W17Section3 - Section 3 Danqing Xu Section Outline I I I Deletion residuals Cooks distance Logistic Regression R Example Case Deletion in Linear

W17Section3 - Section 3 Danqing Xu Section Outline I I I...

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Section 3 Danqing Xu 1/27/2017
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Section Outline I Deletion residuals I Cook’s distance I Logistic Regression R Example
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Case Deletion in Linear Regression Using the notation from the last section, a subscript (i) means “with the ith case deleted,” for examples: I ˆ β ( i ) is the estimate of β computed without case i I X ( i ) is the ( n - 1 ) × p matrix obtained from X by deleting the i th row I Y ( i ) is the ( n - 1 ) × 1 column vector obtained from Y by deleting the i th element In particular, then ˆ β ( i ) = X T ( i ) X ( i ) - 1 X T ( i ) Y ( i )
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Deleted Residual If we let I y i denote the observed response for the i th case, and I ˆ y j ( i ) denote the predicted response for the j th case based on the estimated model with the i th case deleted then the i th deleted residual is defined as: d i = y i - ˆ y i ( i )
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Studentized Residual I Deleted residuals depend on the units of measurement just as the ordinary residuals do. We can solve this problem though by dividing each deleted residual by an estimate of its standard deviation. That’s where “studentized residuals” come into play. I The studentized residual is defined as t i = y i - ˆ y i ( i ) ˆ σ ( i ) 1 + x T i X T ( i ) X ( i ) - 1 x i where x T i (dimension 1 × p ) is the i th row of X matrix (dimension n × p ) I *A statistic divided by its estimated standard deviation is usually called a studentized statistic , in honor of W.S.Gosset, who first wrote about the t-distribution using the pseudonym Student.
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