kendallNA {irrNA}

Kendall's coefficient of concordance W – generalized for randomly incomplete datasets

Description

This function computes Kendall's coefficient of concordance W that is an index of interrater reliability for ordinal ratings. This function also works on incomplete datasets without any imputation of missing values or (row- or cloumn-wise) omissions of data.

Usage

kendallNA(X)

Arguments

X

n*m matrix or dataframe; n objects (rows), k raters (columns)

Details

This function is able to calculate W, also on randomly incomplete (i.e. unbalanced) data sets. Therefor it uses the mean Spearman's \rho of all pairwise comparisons, see Kendall (1962):

W = [1 + mean \rho_S * (k-1)] / k

where k is the mean number of (pairwise) ratings per object and mean \rho_S is calculated weighted, according to Taylor (1987), since the pairwise \rho_S are possibly based on a different number of ratings, what must be reflected in weights.
Thus, an imputation of missing values or (row- or cloumn-wise) omissions of data are obsolete. 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 \rho_S is used, which (already with complete datasets) results in slightly different values than the tie correction explicitly specified for W.
More details are given in Brueckl (2011).

Value

amrho

mean Spearman's \rho

amk

mean number of (pairwise) ratings per object

W

Kendall's coefficient of concordance among raters

chisqu

value of the \chi-squared test statistic

df

degrees of freedom

p

one-tailed type I error probability (statistical significance)

Author(s)

Markus Brueckl

References

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

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

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

iccNA, kendall

Examples

# Example 1:
data(ConsistNA)
# ConsistNA exhibits missing values and a perfect concordance
# between raters:
ConsistNA
# Common W-algorithms fail, since each row as well as each 
# column of ConsistNA exhibits unfilled cells and these missing 
# data are omitted column-wise or row-wise (please install and 
# load the irr package):
#kendall(ConsistNA)
# But the generalization of Kendall's W implemeted in irrNA 
# is able to assess the perfect concordance, assuming that 
# the data were at least ordinally scaled and not tied, e.g. 
# that each rater really ranked the objects that he rated 
# without giving equal ranks to two or more objects.
kendallNA(ConsistNA)
#
# Example 2:
data(IndepNA)
# IndepNA exhibits missing values and zero variance between 
# the raters (just as well as between the objects):
IndepNA
# Common W-algorithms fail:
#kendall(IndepNA)
# kendallNA includes all (rater-pairwise) available data in 
# its calculation (e.g. only Objects 1--4 when Rater1 and 
# Rater2 are correlated):
kendallNA(IndepNA)
#
# Example 3:
data(IndepW)
# IndepW exhibits missing values and a mean Spearman's rho,
# that equals zero:
IndepW
# Again, common W-algorithms fail,
#kendall(IndepW)
# while kendallNA includes all (rater-pairwise) available 
# data:
kendallNA(IndepW)