AI 中文总结
研究相对重要性分析中GD计算量大的问题，提出用ORMs替代，指出RW和GCD的结构限制。通过闭式分析揭示偏差机制，用MAP和SK算法校正GCD偏差得GCD - MAP和GCD - SK，模拟显示其性能提升并给出选择度量的实证指南。
AI 中文摘要
相对重要性（RI）分析量化每个预测变量对线性模型解释方差的贡献。广泛使用的基准通用优势（GD）需要评估\(2^p - 1\)个子模型，随着预测变量数量\(p\)的增加，计算量会变得很大。正交化 - 重分配度量（ORMs），包括相对权重（RW）和格林 - 卡罗尔 - 德萨尔博度量（GCD），通过为正交化预测变量分配重要性并将其重新分配给原始预测变量提供了有效的替代方法。然而，每种方法都有结构限制：RW存在压缩预测变量重要性值差异的水平问题，而GCD存在先验偏差，在观察响应之前系统地偏向某些预测变量。我们表明这种偏差由重分配矩阵的行和控制。在复合对称性下的闭式分析将GCD和RW的重分配与基于GD的基准相关联，表明仅齐次多重共线性不会引起先验偏差，并将RW的水平问题形式化为相对于基准的过度收缩。我们使用交替投影法（MAP）和Sinkhorn - Knopp（SK）算法将GCD的重分配矩阵映射到双重随机矩阵来校正GCD的偏差，得到GCD - MAP和GCD - SK。综合模拟表明，GCD - SK消除了结构行和偏差，大大改进了GCD，并且在第一主成分占主导时通常优于RW。我们以在这些度量之间进行选择的实证指南作为结论。
英文摘要
Relative importance (RI) analysis quantifies each predictor's contribution to the explained variance of a linear model. General Dominance (GD), a widely used benchmark, requires evaluating $2^p-1$ sub-models and becomes computationally intensive as the number of predictors $p$ grows. Orthogonalization-Reallocation Measures (ORMs), including Relative Weights (RW) and the Green--Carroll--DeSarbo measure (GCD), provide efficient alternatives by assigning importance to orthogonalized predictors and reallocating it to the original predictors. Each, however, has a structural limitation: RW exhibits a leveling problem that compresses differences among predictor importance values, whereas GCD exhibits an a priori bias that systematically favors certain predictors before a response is observed. We show that this bias is governed by the row-sums of the reallocation matrix. A closed-form analysis under compound symmetry relates the reallocations underlying GCD and RW to a GD-based benchmark, showing that homogeneous multicollinearity alone does not induce an a priori bias and formalizing RW's leveling problem as excess shrinkage relative to the benchmark. We correct GCD's bias by mapping its reallocation matrix to a doubly stochastic matrix using the Method of Alternating Projections (MAP) and the Sinkhorn--Knopp (SK) algorithm, yielding GCD-MAP and GCD-SK. Comprehensive simulations show that GCD-SK removes the structural row-sum bias, substantially improves upon GCD, and often outperforms RW when the first principal component is dominant. We conclude with empirical guidelines for selecting among the measures.
Comments23 pages, 5 figures
