Cramer's Coefficient
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Cramer's Coefficient
Note about $\chi^2$ Test of Independence:
- when the size of a data set increases, the gap between observed and expected values also increases
- even if the distribution remains unchanged
- thus we reject the independence hypothesis as the size grows
- Crammer's Coefficient provides a solution for that
Definition
The Cramer's coefficient $v$
- $V = \sqrt{ \cfrac{\chi^2}{\chi^2_\text{max} } }$
- with $\chi^2_\text{max} = N \times ( \min(N, P) - 1 )$ where
- $N$ is the number of tuples and $P$ the number of attributes
- $V \in [0, 1]$
- 0 - maximal independence, and 1 - maximal correlation
Example
Consider the same example as for $\chi^2$ Test
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$V = \sqrt{ 3 / 150 } = \sqrt{ 30 / 1500 } \approx 0.14 $
R
TODO: Expand it
cv.test = function(x,y) {
CV = sqrt(chisq.test(x, y, correct=FALSE)$statistic /
(length(x) * (min(length(unique(x)),length(unique(y))) - 1)))
print.noquote("Cramér V / Phi:")
return(as.numeric(CV))
}
So we can get Cramer's V as
helpdata = read.csv("http://www.math.smith.edu/r/data/help.csv")
with(helpdata, cv.test(female, homeless)
or
cv.test <- function(x) {
CV <- sqrt(chisq.test(x, correct=FALSE)$statistic / (sum(x) * min(dim(x) - 1 )))
### The result of the Pearson chi-square (without the Yates correction) is divided by the sum of table cells and...
### ...multiplied by the smalles number of (row or column) cells minus 1.
### The $statistic sends the correct value (the X^2 only) into the sqrt function
print.noquote("Cramér V / Phi:")
return(as.numeric(CV))
}
Links
- http://en.wikipedia.org/wiki/Phi_coefficient
- http://sas-and-r.blogspot.fr/2011/06/example-839-calculating-cramers-v.html
- http://home.hib.no/ansatte/gbj/cramer_v.htm
- http://en.wikipedia.org/wiki/Cram%C3%A9r%27s_V
- !!! http://stats.stackexchange.com/questions/105795/understanding-chi2-and-cram%C3%A9rs-v-results/105827#105827