Skip to contents

Goodman Kruskal's Tau

Source: R/gkTau.R
gkTau.Rd

Calculate Goodman-Kruskal's tau, a measure of association for nominal variables in a two-way table. The function accepts either a contingency table or two vectors.

Usage

gkTau(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  direction = c("row", "column"),
  ...
)

Arguments

x

a vector (typically a factor, character, or numeric vector) containing categorical data, or a contingency table.

y

NULL (default) or a vector with compatible dimensions to x. If supplied, table(x, y, \dots) is calculated.

conf.level

confidence level of the interval. If set to NA (the default), only the point estimate is returned.

sides

character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().

direction

direction of the calculation. Must be "row" (default) or "column". "row" gives tau (R|C), i.e. the row variable is the dependent one and is predicted from the column variable; "column" gives tau (C|R).

...

further arguments are passed to the function table(), allowing i.e. to set useNA. This refers only to the vector interface; supplying them without y is an error.

Value

if conf.level = NA, a numeric scalar. Otherwise a named numeric vector with elements:

est

point estimate of Goodman-Kruskal's tau

lci

lower confidence interval bound

uci

upper confidence interval bound

Details

Goodman-Kruskal tau measures association for cross tabulations of nominal level variables. Goodman-Kruskal tau is based on random category assignment. It measures the percentage improvement in predictability of the dependent variable (column or row variable) given the value of other variables (row or column variables). Goodman-Kruskal tau is the same as Goodman-Kruskal lambda except the calculations of the tau statistic are based on assignment probabilities specified by marginal or conditional proportions. Misclassification probabilities are based on random category assignment with probabilities specified by marginal or conditional proportion.

Goodman Kruskal tau reduces to \(\phi^2\) (see: phi()) in the 2x2-table case.

The measure lies in \([0, 1]\) by construction. Both ends are reached by cancellation, so an estimate within a few machine epsilons of a bound is reported as that bound. Tau is undefined when the dependent variable has fewer than two non-empty categories (the denominator is then zero), which is signalled with an error.

The confidence interval uses the asymptotic standard error of Liebetrau (1983). That variance vanishes at both ends of the range: under exact independence (\(\tau = 0\), where the limiting distribution is a weighted sum of chi-square variables rather than normal) and under perfect prediction (\(\tau = 1\)). Where the estimated standard error is zero the interval would collapse to a single point and thus exclude every other value, which no sample supports; the bounds are returned as NA with a warning instead. Close to either end the normal approximation is poor and the interval is too narrow.

Note

Based on code by Antti Arppe, adapted to conform to package standards.

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.

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.

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. 24–30

See also

lambda(), cramerV(), Association()

Other assoc.nominal: contCoef(), cramerV(), lambda(), mutInf(), phi(), tschuprowT(), uncertCoef(), yule

Examples

# 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 C|R
gkTau(tab, direction="column", conf.level=0.95)
#>         est         lci         uci 
#> 0.041216580 0.009920576 0.072512583 
# Goodman Kruskal's tau R|C
gkTau(tab, direction="row", conf.level=0.95)
#>        est        lci        uci 
#> 0.16523315 0.04921484 0.28125146 

# 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)))

gkTau(tab, direction="row", conf.level=0.95)
#>       est       lci       uci 
#> 0.2156410 0.0000000 0.5537724 
gkTau(tab, direction="column", conf.level=0.95)
#>       est       lci       uci 
#> 0.2156410 0.0000000 0.5537724 
# reduce both to:
phi(tab)^2
#> [1] 0.215641


# example 1 in Liebetrau (1983)

tt <- matrix(c(549,93,233,119,225,455,402,  
               212,124,78,42,41,12,132,
               54,54,33,13,46,7,153), ncol=3,
             dimnames=list(rownames=c("Gov", "Mil", "Edu", "Eco", "Intel", "Rel", "For"), 
                           colnames=c("One", "Two", "Multi")))

gkTau(tt, direction = "row", conf.level = 0.95)
#>        est        lci        uci 
#> 0.02580507 0.02094902 0.03066113 
gkTau(tt, direction = "column", conf.level = 0.95)
#>        est        lci        uci 
#> 0.08601920 0.07243943 0.09959897 


# SPSS
ttt <- matrix(c(225,53,206,3,1,12), nrow=3,
              dimnames=list(rownames=c("right","center", "left"), 
                            colnames=c("us","ussr")))

round(gkTau(ttt, direction = "row", conf.level = 0.95), digits = 3)
#>   est   lci   uci 
#> 0.010 0.000 0.023 
round(gkTau(ttt, direction = "column"), digits = 3)
#> [1] 0.013