KendallW: Kendall's Coefficient of Concordance W

In AndriSignorell/DescTools: Tools for Descriptive Statistics

Kendall's Coefficient of Concordance W

Description

Computes Kendall's coefficient of concordance, a popular measure of association. It is an index of interrater reliability of ordinal data. The coefficient could be corrected for ties within raters.

Usage

KendallW(x, correct = FALSE, test = FALSE, na.rm = NULL)

Arguments

x	n \times m matrix or dataframe, k subjects (in rows) m raters (in columns).
correct	a logical indicating whether the coefficient should be corrected for ties within raters. (Default FALSE)
test	a logical indicating whether the test statistic and p-value should be reported. (Default FALSE)
na.rm	is not longer supported and will be ignored.

Details

The test for Kendall's W is completely equivalent to friedman.test. The only advantage of this test over Friedman's is that Kendall's W has an interpretation as the coefficient of concordance. The test itself is only valid for large samples.
Kendall's W should be corrected for ties, if raters did not use a true ranking order for the subjects. The function warns if ties are present and no correction has been required.

In the presence of NAs the algorithm is switched to a generalized form for randomly incomplete datasets introduced in Brueckl (2011). This approach uses the mean Spearman \rho of all pairwise comparisons (see Kendall, 1962):

W = (1+mean(\rho)*(k-1)) / k

where k is the mean number of (pairwise) ratings per object and mean(\rho) is calculated weighted, according to Taylor (1987), since the pairwise are possibly based on a different number of ratings, what must be reflected in weights. In case of complete datasets, it yields the same results as usual implementations of Kendall's W, except for tied ranks. In case of tied ranks, the (pairwise) correction of s used, which (already with complete datasets) results in slightly different values than the tie correction explicitly specified for W.

Value

Either a single value if test = FALSE or else

a list with class “htest” containing the following components:

statistic	the value of the chi-square statistic.
p.value	the p-value for the test.
method	the character string “Kendall's coefficient of concordance W”.
data.name	a character string giving the name(s) of the data.
estimate	the coefficient of concordance.
parameter	the degrees of freedom df, the number of subjects examined and the number of raters.

Author(s)

Andri Signorell [e-mail]
based on code by Matthias Gamer [e-mail]
and Markus Brueckl [e-mail]

References

Kendall, M.G. (1948) Rank correlation methods. London: Griffin.

Kendall, M.G. (1962). Rank correlation methods (3rd ed.). London: Griffin.

Brueckl, M. (2011). Statistische Verfahren zur Ermittlung der Urteileruebereinstimmung. in: Altersbedingte Veraenderungen der Stimme und Sprechweise von Frauen, Berlin: Logos, 88–103.

Taylor, J.M.G. (1987). Kendall's and Spearman's correlation coefficients in the presence of a blocking variable. Biometrics, 43, 409–416.

See Also

cor, KappaM, CronbachAlpha, ICC, friedman.test

Examples

anxiety <- data.frame(rater1=c(3,3,3,4,5,5,2,3,5,2,2,6,1,5,2,2,1,2,4,3),
                      rater2=c(3,6,4,6,2,4,2,4,3,3,2,3,3,3,2,2,1,3,3,4),
                      rater3=c(2,1,4,4,3,2,1,6,1,1,1,2,3,3,1,1,3,3,2,2))

KendallW(anxiety, TRUE)

# with test results
KendallW(anxiety, TRUE, test=TRUE)

# example from Siegel and Castellan (1988)
d.att <- data.frame(
  id        = c(4,21,11),
  airfare   = c(5,1,4),
  climate   = c(6,7,5),
  season    = c(7,6,1),
  people    = c(1,2,3),
  program   = c(2,3,2),
  publicity = c(4,5,7),
  present   = c(3,4,6),
  interest  = c(8,8,8)
)

KendallW(t(d.att[, -1]), test = TRUE)

# which is perfectly the same as
friedman.test(y=as.matrix(d.att[,-1]), groups = d.att$id)

AndriSignorell/DescTools documentation built on Feb. 5, 2025, 7:02 a.m.