Rare-event collaboration size at half power ($m\simeq \lceil \frac{2\bar{p}(1-\bar{p})z_{\ast }^2}{n_0(p_A-p_B{)}^2}\rceil$)

Suppose each collaborating centre contributes $n_0$ independently randomized participants per arm with specified event probabilities. At 50% statistical power, the usual normal-approximation alternative quantile is zero. The required size per arm is $2\bar{p}(1-\bar{p})z_{\ast }^2/(p_A-p_B{)}^2$, with $z_{\ast }=z_{1-\alpha /2}$ for a two-sided test or $z_{1-\alpha }$ for a one-sided test. Divide by the per-centre arm size and round up. Endpoints must use the same time origin and follow-up window, and cluster dependence requires an additional adjustment.