NAG FL Interface
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 <nag.h>
void 
g08daf_ (const double x[], const Integer *ldx, const Integer *k, const Integer *n, double rnk[], double *w, double *p, Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
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
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) array
Input

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:
$\mathbf{ldx}$ – 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:
${\mathbf{ldx}}\ge {\mathbf{k}}$.

3:
$\mathbf{k}$ – Integer
Input

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

4:
$\mathbf{n}$ – Integer
Input

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) array
Workspace


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:
$\mathbf{ifail}$ – 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 $\mathbf{1}$ 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, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 2$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{ldx}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{k}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{k}}$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{k}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{k}}\ge 2$.
 ${\mathbf{ifail}}=99$
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.
 ${\mathbf{ifail}}=399$
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.
 ${\mathbf{ifail}}=999$
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
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
10.2
Program Data
10.3
Program Results