KendallTauA: Kendall's $\tau_{a}$

Description

Calculate Kendall's tau-a statistic, a measure of association for ordinal factors in a two-way table.
The function has interfaces for a table (matrix) and for single vectors.

Usage

KendallTauA(x, y = NULL, direction = c("row", "column"), conf.level = NA, ...)

Value

a single numeric value if no confidence intervals are requested,

and otherwise a numeric vector with 3 elements for the estimate, the lower and the upper confidence interval

Arguments

x: a numeric vector or a table. A matrix will be treated as table.
y: NULL (default) or a vector with compatible dimensions to x. If y is provided, table(x, y, ...) is calculated.
direction: direction of the calculation. Can be "row" (default) or "column", where "row" calculates Kendall's tau-a (R|C) ("column dependent").
conf.level: confidence level of the interval. If set to NA (which is the default) no confidence interval will be calculated.
...: further arguments are passed to the function table, allowing i.e. to set useNA. This refers only to the vector interface.

Author

Andri Signorell <andri@signorell.net>

Details

Kendall's tau coefficient (sometimes called "Kendall rank correlation coefficient"), is a statistic used to measure the association between two measured quantities. It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities.
Kendall's tau-a is computed as $$ \tau_a(C|R) = \frac{P-Q}{\frac{1}{2} \cdot n \cdot (n-1)}$$ where P equals twice the number of concordances and Q twice the number of discordances. Its range is [-1, 1].
(Note that Kendall tau-a does not take into consideration any ties, which makes it unpractical. Consider using KendallTauB (Tau-b) when ties are present.)

References

Agresti, A. (2002) Categorical Data Analysis. John Wiley & Sons, pp. 57-59.

Hollander, M, Wolfe, D. A., Chicken, E. (2014) Nonparametric Statistical Methods, Third edition, Wiley,

Liebetrau, A. M. (1983) Measures of Association, Sage University Papers Series on Quantitative Applications in the Social Sciences, 07-004. Newbury Park, CA: Sage, pp. 49-56

Examples

Run this code


# example in:
# http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf
# pp. S. 1821

tab <- as.table(rbind(c(26,26,23,18,9),c(6,7,9,14,23)))

# Kendall's tau-a C|R
KendallTauA(tab, direction="column", conf.level=0.95)
# Kendall's tau-a R|C
KendallTauA(tab, direction="row", conf.level=0.95)

# http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf
# pp. 1814 (143)
tab <- as.table(cbind(c(11,2),c(4,6)))

KendallTauA(tab, direction="row", conf.level=0.95)
KendallTauA(tab, direction="column", conf.level=0.95)

# Liebetrau, pp. 52
x <- c(1,2,2,3,3,3,4,5)
y <- c(1,3,2,1,5,3,4,5)

ConDisPairs(table(x, y))
KendallTauA(x, y, conf.level=0.95)

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KendallTauA: Kendall's \(\tau_{a}\)