Methods and formulas for Kendall's coefficients for Attribute Agreement Analysis

Select the method or formula of your choice.

Kendall's coefficient of concordance

Use Kendall's statistic with ordinal data of three or more levels.

In the description of the method, without loss of generality, we assume that a single rating on each subject is made by each rater, and there are k raters per subject. Then, to calculate Kendall's coefficient, the k raters represent the k trials for each rater.

Suppose data are arranged into a k x N table with each row representing the ranks assigned by a particular rater to the N subjects.

Formulas

When the true standard is not known, Minitab estimates Kendall's coefficient by:

Notation

TermDescription
Nthe number of subjects
Σ Ri2the sum of the squared sums of ranks for each of the ranked N subjects
Kthe number of appraisers
TjTj assigns the average of ratings to tied observation
TermDescription
tithe number of tied ranks in the ith grouping of ties
gjthe number of groups of ties in the jth set of ranks

Testing significance of Kendall's coefficient of concordance

To test the significance of Kendall's coefficient, use:

c 2= k (N – 1) W

Notation

TermDescription
c 2is distributed as chi-square with N – 1 degrees of freedom
kthe number of appraisers
Nthe number of subjects
Wthe calculated Kendall's coefficient

Kendall's correlation coefficient

Use Kendall's statistic with ordinal data of three or more levels.

In the description of the method, without loss of generality, we assume that a single rating on each subject is made by each rater, and there are k raters per subject. Then, to calculate Kendall's correlation coefficient, the k raters represent the k trials made by all the raters.

When the true standard is known, Minitab estimates Kendall's correlation coefficient by calculating the average of the Kendall's coefficients between each appraiser and the standard.

The Kendall's correlation coefficient for the agreement of the trials with the known standard is the average of the Kendall correlation coefficients across trials.

Formulas

Minitab calculates Kendall's coefficient between each trial and the standard using:

Notation

TermDescription
TX number of pairs tied on X = 0.5 Σi ni+ (ni+– 1)
TY number of pairs tied on Y = 0.5 Σj n+j (n+j– 1)
Cnumber of concordant pairs = Σi<kΣj<l nij nkl
Dnumber of discordant pairs = Σi<kΣj>l nij nkl
TermDescription
ni+number of observations in the ith row
n+jnumber of observations in the jth column
nij observations in the cell corresponding to ith row and jth column
nkl observations in the cell corresponding to kth row and lth column
n++ total number of observations

Reference

A. Agresti (1984). Analysis of Ordinal Categorical Data, John Wiley & Sons.

Testing significance of Kendall's correlation coefficient

Formula

To test the significance of Kendall's coefficient when the true standard is known, use:

when Tc > 0

use:

when Tc ≤ 0

Notation

TermDescription
Tcthe average of the Kendall correlation coefficients between each appraiser and the standard
Nthe total number of subjects
kthe number of raters
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