标题： 精确的自由度通过投影-秩划分定理 显示英文标题

标题： Exact degrees of freedom via a projector-rank partition theorem

摘要： 当阻塞、嵌套、分块或不相等的重复干扰了一元经典方差分析时，自由度（$\mathrm{df}$）的分配变得不透明。我们通过一个$\mathrm{df}$分区定理解决了这个歧义。 对于$N$次观测和实验中索引所有随机化层的集合$\mathcal{S}$，设$\mathbf{P}_s$表示在随机化层$s \in \mathcal{S}$内平均观测值的幂等投影算子，$\mathcal C_{\overline E}$表示与可能混杂的效果$\overline{E}$相关的因子对比子空间。 我们证明了秩恒等式$N-1 = \sum_{s}\sum_{\overline E}\dim\left(\mathcal C_{\overline E} \cap \operatorname{Im}\mathbf{P}_s\right)$，它同时使所有层次和所有效应实现对角化，从而对任意固定的随机混合、块结构或复制模式得出精确的整数$\mathrm{df}$。 它为复杂的非平衡设计（包括许多传统上依赖数值近似的复杂非平衡设计）$-$得出了封闭形式的$\mathrm{df}$表。 引入了扩展了 Box-Hunter 分辨率概念到多层和不完全设计的混杂度量，如$\mathrm{df}$-保留率$\rho(\overline E)$和相应的缺陷$\delta(\overline{E})$，并提供了信息损失的即时诊断。 诸如Cochran恒等式（当$\lvert \mathcal{S} \rvert =1$时）和平衡分数、Yates的$\mathrm{df}$（用于平衡裂区试验），以及当$\rho(\overline{E}) = 0$时的分辨率$R$等经典结果成为推论。 显示英文摘要

摘要： Degrees of freedom ($\mathrm{df}$) allocation becomes opaque once blocking, nesting, fractionation, or unequal replication disturb the balanced one-stratum classical ANOVA. We resolve this ambiguity with a $\mathrm{df}$ partition theorem. For $N$ observations and a set $\mathcal{S}$ that indexes all randomization strata in the experiment, let $\mathbf{P}_s$ denote the idempotent projector that averages observations within randomization stratum $s \in \mathcal{S}$ and $\mathcal C_{\overline E}$ the factorial-contrast subspace attached to the possibly aliased effect $\overline{E}$. We prove the rank identity $N-1 = \sum_{s}\sum_{\overline E}\dim\left(\mathcal C_{\overline E} \cap \operatorname{Im}\mathbf{P}_s\right)$, which simultaneously diagonalizes all strata and all effects, yielding exact integer $\mathrm{df}$ for any fixed-random mixture, block structure, or replication pattern. It results in closed-form $\mathrm{df}$ tables for complex unbalanced designs$-$including many that traditionally relied on numerical approximations. Aliasing measures such as the $\mathrm{df}$-retention ratio $\rho(\overline E)$ and the corresponding deficiency $\delta(\overline{E})$ that extend Box-Hunter's resolution concept to multi-stratum and incomplete designs, and supply immediate diagnostics of information loss, are introduced. Classical results such as Cochran's identity when $\lvert \mathcal{S} \rvert =1$ and balanced fractions, Yates' $\mathrm{df}$ for balanced split-plots, and resolution-$R$ when $\rho(\overline{E}) = 0$ emerge as corollaries.