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Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory

Cramér's V (φ_{c}) 

In statistics, Cramér's V (sometimes referred to as Cramér's phi or Cramers C 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 chisquared statistic and was published by Harald Cramér in 1946.^{[1]}
Usage and interpretationEdit
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 chisquared 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_{c}^{2} 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 chisquared 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]}
CalculationEdit
Cramér's V is computed by taking the square root of the chisquared 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:
where:
 is the phi coefficient.
 is derived from Pearson's chisquared test
 is the grand total of observations and
 being the number of rows or the number of columns, whichever is less.
The pvalue for the significance of φ_{c} is the same one that is calculated using the Pearson's chisquared test ^{[citation needed]}.
The formula for the variance of φ_{c} is known.^{[3]}
Unlike the contingency tablethe value of V is relatively independent of the number of columns or rows.
See alsoEdit
Other measures of correlation for nominal data:
Other related articles:
ReferencesEdit
 ↑ Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, p282. ISBN 0691080046
 ↑ Sheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press.
 ↑ 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 linksEdit
 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
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