Goodman Kruskal's Gamma

Calculate Goodman Kruskal's Gamma statistic, a measure of association for ordinal factors in a two-way table.
The function has interfaces for a contingency table (matrix) and for single vectors (which will then be tabulated).

GoodmanKruskalGamma(x, y = NULL, conf.level = NA, ...)

Arguments

x

a numeric vector or a contingency table. A matrix will be treated as a table.

y

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

conf.level

confidence level of the interval. If set to NA (which is the default) no confidence intervals will be calculated.

...

further arguments are passed to the function table, allowing i.e. to control the handling of NAs by setting the useNA argument. This refers only to the vector interface, the dots are ignored if x is a contingency table.

Details

The estimator of \(\gamma\) is based only on the number of concordant and discordant pairs of observations. It ignores tied pairs (that is, pairs of observations that have equal values of X or equal values of Y). Gamma is appropriate only when both variables lie on an ordinal scale.
It has the range [-1, 1]. If the two variables are independent, then the estimator of gamma tends to be close to zero. For \(2 \times 2\) tables, gamma is equivalent to Yule's Q (YuleQ).
Gamma is estimated by $$ G = \frac{P-Q}{P+Q}$$ where P equals twice the number of concordances and Q twice the number of discordances.

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

References

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

Brown, M.B., Benedetti, J.K.(1977) Sampling Behavior of Tests for Correlation in Two-Way Contingency Tables, Journal of the American Statistical Association, 72, 309-315.

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.

Author

Andri Signorell <andri@signorell.net>

See also

There's another implementation of gamma in vcdExtra GKgamma
ConDisPairs yields concordant and discordant pairs

Other association measures:
KendallTauA (tau-a), KendallTauB (tau-b), cor (method="kendall") for tau-b, StuartTauC (tau-c), SomersDelta
Lambda, GoodmanKruskalTau (tau), UncertCoef, MutInf

Examples

# example in:
# http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf
# pp. S. 1821 (149)

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

GoodmanKruskalGamma(tab, conf.level=0.95)
#>     gamma    lwr.ci    upr.ci 
#> 0.5313123 0.3479918 0.7146328