DescTools (version 0.99.14)

GoodmanKruskalTauA: Goodman Kruskal's Tau a

Description

Calculate Goodman Kruskal'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

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

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 Goodman Kruskal'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.

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

Details

Goodman and Kruskal's tau-a is a measure of categorical association which is based entirely on the observed data and possesses a clear interpretation in terms of proportional reduction in error. It gives the probabilities of correctly assigning cases to one set of categories improved by the knowledge of another set of categories. The statistic is asymmetric and yields different results predicting row assignments based on columns than from column assignments based on rows. Goodman Kruskal'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. It's range is [0, 1]. Goodman Kruskal tau reduces to $\phi^2$ (see: Phi) in the 2x2-table case. (Note that Goodman Kruskal tau-a does not take into consideration any ties, which makes it unpractical.)

References

Agresti, A. (2002) Categorical Data Analysis. John Wiley & Sons, pp. 57-59. Goodman, L. A., & Kruskal, W. H. (1954) Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732-764. Somers, R. H. (1962) A New Asymmetric Measure of Association for Ordinal Variables, American Sociological Review, 27, 799-811. Goodman, L. A., & Kruskal, W. H. (1963) Measures of association for cross classifications III: Approximate sampling theory. Journal of the American Statistical Association, 58, 310-364. http://support.sas.com/onlinedoc/913/getDoc/en/statug.hlp/freq_sect18.htm http://support.sas.com/onlinedoc/913/getDoc/en/statug.hlp/freq_sect20.htm

See Also

ConDisPairs yields concordant and discordant pairs Other association measures: GoodmanKruskalTauA (Tau a), cor (method="kendall") for Tau b, StuartTauC, GoodmanKruskalGamma Lambda, UncertCoef, MutInf

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)))

# Goodman Kruskal's tau-a C|R
GoodmanKruskalTauA(tab, direction="column", conf.level=0.95)
# Goodman Kruskal's tau-a R|C
GoodmanKruskalTauA(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)))

GoodmanKruskalTauA(tab, direction="row", conf.level=0.95)
GoodmanKruskalTauA(tab, direction="column", conf.level=0.95)
# reduces to:
Phi(tab)^2

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