发表机构
Chulalongkorn University（朱拉隆功大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对序数结局，证明基于秩的样本量计算中方差替代的比值上界为16/9，提供效应特定包络，并阐明其作为设计敏感性证书而非膨胀因子的用途。
AI 中文摘要
基于秩的检验在有效性上无需分布假设，但其所需的样本量取决于两种结局分布的形状，而这些形状在设计阶段是未知的。标准做法是代入在零假设下标定的方差。我们探究这种代入可能错到何种程度。将 $\Pi$ 记为估计相对效应的真实渐近方差与代入方差之比，我们证明在均衡分配下，对于任意类别数的所有序数分布，$\Pi \le 16\theta(1-\theta)/\{2+\theta(1-\theta)\} \le 16/9$：在一阶渐近计算范围内，所需样本量最多可超出计算值的 $77.8\\%$，且一个显式的两点族在每个效应量处均达到该界。对设计而言，起实际作用的是效应特定的包络而非常数：在 $\theta=0.65$ 时上限降至 $1.63$，在 $\theta=0.75$ 时降至 $1.37$，因此为较大效应设定功效的试验相应暴露风险较小。证明是初等的。在五个提取的试验比较和 4,711 个生成的备选方案中，实际成本保持在几个百分点以内，因为极值构型是移位型处理机制不会产生的。这些事实互为补充，并确定了该界的使用方式：它是设计敏感性证书，而非膨胀因子。方案可报告常规样本量以及任何序数构型下一致的最大需求。
英文摘要
Rank-based tests need no distributional assumptions to be valid, but the sample size they require does depend on the shapes of the two outcome distributions, which are unknown at the design stage. Standard practice substitutes a variance calibrated under the null. We ask how wrong that substitution can be. Writing $Π$ for the ratio of the true asymptotic variance of the estimated relative effect to the substituted one, we prove that under balanced allocation $Π\le 16θ(1-θ)/\{2+θ(1-θ)\} \le 16/9$ for all ordinal distributions in any number of categories: within the first-order asymptotic calculation the required sample size can exceed the calculated one by at most $77.8\%$, and an explicit two-point family attains the bound at every effect size. The effect-specific envelope, not the constant, is the operative quantity for a design: the ceiling falls to $1.63$ at $θ=0.65$ and $1.37$ at $θ=0.75$, so a trial powered for a larger effect is correspondingly less exposed. The proof is elementary. In five extracted trial comparisons and 4,711 generated alternatives the realized cost stays within a few percent, because the extremal configuration is one that shift-type treatment mechanisms do not produce. These facts are complementary, and fix how the bound should be used: it is a design sensitivity certificate, not an inflation factor. A protocol can report the conventional sample size alongside the largest requirement consistent with any ordinal configuration.
