Cramér's V

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Cramér's V (φc) $\phi_c = \sqrt{ \frac{\chi^2}{N(k - 1)}}$

In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φc) is a popular[citation needed] 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.[1]

Usage and interpretation

φc is the intercorrelation of two discrete variables[2] 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.[citation needed]

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.[citation needed]

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 .[citation needed]

The formula for the variance of φc is known.[3]