In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φc) is a 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.
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.
Let a sample of size n of the simultaneously distributed variables and for be given by the frequencies
- number of times the values were observed.
The chi-squared statistic then is:
Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the minimum dimension minus 1:
- is the phi coefficient.
- is derived from Pearson's chi-squared test
- is the grand total of observations and
- being the number of columns.
- being the number of rows.
The formula for the variance of V=φc is known.
In R, the function
cramersV() from the
lsr package, calculates V using the chisq.test function from the stats package.
Cramér's V can be a heavily biased estimator of its population counterpart and will tend to overestimate the strength of association. A bias correction, using the above notation, is given by
Other measures of correlation for nominal data:
- The phi coefficient
- Tschuprow's T
- The uncertainty coefficient
- The Lambda coefficient
- The Rand index
- Davies–Bouldin index
- Dunn index
- Jaccard index
- Fowlkes–Mallows index
Other related articles:
- Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, page 282 (Chapter 21. The two-dimensional case). ISBN 0-691-08004-6 (table of content)
- 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)
- Bergsma, Wicher (2013). "A bias correction for Cramér's V and Tschuprow's T". Journal of the Korean Statistical Society. 42 (3): 323–328. doi:10.1016/j.jkss.2012.10.002.
- Bartlett, Maurice S. (1937). "Properties of Sufficiency and Statistical Tests". Proceedings of the Royal Society of London. Series A. 160 (901): 268–282. JSTOR 96803.
- A Measure of Association for Nonparametric Statistics (Alan C. Acock and Gordon R. Stavig Page 1381 of 1381–1386)
- Nominal Association: Phi and Cramer's Vl[dead link] from the homepage of Pat Dattalo.