Cramér's V

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

In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φ_c) 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

φ_c is the intercorrelation of two discrete variables^[2] and may be used with variables having two or more levels. φ_c 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 φ_c 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.

φ_c² 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 φ_c 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 φ_c coefficient is:

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

where:

• φ² is the phi coefficient.
• χ² 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 φ_c is the same one that is calculated using the Pearson's chi-squared test^{[citation needed]}.

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

See also

Other measures of correlation for nominal data:

• The phi coefficient
• Tschuprow's T
• The uncertainty coefficient
• The Lambda coefficient

Other related articles:

• Contingency table
• Effect size

References

1. ^ Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, p282. ISBN 0691080046
2. ^ Sheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press.
3. ^ Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32. (pages 15–16)

• Cramér, H. (1999). Mathematical Methods of Statistics, Princeton University Press

External links

• A Measure of Association for Nonparametric Statistics (Alan C. Acock and Gordon R. Stavig Page 1381 of 1381–1386)
• Nominal Association: Phi, Contingency Coefficient, Tschuprow's T, Cramer's V, Lambda, Uncertainty Coefficient