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 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_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.

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.^[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]

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

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

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