Cramér's V

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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 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.

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.

φc2 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 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 φc will tend to 1 without strong evidence of a meaningful correlation.

V may be viewed as the association between two variables as a percentage of their maximum possible variation. V2 is the mean square canonical correlation between the variables.

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]

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.

The formula for the variance of Vc 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:

Other related articles:

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 )
  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. https://CRAN.R-project.org/package=rcompanion. 
  5. "Lsr: Companion to "Learning Statistics with R"". 2015-03-02. https://CRAN.R-project.org/package=lsr. 
  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. doi:10.1098/rspa.1937.0109. 

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