ANOVA Effect Size

For balanced one-way ANOVA, the effect size is

$$ f = \sqrt{\frac{\frac{1}{k}\sum_{i=1}^{k}(\mu_i - \bar{\mu})^2}{\sigma^2}}, \quad \bar{\mu} = \frac{1}{k}\sum_{i=1}^{k}\mu_i. $$

It measures how much the treatment means vary relative to the common within-group standard deviation.

In STA305, it is linked to the ANOVA noncentrality parameter by

$$ \delta = k n f^2 $$

for a balanced design with $n$ observations per group.

R

library(pwr)
pwr.anova.test(k = k, n = n, f = f,
               sig.level = 0.05, power = NULL)