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Unformatted text preview: .364 0.7157 -0.290 0.7723 2.368 0.0182 * 0.05 ’.’ 0.1 ’ ’ 1 Resid. stand. error: 16.43 on 693 degr. of freedom Multiple R-squared: 0.2117, Adj. R-squared: 0.2038 F-stat.: 26.59 on 7 and 693 DF, p-value: < 2.2e-16 27 Multiple Regression Decomposition and R 2 Via matrix algebra one gets again the following decomposition n n (yi − y )2 = ¯ SYY = i =1 n (yi − yi )2 + ˆ i =1 (ˆi − y )2 = RSS + SSreg y ¯ i =1 where the tted values are ˆ ˆ ˆ yi = β0 + β1 xi 1 + . . . + βp xip ˆ The multiple correlation coecient is R2 = SSreg SYY − RSS RSS = =1− SYY SYY SYY with adjusted p+1 28 ¯ R 2 = R 2 = 1 − is the number of tted parameters n−1 (1 − R 2 ) n−p−1 β0 , . . . , β p . Comments on Multiple Regression Output Summary The Std. Error is the estimated standard deviation of the respective estimate. The t value is the ratio t = Estimate/Std. Pr( > |t|) represents the two-sided p-value for the observed value t, Error. The when testing the hypothesis F statistic tests the hypothesis H0 : β1 = . . . = βp = 0. Hj : βj = 0. of no regression eect at all, i.e., It appears that the intercept ht β0 = 0 and that are signicant predictor variables, i.e., The signicance of dwt should be viewed with caution, in the context of multiple tests performed. 29 gestation and β1 = 0 and β3 = 0. Model Update We delete the non-signicant variables from previous analysis > wtreg0 <- update(wtreg,.~.-age-wt1-dage-dht) > summary(wtreg0) Call: lm(formula = wt ~ gestation + ht + dwt, data = babies, subset = gestation < 999 & age < 99 & ht < 99 & wt1 < 999 & dage < 99 & dht < 99 & dwt < 999) Residuals: Min 1Q -48.230 -10.482 Median 0.394 Note the modied model call. 30 3Q 10.379 Max 56.108 Model Update (Output Continued) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -102.46196 18.91277 -5.418 8.33e-08 gestation 0.44803 0.03893 11.509 < 2e-16 ht 1.31132 0.25456 5.151 3.37e-07 dwt 0.07553 0.02839 2.660 0.00799 --Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Resid. std. error: 16.45 on 697 degrees of freedom Multiple R-squared: 0.2056, Adj. R-squared: 0.2021 F-stat.: 60.12 on 3 and 697 DF, p-value: < 2.2e-16 31 * * * * Examining Normality of Residuals > qqnorm(wtreg0\$resid) > qqline(wtreg0\$resid) 60 Normal Q−Q Plot q q q q q 20 0 −40 −20 Sample Quantiles 40 q q q qq q q q q q qq qq qq qq qq q q qq qq qqq qqq qqq qq q q qq q q q qq qq qq qq qq q q q q qq q q q q qq q q q q q q q q q q q qq q qq qq q qq qq qq q q qq qq qq qq qq qq qq qq qq q q qq qq qq qq qq q q q qq qq qq qq qq qq q q qq q qq q q q qq qq qq qq qq qq qqq qq qq qqq qqq qq qq qq qq qq qq q qq qq qq qq qq qq qq qq q qq qq qq qq q q q q q qq qq qq qq qq qq q qq qq q qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq qq q q qq qq qq q q qq qq qq qq q qq q qq qq q q q q qq qqq qqq qqq qq qq q q q q q q q q q q q qq q q q qq q q q q q q q q q q q qqq qqq qqq qqq qq qq qq qqq qq q qq q qq qq q q q q q qqq q q q q −3 −2 −1 0 Theoretical Quantiles 32 1 2 3...
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