How do I calculate the minimum sample size needed to test:
H_0: all proportions are equal to 0.3 vs.
H_1: at least one proportion is different from the others
with power 80%?
I only have the observed sample sizes of the four samples, and not the data so I cannot calculate the eta-squared, which I believe is the approporiate effect size for an ANOVA-based model. Is it possible to still calculate an effect size to use in a sample size/power calculation if I don't have any sample data to calculate the sum of squares? Can I just use the observed n_1, n_2, n_3, n_4?
Yes, you can still estimate the required sample size using an assumed effect size. Since you're testing whether at least one proportion differs from 0.3, you can use Cohen’s w as the effect size for a chi-square goodness-of-fit test. If you don’t have the exact data, you can estimate w based on expected proportions (all 0.3) and a reasonable deviation for at least one category. Once you have w, you can use a power analysis formula or software like G*Power to determine the minimum required sample size for 80% power. Your observed sample sizes (n₁, n₂, n₃, n₄) can help inform the total required N but won’t directly give you the effect size.