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Unformatted text preview: Chapter 12 Multiple Regression Multiple Regression • Numeric Response variable ( y ) • k Numeric predictor variables ( k < n ) • Model: Y = β + β 1 x 1 + ⋅ ⋅ ⋅ + β k x k + ε • Partial Regression Coefficients: β i ≡ effect (on the mean response) of increasing the i th predictor variable by 1 unit, holding all other predictors constant • Model Assumptions (Involving Error terms ε ) – Normally distributed with mean 0 – Constant Variance σ 2 – Independent (Problematic when data are series in time/space) Example  Effect of Birth weight on Body Size in Early Adolescence • Response: Height at Early adolescence ( n =250 cases) • Predictors ( k =6 explanatory variables) • Adolescent Age ( x 1 , in years  1114) • Tanner stage ( x 2 , units not given) • Gender ( x 3 =1 if male, 0 if female) • Gestational age ( x 4 , in weeks at birth) • Birth length ( x 5 , units not given) • Birthweight Group ( x 6 =1,...,6 <1500 g (1), 1500 1999 g (2), 20002499 g (3), 25002999 g (4), 3000 3499 g (5), >3500 g (6)) Source: Falkner, et al (2004) Least Squares Estimation • Population Model for mean response: k k x x Y E β β β + + + = 1 1 ) ( • Least Squares Fitted (predicted) equation, minimizing SSE : ∑  = + + + = 2 ^ ^ 1 1 ^ ^ ^ Y Y SSE x x Y k k β β β • All statistical software packages/spreadsheets can compute least squares estimates and their standard errors Analysis of Variance • Direct extension to ANOVA based on simple linear regression • Only adjustments are to degrees of freedom: – DF R = k DF E = n( k+ 1) Source of Variation Sum of Squares Degrees of Freedom Mean Square F Model SSR k MSR = SSR / k F = MSR / MSE Error SSE n( k +1) MSE = SSE /( n( k +1)) Total TSS n1 TSS SSR TSS SSE TSS R = = 2 Testing for the Overall Model  Ftest • Tests whether any of the explanatory variables are associated with the response • H : β 1 = ⋅ ⋅ ⋅ = β k =0 (None of the x s associated with y ) • H A : Not all β i = 0 ) ( : : . . )) 1 ( /( ) 1 ( / : . . ) 1 ( , , 2 2 obs k n k obs obs F F P val P F F R R k n R k R MSE MSR F S T ≥ ≥ + = = + α Example  Effect of Birth weight on Body Size in Early Adolescence • Authors did not print ANOVA, but did provide following: • n =250 k =6 R 2 =0.26 • H : β 1 = ⋅ ⋅ ⋅ = β 6 =0 H A : Not all β i = 0 ) 2 . 1 4 ( : 1 3 . 2 : . . 2 . 1 4 0 03 0 . 0 4 33 . )) 1 6 ( 2 50 /( ) 2 6 . 1 ( 6 / 2 6 . )) 1 ( /( ) 1 ( / : . . 2 4 3 , 6 , 2 2 ≥ = ≥ = = + = = + = = F P va l P F F R R k n R k R M S E M S R F S T o b s o b s α Testing Individual Partial Coefficients  ttests • Wish to determine whether the response is associated with a single explanatory variable, after controlling for the others • H : β i = 0 H A : β i ≠ 0 (2sided alternative) )  ( 2 :   : . ....
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 Fall '08
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 Regression Analysis, dummy variables, predictors, rizatriptan

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