5.5 Effect sizes for chi-squares
This is the first time we’re coming across effect sizes for any test - thankfully, we start with relatively easy ones to wrap your head around. We will cover two effect sizes: phi (\(\phi\)) and Cramer’s V, both of which apply when conducting a test of independence.
5.5.1 Phi
Phi (\(\phi\)) is an effect size for chi-squares that applies only to 2x2 designs. The formula for phi is:
\[ \phi = \sqrt{\frac{\chi^2}{n}} \]
Essentially, it is the chi-square test statistic divided by the sample size, which is then square rooted. Again, it only works for 2x2 designs - i.e. each categorical variable can only have two categories within it.
5.5.2 Cramer’s V
Cramer’s V is another effect size for chi-squares, but one that can be used for anything beyond a 2x2 design as well. The formula for Cramer’s V is similar:
\[ V = \sqrt{\frac{\chi^2}{n(k-1)}} \] Here, k refers to the number of groups in the variable with the lowest number of groups. So for example, in a 2x3 design, one variable has 2 levels and the other has 3; k = 2 in this instance.
Phi and Cramer’s V can both be calculated in R with the following functions from the effectsize package, a handy package that will calculate many common effect size measures. Like chisq.test(), both functions will work if you give them a contingency table. Helpfully, both functions calculate 95% CIs.
5.5.3 Interpretation
Phi and Cramer’s V are essentially both correlation coefficients (more on this in Week 10). Both phi and Cramer’s V can only be between 0 and 1. For this subject, the size of Cramer’s V and phi can be interpreted as follows:
- If the effect size = .10, the effect is small
- If effect size = .30, the effect is medium
- If effect size = .50 or above, the effect is large