User:Talgalili/Cramér's V (statistics)

In statistics, Cramér's V (sometimes referred to as Cramér's φ and denoted as φc) is the most popular of the chi-square-based measures of association between two nominal variables, giving a value between 0 and +1 (inclusive). It was developed by Harald Cramér.

Usage
φc is the intercorrelation of two discrete variables and may be computed for any value of r (rows) or c (columns) (nominal, ordered, and so on).

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 have equal marginals. However, as chi-square values tend to increase with the number of cells, the greater the difference between r and c, the more likely φc will tend to 1 without strong evidence of a meaningful correlation.

Cramér's V may also be applied to 'goodness of fit' chi-square models (i.e. those where c=1). In this case it functions as a measure of tendency towards a single outcome (i.e. out of k outcomes).

V is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2 x 2). Cramér's V may be used with variables having more than two levels.

Calculation
Cramér's V is computed by taking the square root of the chi-square 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 formulae for the V coefficient is:


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

where:
 * χ2 is derived from Pearson's chi-square test
 * N is the grand total of observations and
 * k being the number of rows or the number of columns, whichever is less.