Kendall's W

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

${\displaystyle R_{i}=\sum _{j=1}^{m}r_{i,j},}$

and the mean value of these total ranks is

${\displaystyle {\bar {R}}={\frac {1}{n}}\sum _{i=1}^{n}R_{i}.}$

The sum of squared deviations, S, is defined as

${\displaystyle S=\sum _{i=1}^{n}(R_{i}-{\bar {R}})^{2},}$

and then Kendall's W is defined as[1]

${\displaystyle W={\frac {12S}{m^{2}(n^{3}-n)}}.}$

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 ${\displaystyle m \choose {2}}$ possible pairs of rankings between judges

${\displaystyle {\bar {r}}_{s}={\frac {mW-1}{m-1}}}$

In some cases, the importance of the raters (experts) might not be the same as each other. In this case, the Weighted Kendall's W should be used.[4] Suppose that object ${\displaystyle i}$ is given the rank ${\displaystyle r_{ij}}$ by judge number ${\displaystyle j}$, where there are in total ${\displaystyle n}$ objects and ${\displaystyle m}$ judges. Also, the weight of judge ${\displaystyle j}$ is shown by ${\displaystyle \vartheta _{j}}$ (in real-world situation, the importance of each rater can be different). Indeed, the weight of judges is ${\displaystyle \vartheta _{j}(j=1,2,...,m)}$. Then, the total rank given to object ${\displaystyle i}$ is

${\displaystyle R_{i}=\sum _{j=1}^{m}\vartheta _{j}r_{ij}}$

and the mean value of these total ranks is,

${\displaystyle {\bar {R}}={\frac {1}{n}}\sum _{i=1}^{n}R_{i}}$

The sum of squared deviations, ${\displaystyle S}$, is defined as,

${\displaystyle S=\sum _{i=1}^{n}(R_{i}-{\bar {R}})^{2}}$

and then Weighted Kendall's W is defined as,

${\displaystyle W_{w}={\frac {12S}{(n^{3}-n)}}}$

The above formula is suitable when we do not have any tie rank.

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)[5]

${\displaystyle \chi ^{2}=m(n-1)W}$

Where the test statistic takes a chi-squared distribution with ${\displaystyle df=n-1}$ degrees of freedom.

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

${\displaystyle \chi ^{2}={\frac {\lambda (n^{2}-1)}{k+1}}W}$

Where again, there are ${\displaystyle df=n-1}$ degrees of freedom.

Legendre[6] 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 ${\displaystyle m<20}$. Marozzi[7] extended this by also considering the F test, as proposed in the original publication introducing the W statistic by Kendall & Babington Smith (1939):

${\displaystyle F={\frac {W(m-1)}{1-W}}}$

Where the test statistic follows an F distribution with ${\displaystyle v_{1}=n-1-(2/m)}$ and ${\displaystyle v_{2}=(m-1)v_{1}}$ degrees of freedom. Marozzi found the F test performs approximately as well as the permutation test method, and may be preferred to when ${\displaystyle m}$ is small, as it is computationally simpler.

Kendall's W and Weighted Kendall's W are implemented in MATLAB,[8] SPSS, R,[9] and other statistical software packages.

• Maurice Kendall
• Kendall's tau
• Spearman's rank correlation coefficient
• Friedman test

• 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003). Nonparametric Statistical Inference (4th ed.). New York: Marcel Dekker. pp. 476–482. ISBN 978-0-8247-4052-8.