Chapter04_06 - Model Adequacy Checking(Chap 4 6 STAT 563...

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Model Adequacy Checking (Chap 4 & 6) STAT 563 Spring 2007
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Model Assumptions Recall the five assumptions Response and the predictors are approximately linear related The error term has zero mean The error term has constant variance σ 2 The errors are uncorrelated The errors are normally distributed Assumptions (4) and (5) together imply that the errors are independent random variables Assumption (5) is required for hypothesis testing and interval estimation
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Use of Residuals in checking model assumptions
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Hat Matrix Recall for the model Y=X β + ε , we estimate β via least squares as With the fitted values computed as Is an n x n matrix called hat matrix . (it puts a hat on Y) Y X X X ' ) ' ( ˆ 1 - = β ' ) ' ( ' ) ' ( ˆ ˆ 1 1 X X X X H where Y H Y X X X X X Y - - = = = = β
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Hat Matrix H, the “hat” matrix, is a projection matrix of Y onto the (p+1)-dimensional space defined by the columns of X. Y H Y = ˆ Y X 1 X 2
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Note All symmetric idempotent matrices are projection matrices Also note that so that h ij measures the influence of Y j on ) ) ( ( 0 ' ) ( ' ˆ )' ˆ ( ˆ ˆ 2 Φ = - = - = - = Φ = - = - - H H H H H H I because Y Y Y H H I Y Y Y Y Since Y Y Y H is idempotent j i j ij i ii n in i i i Y h Y h Y h Y h Y h Y + = + + + = .... ˆ 2 2 1 1 i Y ˆ
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Note The h ii values are often called “leverage values” . The leverage values are only a function of the predictor variables so the indicate when a set of predictor variables will induce a large effect on the corresponding predicted value of the response Rule of Thumb If h ii > 2(p+1)/n, the i th response has an undue influence on the i th predicted response (2 times average, see below) Rationale: n p n H tr h ii / ) 1 ( / ) ( + = =
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Hat Matrix Properties j i h a from h b h h h h a H H ij n j ij ii ii n j ij ii - = = = = , 1 1 )) ( ( 1 0 ) ( 1 0 ) ( 1 2 2 1 2 2
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Hat Matrix For the centered model, we can define the centered Hat matrix as Where the centered model is written as ' 1 ' ) ( C C C C C X X X X H - = ε β α ε β β β ε β β + + = + - + + = + + = = C C i j ij j j j i ij k j j i X Y x x x x y 1 ) ( 0 1 0
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Centered Model Write the prediction as From above, note that . ˆ ' 1 ) ( 1 ˆ 1 ˆ ˆ ' 1 ' y and s of matrix a is J where Y X X X X J n X Y C C C C C C = + = + = - α β α C C H J n H Y H J n Y + = + = 1 1 ˆ
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Centered Model If we let h ik and h cik be the elements of these matrices, we see that We can also see that The diagonal elements of H and H C are bounded as cik ik h n h + = 1 k n k cik k n k ik i y h y y h y = = + = = 1 1 ˆ 1 0 1 1 cii ii h h n
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Hat Matrix Hat matrix is unaffected by the non-singular transformation on the predictors Let T be non-singular (ie, inverse exists), and let Z=XT H X X X X X T T X X XTT X T XT X T XT Z Z Z Z H Z = = = = = - - - - - - ' ) ' ( ' ' ) ' ( ) ' ( ' ' ) ' ' ( ' ) ' ( 1 1 1 1 1 1
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Hat Matrix We are interested in two transformations: Where P is the matrix of eigenvectors of the correlation matrix Note that X S =X C T S Hence the first transformation scales the centered predictor but does not alter the Hat matrix.
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