# 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.

## 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.

Kendall and Gibbons (1990) also show *W* is linearly related to the mean value of the Spearman's rank correlation coefficients between all possible pairs of rankings between judges

## Incomplete Blocks

When the judges evaluate only some subset of the *n* objects, and when the correspondent block design is a (n, m, r, p, λ)-design (note the different notation). In other words, when

- each judge ranks the same number
*p*of objects for some , - every object is ranked exactly the same total number
*r*of times, - and each pair of objects is presented together to some judge a total of exactly λ times, , a constant for all pairs.

Then Kendall's *W* is defined as [2]

If and so that each judge ranks all *n* objects, the formula above is equivalent to the original one.

## 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.[3]

## Significance Tests

In the case of complete ranks, a commonly used significance test for *W* against a null hypothesis of no agreement (i.e. random rankings) is given by Kendall and Gibbons (1990)[4]

Where the test statistic takes a chi-squared distribution with degrees of freedom.

In the case of incomplete rankings (see above), this becomes

Where again, there are degrees of freedom.

Legendre[5] compared via simulation the power of the chi-square and permutation testing approaches to determining significance for Kendall's *W*. Results indicated the chi-square method was overly conservative compared to a permutation test when . Marozzi[6] extended this by also considering the *F* test, as proposed in the original publication introducing the *W* statistic by Kendall & Babington Smith (1939):

Where the test statistic follows an F distribution with and degrees of freedom. Marozzi found the *F* test performs approximately as well as the permutation test method, and may be preferred to when is small, as it is computationally simpler.

## Notes

- Dodge (2003): see "concordance, coefficient of"
- Gibbons & Chakraborti (2003)
- Siegel & Castellan (1988, p. 266)
- Kendall, Maurice G. (Maurice George), 1907-1983. (1990).
*Rank correlation methods*. Gibbons, Jean Dickinson, 1938- (5th ed.). London: E. Arnold. ISBN 0195208374. OCLC 21195423.CS1 maint: multiple names: authors list (link) - 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.

## 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. - Kendall, M. G., & Gibbons, J. D. (1990). Rank correlation methods. New York, NY : Oxford University Press.
- 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 - 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 978-0-07-057357-4. - Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003).
*Nonparametric Statistical Inference*(4th ed.). New York: Marcel Dekker. pp. 476–482. ISBN 978-0-8247-4052-8.