 Sample size for comparing two proportions

For equal independent binomial samples, the displayed normal-approximation size per group targets a two-sided significance level $\alpha$ and power $1-\beta$ at specified alternative rates, with $\bar{p}=(p_1+p_0)/2$. Its first variance term sets the null critical value and its second the alternative distribution. A numerical sample size is not determined by an effect ratio alone; power, sidedness, allocation and baseline rates must also be specified. Rare-event discreteness and clustered designs can materially change the achieved power.