exit: the value of Kendall's coefficient of concordance, $W$.
 7: $\mathbf{p}$ – Real (Kind=nag_wp)Output

On exit: the approximate significance, $p$, of $W$.
 8: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{n}}<2$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{ldx}}<{\mathbf{k}}$. 
 ${\mathbf{ifail}}=3$

On entry,  ${\mathbf{k}}\le 1$. 
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation 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
g08daf is not threaded in any implementation.
The time taken by g08daf is approximately proportional to the product $nk$.
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
$\left(p=0.008<0.01\right)$, indicating that the null hypothesis of there being no agreement between the three rankings
X,
Y,
Z may be rejected with reasonably high confidence.
10.1
Program Text
Program Text (g08dafe.f90)
10.2
Program Data
Program Data (g08dafe.d)
10.3
Program Results
Program Results (g08dafe.r)