Kendall's Coefficient of Concordance W

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: deprecated and ignored

Value

if test = FALSE, a numeric scalar containing Kendall's W. Otherwise an object of class "htest" with components:

statistic: chi-squared test statistic
p.value: p-value for the test
method: description of the test and coefficient variant
data.name: name of the data
estimate: coefficient of concordance W
parameter: degrees of freedom and numbers of subjects and raters

Details

The test for Kendall's W is completely equivalent to stats::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.

Note

Based on code by Matthias Gamer and Markus Brueckl, adapted to conform to package standards.

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.

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)
#> [1] 0.5396569

# with test results
kendallW(anxiety, TRUE, test=TRUE)
#> 
#> 	Kendall's coefficient of concordance W (with ties correction)
#> 
#> data:  anxiety
#> Kendall chi-squared = 30.76, df = 19, subjects = 20, raters = 3,
#> p-value = 0.04288
#> alternative hypothesis: W is greater 0
#> sample estimates:
#>         W 
#> 0.5396569 
#> 

# 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)
#> 
#> 	Kendall's coefficient of concordance W
#> 
#> data:  t(d.att[, -1])
#> Kendall chi-squared = 13.778, df = 7, subjects = 8, raters = 3, p-value
#> = 0.05528
#> alternative hypothesis: W is greater 0
#> sample estimates:
#>         W 
#> 0.6560847 
#> 

# which is perfectly the same as
friedman.test(y=as.matrix(d.att[,-1]), groups = d.att$id)
#> 
#> 	Friedman rank sum test
#> 
#> data:  as.matrix(d.att[, -1])
#> Friedman chi-squared = 13.778, df = 7, p-value = 0.05528
#>