Cramér's V

In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φc) is a popular measure of association between two nominal variables, giving a value between 0 and +1 (inclusive). It is based on Pearson's chi-squared statistic and was published by Harald Cramér in 1946.

Usage and interpretation
φc is the intercorrelation of two discrete variables and may be used with variables having two or more levels. φc is a symmetrical measure, it does not matter which variable we place in the columns and which in the rows. Also, the order of rows/columns doesn't matter, so φc may be used with nominal data types or higher (ordered, numerical, etc)

Cramér's V may also be applied to goodness of fit chi-squared models when there is a 1×k table (e.g: r=1). In this case k is taken as the number of optional outcomes and it functions as a measure of tendency towards a single outcome.

Cramér's V varies from 0 (corresponding to no association between the variables) to 1 (complete association) and can reach 1 only when the two variables are equal to each other.

φc2 is the mean square canonical correlation between the variables.

In the case of a 2×2 contingency table Cramér's V is equal to the Phi coefficient.

Note that as chi-squared values tend to increase with the number of cells, the greater the difference between r (rows) and c (columns), the more likely φc will tend to 1 without strong evidence of a meaningful correlation.

Calculation
Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (k is the smaller of the number of rows r or columns c).

The formula for the φc coefficient is:


 * $$\phi_c = \sqrt{\frac{\varphi^2}{(k-1)}} = \sqrt{ \frac{\chi^2}{N(k - 1)}}$$

where:
 * $$\varphi^2$$ is the phi coefficient.
 * $$\chi^2$$ is derived from Pearson's chi-squared test
 * $$N$$ is the grand total of observations and
 * $$k$$ being the number of rows or the number of columns, whichever is less.

The p-value for the significance of φc is the same one that is calculated using the Pearson's chi-squared test.

The formula for the variance of φc is known.