提出新方法ROD，让观测数据回归结果更稳定可靠。
Retrospective Orthogonal Design: Response-Surface Reconstruction from Observational Data
- 构建概率平衡格点，用加权张量积对比重建响应面
- 在6480种模拟中表现优于或媲美多项式回归，尤其适合非线性交互
- 结果不依赖变量顺序，适合需严格可重复性的实证研究
当多重共线性存在时，观测数据的回归估计易受模型设定影响，而顺序平方和（SS）则依赖项的输入顺序。本文提出回顾正交设计（ROD），在概率平衡格点上重建条件均值曲面。ROD保留观测单元均值，填补缺失单元，应用加权张量积对比，并通过弗鲁恩特哈尔多面体上的分段线性插值评估重建表面。分辨率与补全由秩允许候选者的验证联合选择，随后在未触碰测试集上重新拟合与评估。对于允许的格点，满足 $ \mathbf{X}^\top\mathbf{W}\mathbf{X}=c\mathbf{I}$，确保对比效应不变且平方和唯一、与顺序无关。响应自由投影校准将固定重建映射至指定科学基底，纠正有限分辨率恢复损失。在涵盖九种数据生成过程的6480种模拟条件下，ROD在五种过程中表现匹配或超越多项式回归，在阈值、符号交互及局部表面任务中表现最佳。对于二次交互过程，平均外样本 $R^2$ 差异仅 $0.0001$，校准后系数偏差始终较小。基于Rao的信息调整提供考虑依赖关系的样本量规划指导。在加权明岑应用中，ROD取得最高外样本 $R^2$ 点估计，与多项式回归置信区间高度重叠，并实现与变量进入顺序无关的完整平方和分配。
原文摘要 · Abstract (English)
Regression estimates from observational data can depend on specification under multicollinearity, while sequential sums of squares (SS) depend on term order. We introduce Retrospective Orthogonal Design (ROD), which reconstructs conditional mean surfaces on a probability-balanced lattice. ROD preserves observed cell means, completes unsupported cells, applies weighted tensor-product contrasts, and evaluates the reconstructed surface through piecewise-affine interpolation over Freudenthal polyhedra. Resolution and completion are selected jointly by validation among rank-admissible candidates, followed by refitting and evaluation on an untouched test set. For an admissible lattice, $\mathbf{X}^{\top}\mathbf{W}\mathbf{X}=c\mathbf{I}$, yielding specification-invariant contrast effects and unique, order-independent SS within the retained contrast space. Response-free projection calibration maps the fixed reconstruction onto a declared scientific basis and corrects finite-resolution recovery loss. Across 6,480 simulation conditions spanning nine data-generating processes, ROD matched or exceeded polynomial regression in five processes and performed strongest on threshold, sign-interaction, and localized surfaces. For the quadratic-interaction process, mean out-of-sample $R^2$ differed by only $0.0001$, while calibrated coefficient bias remained small across prespecified targets. A Rao-based information adjustment provides dependence-aware sample-size guidance for ROD planning. In a weighted Mincer application, ROD produced the highest out-of-sample $R^2$ point estimate, with substantial interval overlap with polynomial regression, and provided exhaustive SS allocations invariant to term-entry order.
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