Measurements about detecting influential observation in multivariate linear regression models

IX SUMMARY Linear regression models are divided into two different groups, one is called univariate linear regression, the other is called multivariate linear regression. In the univariate linear regression models; Y is a nxl vector, X is an nxp matrix, 8 is a nxl vector. However; in the multivariate linear regression models Y is an nxr matrix and s is an nxr matrix (r>l). For these two models the assumptions are the same. These assumptions are explained in details in part 1. Regression models are assessed by the variates, observations and assumptions. The variates which do not represent the model exactly may cause gross changes on the regression coefficients and fitted model. Therefore the assessment of the variates and these observations are very significant for the model adequacy. For this purpose, many diagnostic methods (or influence measures) have been proposed. Some of these are; Dt (Cook's (1977)), Hat matrix (H) (Hoaglin and Welsch's (1978)), COVRATIOu FVARATIOt, DFF1TS,, DFBETASi (Belsley et al 's (1980)), AP, (Andrews andPregibon 's (1978)). In these diagnostic methods, influential case (or cases) is (are) omitted from the model or the perturbation is introduced to the model or the perturbation is introduced to the case (or cases) and then influential observations are assessed based on these measures.While D,, DFFITSu DFBETAS,, COVRATIOt, FVARATIO,, AP, are used in the case of assessment of single-case influence (*' is the case or row of X and Y)> Jj class J, (f;u,v,c) = [e, '(I - Ht )'u HIve1 ]f(m,n,p) / c is used for the case of assessment of multiple-case influence. For the multivariate regression models, Jf and Jf class JItr(f;a,b) = f(.)tr[HIQ1(I-Hl-Qir(I-HI)b] JI**(f;a,b) = f(.)det[(I-H1-QI)°(I-Hl)b} defined by Barrett and Ling (1992) are used for the assessment of multiple case influence. These classes offer considerable computational savings. Cook (1986) gave a general method for assessing the influence of local departures from assumptions in likelihood based models. With this method, ifa minor perturbations in the model leads to a major change in the results of the analysis, then there is evidence of difficulty. In order to assess local influence Cook suggests using the normal curvature of the likelihood dislacement surface. However in this method there are some problems such as lack of the definition of the parameters, computational difficulties. Therefore Billor and Loynes (1992) proposed an alternative method, LD*. This measure in univariate case is examined and applied to the multivariate case in order to assess local influence in part 4.