R_MultipleReg.pdf - Multiple regression coeﬃcients Recall...

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Multiple regression coeﬃcients Recall Body Fat Example (Table7.1) and consider the regression model Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + ϵ. > #Table7.1 Body Fat Example > summary(lm(y~x1+x2+x3)) Call: lm(formula = y ~ x1 + x2 + x3) Residuals: Min 1Q Median 3Q Max -3.7263 -1.6111 0.3923 1.4656 4.1277 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 117.085 99.782 1.173 0.258 x1 4.334 3.016 1.437 0.170 x2 -2.857 2.582 -1.106 0.285 x3 -2.186 1.595 -1.370 0.190 Residual standard error: 2.48 on 16 degrees of freedom Multiple R-squared: 0.8014, Adjusted R-squared: 0.7641 F-statistic: 21.52 on 3 and 16 DF, p-value: 7.343e-06 Now let e ( X 1 | X 2 , X 3 ) be the residual of the regression of X 1 on X 2 , X 3 . Then the regression coeﬃcient ˆ α 1 from the model Y = α 0 + α 1 e ( X 1 | X 2 , X 3 ) + ϵ . is the same as ˆ β 1 .This is because e ( X 1 | X 2 , X 3 ) is orthogonal to the space spanned by { 1 , X 2 , X 3 } and span { 1 , X 1 , X 2 , X 3 } can be generated by the or- thogonal bases A 1 (1 , X 2 , X 3 ) , A 2 (1 , X 2 , X 3 ) , A 3 (1 , X 2 , X 3 ) and e ( X 1 | X 2 , X 3 ) = X 1 - A 0 (1 , X 2 , X 3), for some linear functions A j . More generally, the k -th multiple regression coeﬃcient is the univariate

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Unformatted text preview: regression coeﬃcient of Y on e ( X k | X 1 , · · · , X k − 1 , X k +1 , · · · , X p − 1 ). There-fore, the multiple regression coeﬃcient ˆ β k represents the additional contribu-tion of X k on Y , after X k has been adjusted for 1 , X 1 , · · · , X k − 1 , X k +1 , · · · , X p − 1 . 1 &gt; m1.23&lt;-lm(x1~x2+x3) &gt; summary(lm(y~resid(m1.23))) Call: lm(formula = y ~ resid(m1.23)) Residuals: Min 1Q Median 3Q Max-9.6795 -2.8254 0.4596 3.4210 7.6207 Coefficients: Estimate Std. Error t value Pr(&gt;|t|) (Intercept) 20.195 1.158 17.441 1.01e-12 *** resid(m1.23) 4.334 6.297 0.688 0.5---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 5.178 on 18 degrees of freedom Multiple R-squared: 0.02565, Adjusted R-squared: -0.02848 F-statistic: 0.4738 on 1 and 18 DF, p-value: 0.5 2...
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