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I am trying to understand the value Cramer's V provides.

I found the following sentence (from here):

"V may be viewed as the association between two variables as a percentage of their maximum possible variation. V^2 is the mean square canonical correlation between the variables."

But I don't think I fully understand it.

Can someone please help explain it a bit further (or with a more detailed example)?

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The quote is correct. If you have two categorical variables and you recode them into two sets of dummy variables and then perform canonical correlation analysis (CCA) on these two sets (leaving out any one dummy from each set as redundant) - you will get canonical correlations (see CCA algorithm, to compute) the average of the squares of which is exactly the Cramer's V squared between the original categorical variables.

An example.

Two nominal variables A (3 categories) and B (4 categories) were recoded into dummy sets.
A   B  A1  A2  A3  B1  B2  B3  B4
1   1   1   0   0   1   0   0   0
1   1   1   0   0   1   0   0   0
1   2   1   0   0   0   1   0   0
1   2   1   0   0   0   1   0   0
1   4   1   0   0   0   0   0   1
2   1   0   1   0   1   0   0   0
2   1   0   1   0   1   0   0   0
2   2   0   1   0   0   1   0   0
2   2   0   1   0   0   1   0   0
2   2   0   1   0   0   1   0   0
2   2   0   1   0   0   1   0   0
2   2   0   1   0   0   1   0   0
2   3   0   1   0   0   0   1   0
2   3   0   1   0   0   0   1   0
2   4   0   1   0   0   0   0   1
2   4   0   1   0   0   0   0   1
3   1   0   0   1   1   0   0   0
3   1   0   0   1   1   0   0   0
3   2   0   0   1   0   1   0   0
3   4   0   0   1   0   0   0   1

Throwing one arbitrary dummy from each set out, compute correlations and perform CCA on one set (2 variables) vs the other set (3 variables).

You'll extract two pair of canonical latent roots with correlations:
Canonical correlations and Eigenvalues: 
        Can Corr     Eigenval 
1       .3921542     .1817327 
2       .0859611     .0074443

(.3921542^2 + .0859611^2) / 2 = 0.08059 = squared Cramer's V between A and B.      

Note also that if one of the two categorical variables is dichotomous, squared Cramer's V is as well equivalent to R-square of linear regression of it by the dummies from the second variable.

If you forget about CCA of dummy variables and think about CCA in general, that is, about CCA of any numeric quantitative variables, then you may further know that the mean (or sum, to be exact) squared canonical correlation is what is known by name Pillai's trace - the statistic that has the same meaning in multivariate regression as R-square has in univariate regression. Thus, Cramer's V squared clearly appears to be homologous to multivariate R-square (Pillai trace); V being for two categorical variables and R-square being for two sets of quantitative variables. This fact sheds light on the phrase ...as a percentage of their maximum possible [shared] variation.

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