Coefficient of Concordance

Proposed by Maurice G. Kendall and Bernard Babington Smith, Kendall's coefficient of concordance (W) is a measure of the agreement among several (m) quantitative or semiquantitative variables that are assessing a set of n objects of interest. In the social sciences, the variables are often people, called judges, assessing different subjects or situations. In community ecology, they may be species whose abundances are used to assess habitat quality at study sites. In taxonomy, they may be characteristics measured over different species, biological populations, or individuals.

There is a close relationship between Milton Friedman's two-way analysis of variance without replication by ranks and Kendall's coefficient of concordance. They address hypotheses concerning the same data table, and they use the same χ2 statistic for testing. They differ only in the formulation of their respective null hypothesis. Consider Table 1, which contains illustrative data. In Friedman's test, the null hypothesis is that there is no [Page 165]real difference among the n objects (sites, rows of Table 1) because they pertain to the same statistical population. Under the null hypothesis, they should have received random ranks along the various variables, so that their sums of ranks should be approximately equal. Kendall's test focuses on the m variables. If the null hypothesis of Friedman's test is true, this means that the variables have produced rankings of the objects that are independent of one another. This is the null hypothesis of Kendall's test.

Computing Kendall's W

There are two ways of computing Kendall's W statistic (first and second forms of Equations 1 and 2); they lead to the same result. S or S′ is computed first from the row-marginal sums of ranks Ri received by the objects:

where S is a sum-of-squares statistic over the row sums of ranks Ri and

is the mean of the Ri values. Following that, Kendall's W statistic can be obtained from either of the following formulas:

or

where n is the number of objects and m is the number of variables. T is a correction factor for tied ranks:

in which tk is the number of tied ranks in each (k) of g groups of ties. The sum is computed over all groups of ties found in all m variables of the data table. T = 0 when there are no tied values.

Kendall's W is an estimate of the variance of the row sums of ranks Ri divided by the maximum possible value the variance can take; this occurs when all variables are in total agreement. Hence 0≤W≤1, 1 representing perfect concordance. To derive the formulas for W (Equation 2), one has to know that when all variables are in perfect agreement, the sum of all sums of ranks in the data table (right-hand column of Table 1) is mn(n + 1)/2 and that the sum of squares of the sums of all ranks is m2n(n + 1)(2n + 1)/6 (without ties).

There is a close relationship between Charles Spearman's correlation coefficient rs and Kendall's W statistic: W can be directly calculated from the [Page 166]mean

of the pairwise Spearman correlations rs using the following

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