Example 8.39: calculating Cramer’s V

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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)
[1] Cramér V / Phi:
[1] 0.09765063

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