Cramér's V

Calculation

Let a sample of size n of the simultaneously distributed variables ${\displaystyle A}$ and ${\displaystyle B}$ for ${\displaystyle i=1,\dots ,r;j=1,\dots ,k}$ 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:

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

where:

• ${\displaystyle \phi }$ 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

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

where

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

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