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Section 3Danqing Xu1/27/2017
Section OutlineIDeletion residualsICook’s distanceILogistic Regression R Example
Case Deletion in Linear RegressionUsing the notation from the last section, a subscript (i) means“with the ith case deleted,” for examples:Iˆβ(i)is the estimate ofβcomputed without caseiIX(i)is the(n-1)×pmatrix obtained fromXby deleting theith rowIY(i)is the(n-1)×1 column vector obtained fromYbydeleting theith elementIn particular, thenˆβ(i)=XT(i)X(i)-1XT(i)Y(i)
Deleted ResidualIf we letIyidenote the observed response for theith case, andIˆyj(i)denote the predicted response for thejth case based onthe estimated model with theith case deletedthen theith deleted residual is defined as:di=yi-ˆyi(i)
Studentized ResidualIDeleted residuals depend on the units of measurement just asthe ordinary residuals do. We can solve this problem though bydividing each deleted residual by an estimate of its standarddeviation. That’s where “studentized residuals” come into play.IThe studentized residual is defined asti=yi-ˆyi(i)ˆσ(i)1+xTiXT(i)X(i)-1xiwherexTi(dimension 1×p) is theith row ofXmatrix(dimensionn×p)I*A statistic divided by its estimated standard deviation isusually called astudentized statistic, in honor of W.S.Gosset,who first wrote about the t-distribution using the pseudonymStudent.