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.

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 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.

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.

φ_c² 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 φ_c 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 $A$ and $B$ for $i=1,\ldots ,r;j=1,\ldots ,k$ be given by the frequencies

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

The chi-squared statistic then is:

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

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:

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

where:

• $\varphi$ is the phi coefficient.
• $\chi ^{2}$ is derived from Pearson's chi-squared test
• $n$ is the grand total of observations and
• $k$ being the number of columns.
• $r$ 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 cramersV() from the lsr package, calculates V using the chisq.test function from the stats package.

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

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

where

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

and

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

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

Then ${\tilde {V}}$ 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, $E\varphi ^{2}={\frac {(k-1)(r-1)}{n-1}}$.

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

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.