## Friday, June 3, 2011

### Example 8.39: calculating Cramer's V

Cramer's V is a measure of association for nominal variables. Effectively it is the Pearson chi-square statistic rescaled to have values between 0 and 1, as follows:

V = sqrt(X^2 / [nobs * (min(ncols, nrows) - 1)])

where X^2 is the Pearson chi-square, nobs represents the number of observations included in the table, and where ncols and nrows are the number of rows and columns in the table, respectively. For a 2 by 2 table, of course, this is just the square root of chi-square divided by the number of observations, which is also known as the phi coefficient.

As an example, we'll revisit the table of homelessness vs. gender we present in Section 2.6.3.

SAS
In SAS, Cramer's V is provided when the chisq option to the tables statement is used, in proc freq.
`proc freq data = "c:\book\help.sas7bdat";  tables female*homeless / chisq;run;`

resulting in
`             Statistics for Table of FEMALE by HOMELESS       Statistic                     DF       Value      Prob       ------------------------------------------------------       Chi-Square                     1      4.3196    0.0377       Likelihood Ratio Chi-Square    1      4.3654    0.0367       Continuity Adj. Chi-Square     1      3.8708    0.0491       Mantel-Haenszel Chi-Square     1      4.3101    0.0379       Phi Coefficient                      -0.0977       Contingency Coefficient               0.0972       Cramer's V                           -0.0977`

where (as usual) several additional values are also included. The negative value shown for Cramer's V is odd-- it's unclear what rationale should be used for using the negative root. According to the documentation, this is only a possibility for 2 by 2 tables.

R
As far as we know, Cramer's V is not included in base R. Of course, it is easy to assemble directly. We found one version on line. However, this requires a table as input, so we've rewritten it here to accept vector input instead.

Here's the function, which uses unique() (section 1.4.16) to extract the values of the rows and columns and length() (Section 1.4.15) to find their number and the number of observations. A more bullet-proof version of the function would check to ensure the two vectors are of equal length (or allow the input in a variety of formats).
`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) Cramér V / Phi: 0.09765063`

Anonymous said...

be good to have ci's as well for Cramer's V

Nick Horton said...

This could certainly be done easily using a bootstrapping procedure:

require(mosaic)

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))
}

with(helpdata, cv.test(female, homeless))

res = do(5000)* with(resample(helpdata), cv.test(female, homeless))
qdata(c(.025, .975), res\$result)

> with(helpdata, cv.test(female, homeless))
 Cramér V / Phi:
 0.09765063

> qdata(c(.025, .975), res\$result)
2.5% 97.5%
0.01315987 0.18717591

Also note that the "vcd" package has an "assocstats()" function which calculates Cramer's V (and other statistics.

Anonymous said...

you call that easy? and where is the interpretation?

Nick Horton said...

My interpretation is that we observed a Cramer's V of 0.098 (very weak association). We're 95% confidence that the true V is captured by the interval 0.013 to 0.187.

thelatemail said...

You could get similar results using a conversion to Fisher Z and then back again. I.e.:

#"mat" being a r x c matrix/table
chicalc <- chisq.test(mat)\$statistic

# calculate Cramer's v -
K <- min(nrow(mat),ncol(mat))
crv <- sqrt(chicalc / sum(mat)*(K-1))

# convert the Cramer's V to a Fisher's Z
fz <- 0.5 * log((1 + crv)/(1 - crv))

# calculate 95% conf.int around Fisher Z
conf.level <- 0.05
se <- 1/sqrt(sum(mat)-3) * qnorm(1-(conf.level/2))
cifz <- fz + c(-se,se)

# convert it back to conf.int around Cramer's V
cicrv <- (exp(2 * cifz) - 1)/(1 + exp(2 * cifz))