Cramer's V - example and intuition

Question

Hy, I self study statistics and I come across Cramer's V and I search on Google. The result I find didn't explain me so I please to answer this questions. What is measured by Cramer's V? What is an example when Cramer's V is equal to 0 and an example when is 1?

Answer 1

Cramér's V is a measure of association.

If you have a $k\times m$ contingency table, you can apply a $\chi^2$ calculation and test with $(k-1)(m-1)$ degrees of freedom which may lead you to believe the two sets of factors are not independent, but that does not tell you how associated the values are. The calculated $\chi^2$ value depends on the overall sample size (the larger the sample size, the higher $\chi^2$ is and so the more likely you are to reject the null hypothesis) and is affected by the number of rows and columns in the table. Cramér's V attempts to adjust for these.

The first step is to take into account the size of the data set. If there are $n$ observations in total then consider $\frac{\chi^2}{n}$. This can range from $0$ to $\min(k-1,m-1)$. So now consider $\frac{1}{\min(k-1,m-1)}\frac{\chi^2}{n}$, which can ranges from $0$ to $1$. It turns out that if $k=m=2$ then this is equivalent to $r^2$, the square of a correlation coefficient, so it might be worth considering taking the square root, still with a possible range of $0$ to $1$.

So (at least before adjusting for other issues), Cramér's V is defined as $$V=\sqrt{\frac{1}{\min(k-1,m-1)}\frac{\chi^2}{n}}$$

Here are a couple of invented examples based on the colours of shapes

    circle  star square
red      0   0   100
yellow   0  300   0
green    0   0   400
blue    200  0    0

If you calculate this you get $\chi^2=2000$ and $n=1000$ and $\min(k-1,r-1)=2$, so $V=\sqrt{\frac12 \frac{2000}{1000}}=1$. This is plausible, as knowing the colour tells you the shape with certainty.

By contrast

    circle  star square
red     20   30   50
yellow  60   90  150
green   80  120  200
blue    40   60  100 

If you calculate this you get $\chi^2=0$ and $n=1000$ and $\min(k-1,r-1)=2$, so $V=\sqrt{\frac12 \frac{0}{1000}}=0$. This is plausible, as knowing the colour tells nothing additional about the possible shape.