Cramér's V
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.[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 (notably, ordered or numerical).
Cramér's V varies from 0 (corresponding to no association between the variables) to 1 (complete association) and can reach 1 only when each variable is completely determined by the other. It may be viewed as the association between two variables as a percentage of their maximum possible variation.
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 absolute value of Phi coefficient.
Calculation
Let a sample of size n of the simultaneously distributed variables [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] for [math]\displaystyle{ i=1,\ldots,r; j=1,\ldots,k }[/math] be given by the frequencies
- [math]\displaystyle{ n_{ij}= }[/math] number of times the values [math]\displaystyle{ (A_i,B_j) }[/math] were observed.
The chi-squared statistic then is:
- [math]\displaystyle{ \chi^2=\sum_{i,j}\frac{(n_{ij}-\frac{n_{i.}n_{.j}}{n})^2}{\frac{n_{i.}n_{.j}}{n}}\;, }[/math]
where [math]\displaystyle{ n_{i.}=\sum_jn_{ij} }[/math] is the number of times the value [math]\displaystyle{ A_i }[/math] is observed and [math]\displaystyle{ n_{.j}=\sum_in_{ij} }[/math] is the number of times the value [math]\displaystyle{ B_j }[/math] is observed.
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:
- [math]\displaystyle{ V = \sqrt{\frac{\varphi^2}{\min(k - 1,r-1)}} = \sqrt{ \frac{\chi^2/n}{\min(k - 1,r-1)}}\;, }[/math]
where:
- [math]\displaystyle{ \varphi }[/math] is the phi coefficient.
- [math]\displaystyle{ \chi^2 }[/math] is derived from Pearson's chi-squared test
- [math]\displaystyle{ n }[/math] is the grand total of observations and
- [math]\displaystyle{ k }[/math] being the number of columns.
- [math]\displaystyle{ r }[/math] being the number of rows.
The p-value for the significance of V is the same one that is calculated using the Pearson's chi-squared test.[citation needed]
The formula for the variance of V=φc is known.[3]
In R, the function cramerV() from the package rcompanion[4] calculates V using the chisq.test function from the stats package. In contrast to the function cramersV() from the lsr[5] package, cramerV() also offers an option to correct for bias. It applies the correction described in the following section.
Bias correction
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[6]
- [math]\displaystyle{ \tilde V = \sqrt{\frac{\tilde\varphi^2}{\min(\tilde k - 1,\tilde r - 1)}} }[/math]
where
- [math]\displaystyle{ \tilde\varphi^2 = \max\left(0,\varphi^2 - \frac{(k-1)(r-1)}{n-1}\right) }[/math]
and
- [math]\displaystyle{ \tilde k = k - \frac{(k-1)^2}{n-1} }[/math]
- [math]\displaystyle{ \tilde r = r - \frac{(r-1)^2}{n-1} }[/math]
Then [math]\displaystyle{ \tilde V }[/math] estimates the same population quantity as Cramér's V but with typically much smaller mean squared error. The rationale for the correction is that under independence, [math]\displaystyle{ E[\varphi^2]=\frac{(k-1)(r-1)}{n-1} }[/math].[7]
See also
Other measures of correlation for nominal data:
- The Percent Maximum Difference[8]
- 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:
References
- ↑ 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)
- ↑ "Rcompanion: Functions to Support Extension Education Program Evaluation". 2019-01-03. https://CRAN.R-project.org/package=rcompanion.
- ↑ "Lsr: Companion to "Learning Statistics with R"". 2015-03-02. https://CRAN.R-project.org/package=lsr.
- ↑ 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. doi:10.1098/rspa.1937.0109. Bibcode: 1937RSPSA.160..268B.
- ↑ Tyler, Scott R.; Bunyavanich, Supinda; Schadt, Eric E. (2021-11-19). "PMD Uncovers Widespread Cell-State Erasure by scRNAseq Batch Correction Methods" (in en). BioRxiv: 2021.11.15.468733. doi:10.1101/2021.11.15.468733. https://www.biorxiv.org/content/10.1101/2021.11.15.468733v1.
External links
- 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 from the homepage of Pat Dattalo.
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