naginterfaces.library.nonpar.concordance_​kendall

naginterfaces.library.nonpar.concordance_kendall(x)

concordance_kendall calculates Kendall’s coefficient of concordance on $k$ independent rankings of $n$ objects or individuals.

For full information please refer to the NAG Library document for g08da

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g08/g08daf.html

Parameters

xfloat, array-like, shape $(k,n)$

$x[\text{i}-1,\text{j}-1]$ must be set to the value $x_{\text{i}\text{j}}$ of object $\text{j}$ in comparison $\text{i}$, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,k$.

Returns

wfloat

The value of Kendall’s coefficient of concordance, $W$.

pfloat

The approximate significance, $p$, of $W$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $3$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 2$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

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

$$x_{i1},x_{i2},\dots ,x_{in}\text{.}$$

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 $k(n+1)/2$. 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 $k,2k,\dots ,nk$ (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.

concordance_kendall 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 ${\chi }_{n-1}^2$ 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