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, ...)`

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

- 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.

Andri Signorell <andri@signorell.net>

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.

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.

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`

```
# 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)
```

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