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Chp6 - Multiple Regression Multiple Linear Regression Model...

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Multiple Regression Multiple Linear Regression Model Statistical Inference: estimation and hypothesis testing Model Diagnostics 1
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Multiple Linear Regression Models General linear models with p - 1 predictors Y = β 0 + X 1 β 1 + · · · + X p - 1 β p - 1 + = E ( Y ) + ; (1) where E ( ) = 0; Var( ) = σ 2 . The term linear refers to the fact the model (1) is linear in the param- eters β k . According to (1), ∂E ( Y ) ∂X k = β k , which can be interpreted as the change of rate in E ( Y ) with respect to X k when the other variables are held fixed. 2
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After we observe n samples, the model (1) can be writen in matrix notation as Y = X β + ; where X = 1 X 11 X 12 · · · · · · X 1 ,p - 1 1 X 21 X 22 · · · · · · X 2 ,p - 1 . . . . . . . . . . . . 1 X n 1 X n 2 · · · · · · X n,p - 1 = [ X 0 X 1 · · · X p - 1 ] X j = X 1 j X 2 j . . . X nj , Y = Y 1 Y 2 . . . Y n , β = β 0 β 1 . . . β p - 1 , = 1 2 . . . n E ( ) = 0 , Var( ) = σ 2 I n , E ( Y ) = X β 3
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Some Examples of General Linear Models Qualitative predictors E ( Y ) = β 0 + X 1 β 1 + X 2 β 2 where X 1 = Age ; X 2 = 1 if male; 0 if female , which is equivalent to fit the following two seperate linear models For male: E ( Y ) = ( β 0 + β 2 ) + X 1 β 1 ; For female: E ( Y ) = β 0 + X 1 β 1 Interaction E ( Y ) = β 0 + X 1 β 1 + X 2 β 2 + X 1 X 2 β 3 4
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Polynomial regression E ( Y ) = β 0 + 1 + X 2 β 2 + · · · + X p - 1 β p - 1 Transformed variables Logit model logit ( E ( Y )) = β 0 + X 1 β 1 + X 2 β 2 + · · · + X p - 1 β p - 1 ; E ( Y ) = 1 + e - ( β 0 + X 1 β 1 + X 2 β 2 + ··· + X p - 1 β p - 1 ) - 1 Probit model Φ - 1 ( E ( Y )) = β 0 + X 1 β 1 + X 2 β 2 + · · · + X p - 1 β p - 1 5
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-6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 x y 1 1 + e - x Φ ( x 29 6
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Piecewise inear regression model E ( Y ) = β 0 + 1 + ( X - c ) + β 2 , where ( X - c ) + = max( X - c, 0). 1- d tree regression model E ( Y ) = m X k =1 β k I ( C k - 1 X < C k ) , where C 0 < C 1 < · · · < C m and I ( C k - 1 X < C k ) = 1 if C k - 1 X < C k and 0 otherwise.
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