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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 = + 1 x 1 + + k x k + Partial Regression Coefficients: 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 ) Normally distributed with mean 0 Constant Variance 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 + + + = 1 1 ) ( Least Squares Fitted (predicted) equation, minimizing SSE :  = + + + = 2 ^ ^ 1 1 ^ ^ ^ Y Y SSE x x Y k k 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 : 1 = = k =0 (None of the x s associated with y ) H A : Not all 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  + = = + 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 : 1 = = 6 =0 H A : Not all 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 Testing Individual Partial Coefficients  ttests Wish to determine whether the response is associated with a single explanatory variable, after controlling for the others H : i = 0 H A : i 0 (2sided alternative) )  ( 2 :   : . ....
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 Fall '08
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