= Rare-event collaboration size at half power
{title2=$m\simeq\left\lceil\frac{2\bar p(1-\bar p)z_*^2}{n_0(p_A-p_B)^2}\right\rceil$}
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 $2\bar p(1-\bar p)z_*^2/(p_A-p_B)^2$, with $z_*=z_{1-\alpha/2}$ 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.
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