# Kendall's W

**Kendall's W** (also known as

**Kendall's coefficient of concordance**) is a non-parametric statistic. It is a normalization of the statistic of the Friedman test, and can be used for assessing agreement among raters. Kendall's

*W*ranges from 0 (no agreement) to 1 (complete agreement).

Suppose, for instance, that a number of people have been asked to rank a list of political concerns, from most important to least important. Kendall's *W* can be calculated from these data. If the test statistic *W* is 1, then all the survey respondents have been unanimous, and each respondent has assigned the same order to the list of concerns. If *W* is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of *W* indicate a greater or lesser degree of unanimity among the various responses.

While tests using the standard Pearson correlation coefficient assume normally distributed values and compare two sequences of outcomes at a time, Kendall's *W* makes no assumptions regarding the nature of the probability distribution and can handle any number of distinct outcomes.

*W* is linearly related to the mean value of the Spearman's rank correlation coefficients between all pairs of the rankings over which it is calculated.

## Definition

Suppose that object *i* is given the rank *r _{i,j}* by judge number

*j*, where there are in total

*n*objects and

*m*judges. Then the total rank given to object

*i*is

and the mean value of these total ranks is

The sum of squared deviations, *S*, is defined as

and then Kendall's *W* is defined as^{[1]}

If the test statistic *W* is 1, then all the judges or survey respondents have been unanimous, and each judge or respondent has assigned the same order to the list of objects or concerns. If *W* is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of *W* indicate a greater or lesser degree of unanimity among the various judges or respondents.

Legendre^{[2]} discusses a variant of the *W* statistic which accommodates ties in the rankings and also describes methods of making significance tests based on *W*. Legendre compared via simulation the Friedman test and its permutation version. Unfortunately,
the simulation study of Legendre was very limited because it considered neither the copula aspect nor the *F* test. Kendall *W* is a rank-based correlation measure, and therefore it is not affected by the marginal distributions of the underlying variables, but only by the copula of the multivariate distribution. Marozzi^{[3]} extended the simulation study of Legendre by considering the copula aspect as well as the *F* test. It is shown that the Friedman test is too conservative and less powerful than both the *F* test and the permutation test for concordance which always have a correct size and behave alike. The *F* test should be preferred because it is computationally much easier. Surprisingly, the power function of the
tests is not much affected by the type of copula.

## Correction for ties

When tied values occur, they are each given the average of the ranks that would have been given had no ties occurred. For example, the data set {80,76,34,80,73,80} has values of 80 tied for 4th, 5th, and 6th place; since the mean of {4,5,6} = 5, ranks would be assigned to the raw data values as follows: {5,3,1,5,2,5}.

The effect of ties is to reduce the value of *W*; however, this effect is small unless there are a large number of ties. To correct for ties, assign ranks to tied values as above and compute the correction factors

where *t _{i}* is the number of tied ranks in the

*i*th group of tied ranks, (where a group is a set of values having constant (tied) rank,) and

*g*is the number of groups of ties in the set of ranks (ranging from 1 to

_{j}*n*) for judge

*j*. Thus,

*T*is the correction factor required for the set of ranks for judge

_{j}*j*, i.e. the

*j*th set of ranks. Note that if there are no tied ranks for judge

*j*,

*T*equals 0.

_{j}With the correction for ties, the formula for *W* becomes

where *R _{i}* is the sum of the ranks for object

*i*, and is the sum of the values of

*T*over all

_{j}*m*sets of ranks.

^{[4]}

## See also

## Notes

- ↑ Dodge (2003): see "concordance, coefficient of"
- ↑ Legendre (2005)
- ↑ Marozzi, Marco (2014). "Testing for concordance between several criteria".
*Journal of Statistical Computation and Simulation*.**84**(9): 1843-1850. doi:10.1080/00949655.2013.766189. Retrieved 30 September 2016. - ↑ Siegel & Castellan (1988, p. 266)

## References

- Kendall, M. G.; Babington Smith, B. (Sep 1939). "The Problem of
*m*Rankings".*The Annals of Mathematical Statistics*.**10**(3): 275–287. doi:10.1214/aoms/1177732186. JSTOR 2235668. - Corder, G.W., Foreman, D.I. (2009).
*Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach*Wiley, ISBN 978-0-470-45461-9 - Dodge, Y (2003)
*The Oxford Dictionary of Statistical Terms*, OUP. ISBN 0-19-920613-9

^{[1]}

- Legendre, P (2005) Species Associations: The Kendall Coefficient of Concordance Revisited.
*Journal of Agricultural, Biological and Environmental Statistics*, 10(2), 226–245. - Siegel, Sidney; Castellan, N. John, Jr. (1988).
*Nonparametric Statistics for the Behavioral Sciences*(2nd ed.). New York: McGraw-Hill. p. 266. ISBN 0-07-057357-3.

- ↑ Marozzi, Marco (2014). "Testing for concordance between several criteria".
*Journal of Statistical Computation and Simulation*.**84**(9): 1843-1850. doi:10.1080/00949655.2013.766189. Retrieved 30 September 2016.