Supplementary Exercise 11.16 of IPS7e: -------------------------------------- Minitab commands and output: --- WOpen "R:\Chapter 11\ex11_015.mtw". Regress 'GPA' 2 'IQ' 'C3'; Constant; Brief 2. The regression equation is GPA = - 2.83 + 0.0822 IQ + 0.163 C3 Predictor Coef SE Coef T P Constant -2.829 1.507 -1.88 0.064 IQ 0.08220 0.01508 5.45 0.000 C3 0.16289 0.05752 2.83 0.006 S = 1.564 R-Sq = 45.9% R-Sq(adj) = 44.5% Analysis of Variance Source DF SS MS F P Regression 2 155.943 77.971 31.87 0.000 Residual Error 75 183.484 2.446 Total 77 339.427 Source DF Seq SS IQ 1 136.319 C3 1 19.624 Unusual Observations Obs IQ GPA Fit SE Fit Residual St Resid 8 97 2.412 5.959 0.260 -3.547 -2.30R 22 109 1.760 6.457 0.394 -4.697 -3.10R 48 102 3.936 7.184 0.253 -3.248 -2.10R 51 103 0.530 6.778 0.196 -6.248 -4.03R 54 72 7.295 4.556 0.607 2.739 1.90 X R denotes an observation with a large standardized residual X denotes an observation whose X value gives it large influence. Comments and answers to questions: ---------------------------------- The fitted regression equation is GPAhat = -2.83 + 0.0822*IQ + 0.163*C3 The t-statistic for testing the regression coefficient for C3 equal to zero is t=2.83 (P=0.006). We conclude that C3 does contribute significantly to the model. The increase caused by C3 in R^2 is 45.9-40.2 = 5.7%. Note that the residuals analysis is deferred to Exercise 11.34. (b) Minitab command and output: --- Regress 'GPA' 3 'IQ' 'SC' 'C3'; Constant; Brief 2. The regression equation is GPA = - 3.49 + 0.0761 IQ + 0.0369 SC + 0.0670 C3 Predictor Coef SE Coef T P Constant -3.491 1.558 -2.24 0.028 IQ 0.07612 0.01549 4.91 0.000 SC 0.03691 0.02456 1.50 0.137 C3 0.06701 0.08558 0.78 0.436 S = 1.551 R-Sq = 47.5% R-Sq(adj) = 45.4% Analysis of Variance Source DF SS MS F P Regression 3 161.378 53.793 22.36 0.000 Residual Error 74 178.049 2.406 Total 77 339.427 Source DF Seq SS IQ 1 136.319 SC 1 23.583 C3 1 1.476 Unusual Observations Obs IQ GPA Fit SE Fit Residual St Resid 8 97 2.412 6.110 0.276 -3.698 -2.42R 22 109 1.760 5.678 0.649 -3.918 -2.78RX 48 102 3.936 7.379 0.282 -3.443 -2.26R 51 103 0.530 6.775 0.194 -6.245 -4.06R R denotes an observation with a large standardized residual X denotes an observation whose X value gives it large influence. Comments and answers to questions: ---------------------------------- We already saw in Supplementary exercise 11.15 that IQ and SC are both strongly significant predictors of GPA in a model containing only these two variables. The fitted regression model involving additionally the C3 variable is GPAhat = -3.49 + 0.0761*IQ + 0.0369*SC + 0.0670*C3 The regression coefficient for IQ is almost unchanged upon adding C3, but the coefficient for SQ has dropped to about 70% of its previous value. The coefficient for C3 is clearly non-significant (t=0.78, P=0.44), with a higher P-value than for SQ which has become non-significant as well. The R^2 has increased to 47.5%, a marginal increase only over the 47.1% of the model with IQ and SC. We conclude that C3 does not have significant or practically useful effect on prediction of GPA in a model that already includes IQ and SC. (c) The coefficient for C3 was 0.163 in the model without SC, and 0.067 in the model including SC. The explanation for this substantial change is a strong collinearity between C3 and SC - their correlation is 0.80. As SC and C3 therefore to a large extent explain the same thing, C3 is not a useful predictor in a model where SC is already present, but it can be useful when SC is absent.