Cramér's V

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.^[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 (notably, ordered or numerical).

Cramér's V may also be applied to goodness of fit chi-squared models when there is a 1 × k table (in this case 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.^{[citation needed]}

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_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 absolute value of 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.^[3]

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.^{[citation needed]}

The formula for the variance of V=φ_c is known.^[4]

In R, the function cramerV() from the package rcompanion^[5] calculates V using the chisq.test function from the stats package. In contrast to the function cramersV() from the lsr^[6] 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^[7]

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

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, ${\displaystyle E[\varphi ^{2}]={\frac {(k-1)(r-1)}{n-1}}}$ .^[8]

See also

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:

• 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 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. Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Routledge. https://doi.org/10.4324/9780203771587 p78-81.
4. Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32. (pages 15–16)
5. "Rcompanion: Functions to Support Extension Education Program Evaluation". 2019-01-03.
6. "Lsr: Companion to "Learning Statistics with R"". 2015-03-02.
7. 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.
8. 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.

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.