GENERAL CLASSES OF INFLUENCE MEASURES FOR MULTIVARIATE REGRESSION

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)-1E(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.