General Classes of Influence Measures for Multivariate Regression
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多元回归影响度量的一般类别
- 研究问题
- 如何将一元最小二乘回归中的观测子集影响度量推广到多元回归，并区分多个观测共同影响中的杠杆与残差作用？
- 方法与思路
- 作者用帽子矩阵和残差相关矩阵的子矩阵刻画两类影响度量，联系删除观测后的变化并分离杠杆与残差成分。
- 材料陈述的贡献
- 摘要称多数已有影响度量可纳入所提出的两类框架，且这种刻画有利于计算观测子集的影响度量。
- 阅读关注点
- 应核对两类度量的定义、矩阵可逆性条件、杠杆与残差分离的含义，以及哪些已有诊断方法不属于该框架。
- 阅读边界
- 材料为版本未注明的OpenAlex索引摘要；矩阵公式排版可能丢失上下标，需查阅原PDF，不能据此重构或实现公式。
原文摘要
Many of the existing measures for influential subsets in univariate ordinary least squares (OLS) regression analysis have natural extensions to the multivariate regression setting. Such measures may be characterized by functions of the submatrices H I of the hat matrix H, where I is an index set of deleted cases, and Q I , the submatrix of Q = E(E T E)−1 E T , where E is the matrix of ordinary residuals. Two classes of measures are considered: f(·)tr[H I Q I (I − H I − Q I ) a (I − H I ) b ] and f(·)det[(I − H I − Q I ) a (I − H I ) b ], where f is a scalar function of the dimensions of matrices and a and b are integers. These characterizations motivate us to consider separable leverage and residual components for multiple-case influence and are shown to have advantages in computing influence measures for subsets. In the recent statistical literature on regression analysis, much attention has been given to problems of detecting observations that, individually or jointly, exert a disproportionate influence on the outcome of univariate linear regression analysis and to assessing the influence of such cases, individually or jointly. By far the most popular approach is that of measuring the change in some feature of the analysis upon deletion of one or more of the cases. Various measures have been proposed that emphasize different aspects of influence on the regression. For a review of such methods, see Cook (1977, 1979), Belsley, Kuh, and Welsch (1980), Cook and Weisberg (1982), and Chatterjee and Hadi (1986, 1988). In this article we generalize some of the univariate measures of influence to the multivariate regression setting and then show that the generalized measures are special cases of two general classes of influence measures. There are other approaches to influence measures in regression diagnostics (see, for example, Cook 1986) that are not special cases of our general classes. The majority of the existing measures, however, are.