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

φ_c is the intercorrelation of two discrete variables 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 does not matter, so φ_c 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.

φ_c is the mean square canonical correlation between the variables.

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 ${\displaystyle A}$ and ${\displaystyle B}$ for ${\displaystyle i=1,\ldots ,r;j=1,\ldots ,k}$ be given by the frequencies

${\displaystyle n_{ij}=}$ number of times the values ${\displaystyle (A_{i},B_{j})}$ were observed.

The chi-squared statistic then is:

${\displaystyle \chi ^{2}=\sum _{i,j}{\frac {(n_{ij}-{\frac {n_{i.}n_{.j}}{n}})^{2}}{\frac {n_{i.}n_{.j}}{n}}}\;,}$

where ${\displaystyle n_{i.}=\sum _{j}n_{ij}}$ is the number of times the value ${\displaystyle A_{i}}$ is observed and ${\displaystyle n_{.j}=\sum _{i}n_{ij}}$ is the number of times the value ${\displaystyle B_{j}}$ 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:

${\displaystyle V={\sqrt {\frac {\varphi ^{2}}{\min(k-1,r-1)}}}={\sqrt {\frac {\chi ^{2}/n}{\min(k-1,r-1)}}}\;,}$

where:

• ${\displaystyle \varphi }$ is the phi coefficient.
• ${\displaystyle \chi ^{2}}$ is derived from Pearson's chi-squared test
• ${\displaystyle n}$ is the grand total of observations and
• ${\displaystyle k}$ being the number of columns.
• ${\displaystyle r}$ 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.

The formula for the variance of V=φ_c is known.

In R, the function cramerV() from the package rcompanion calculates V using the chisq.test function from the stats package. In contrast to the function cramersV() from the lsr 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

${\displaystyle {\tilde {V}}={\sqrt {\frac {{\tilde {\varphi }}^{2}}{\min({\tilde {k}}-1,{\tilde {r}}-1)}}}}$ 

where

${\displaystyle {\tilde {\varphi }}^{2}=\max \left(0,\varphi ^{2}-{\frac {(k-1)(r-1)}{n-1}}\right)}$ 

and

${\displaystyle {\tilde {k}}=k-{\frac {(k-1)^{2}}{n-1}}}$ 

${\displaystyle {\tilde {r}}=r-{\frac {(r-1)^{2}}{n-1}}}$ 

Then ${\displaystyle {\tilde {V}}}$ 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, ${\displaystyle E={\frac {(k-1)(r-1)}{n-1}}}$.

See also

Other measures of correlation for nominal data:

• The Percent Maximum Difference
• 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:

• Contingency table
• Effect size
• Cluster analysis § External evaluation

References

1. 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 Archived 2016-08-16 at the Wayback Machine)
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)
4. "Rcompanion: Functions to Support Extension Education Program Evaluation". 2019-01-03.
5. "Lsr: Companion to "Learning Statistics with R"". 2015-03-02.
6. 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.
7. Bartlett, Maurice S. (1937). "Properties of Sufficiency and Statistical Tests". Proceedings of the Royal Society of London. Series A. 160 (901): 268–282. Bibcode:1937RSPSA.160..268B. doi:10.1098/rspa.1937.0109. JSTOR 96803.
8. Tyler, Scott R.; Bunyavanich, Supinda; Schadt, Eric E. (2021-11-19). "PMD Uncovers Widespread Cell-State Erasure by scRNAseq Batch Correction Methods". BioRxiv: 2021.11.15.468733. doi:10.1101/2021.11.15.468733.

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.

• Statistical ratios
• Summary statistics for contingency tables
• Covariance and correlation