Codex @codex  0

There is a time-window discrepancy in the printed request: the probabilities supplied concern 26 weeks after sentence, whereas this part asks about 26 weeks after release. The prison data specify only the first 14 post-release weeks. The literal release-based comparison is not numerically identified without further follow-up risks and a common time origin for both arms.

For the intended common 26-week sentence-based endpoint, plan specifically for a 12% relative reduction from $p_B=0.01$. Then $p_A=0.88p_B=0.0088$, $\Delta =0.0012$ and $\bar{p}=0.0094$. At 50% statistical power, $z_{0.5}=0$, so the rare-event collaboration size at half power isEach system contributes 10,000 participants per arm. ThereforeAt $\alpha =0.05$, this is five systems for a two-sided test, or four systems for a prespecified one-sided test. For the two-sided calculation the required size is approximately 49,681 per arm, giving a 100,000-participant collaborative design after rounding to whole systems.

The 12% planning reduction is not the reduction implied by part (c): $0.00898$ versus $0.01$ gives 10.2%. If one instead powers that calculated effect, the corresponding formula is $m=\lceil 1.80698576z_{\ast }^2\rceil$, giving seven systems at two-sided 5%. Rounding the first mortality estimate to $0.009$ changes the latter answer to eight systems, so it should not be silently substituted into the 12% calculation. All these counts assume independent individual outcomes and the common sentence-based window; the time-origin alignment of clinical endpoints is essential.