Sample size for comparing two proportions ($n\simeq [z_{1-\alpha /2}\sqrt{2\bar{p}(1-\bar{p})}+z_{1-\beta }\sqrt{p_1(1-p_1)+p_0(1-p_0)}{]}^2/(p_1-p_0{)}^2$)

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.