gkGamma: Calculates the Goodman and Kruskal gamma measure of...
In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

gkGamma

R Documentation

Calculates the Goodman and Kruskal gamma measure of association.

Description

Goodman and Kruskal introduce a second coefficient, gamma (gkGamma) as a measure of association for two ordered categorical variables which do not necessarily have the same number of states. Thus it is useful as a measure of association between an observed variable and a latent proficiency variable.

Usage

gkGamma(tab)

Arguments

tab

A matrix whose rows and columns represent rating two different classifications and whose cells represent observed (or expected) counts.

Details

Let A and B be ordinal variables (both oriented in the same direction either high to low or low to high). Then tab is a cross-tabulation of the two variables (e.g., what is produced by the table funciton). In the case that one of the variables is a classification by a Bayesian network, this could also be an expected tabulation. It also could be normalized by dividing by the total to give probabilities.

Let (a,b) and (a',b') be two randomly chosen data points. Classify this pair according to the following rule:

Concordant (same): If both relationships point the same way, that is either both a>a` and b>b' or both a<a` and b<b', then the relationship is concordant.
Discordant (different): If both relationships point in opposite ways, that is either both a>a` and b<b' or both a<a` and b>b', then the relationship is discordant.
Discordant (different): If both relationships point in opposite ways, that is either both a>a` and b<b' or both a<a` and b>b', then the relationship is discordant.
Tied: If one or both variables are the same, that is either a = a` or b =b' or both, then the relationship is tied.

Let \Pi_s be the proportion of pairs which are concordant, \Pi_d be the proportion of pairs which are discordant, and \Pi_t be the proportion of pairs which are tied. Then gamma is defined as the proportion of concordant pairs minus the proportion of discordant pairs normalized by the number of non-tied pairs, that is:

\gamma = \frac{\Pi_s - \Pi_d}{1-\Pi_i}

Like a correlation coefficient it runs from -1 (perfect negative associations) through 0 (no association) to +1 (perfect association). It is comparable to Kendall's tau (see cor(,method="kendall")).

Value

A numeric scalar. Like a correlation coefficient it runs from -1 (perfect negative associations) through 0 (no association) to +1 (perfect association). It is comparable to Kendall's tau (see cor(,method="kendall")).

Author(s)

Russell Almond

References

Goodman, Leo A., Kruskal, William H. (1954). Measures of Association for Cross Classifications. Journal of the American Statistical Association. 49 (268), 732–764.

Examples


## Example from Goodman and Kruskall (1963)
nab <- matrix(c(8,0,0,5,8,4,3,1,14,3,0,4),3,4)
stopifnot(all.equal(.6122,gkGamma(nab),tolerance=.0001))

## Using built-in data set US Judges, rounding to produce ordered
##   categories. 

table(round(USJudgeRatings[,1:2]))
gkGamma(table(round(USJudgeRatings[,1:2])))
## Kendall's Tau for comparison
cor(round(USJudgeRatings[,1:2]),method="kendall")

ralmond/CPTtools documentation built on Dec. 27, 2024, 7:15 a.m.