Stat 401 B – Lecture 311Model Selection Response: Highway MPG Explanatory: 13 explanatory variables Indicator variables for types of car – Sports Car, SUV, Wagon, Minivan2Explanatory Variables Engine size (liters) Cylinders (number) Horsepower Weight Wheel Base Length Width3“Best” Model The 7-variable model with SUV, Minivan, All Wheel, Engine, Horsepower, Weight and Wheel BaseAppears to be the “best” model.Stat 401 B – Lecture 314Prediction EquationPredicted Highway MPG = 30.74 – 3.15*SUV – 3.28*Minivan –2.08*All Wheel – 1.65*Engine –0.0226*Horsepower –0.0029*Weight + 0.163*Wheel Base5Summary All variables add significantly. R2= 0.705 adj R2= 0.682 RMSE = 3.430786 Cp= 4.90116-20-15-10-505101520ResidualBest Model15 20 25 30 35Predicted Highw ay MPGStat 401 B – Lecture 317.01.05.10.25.50.75.90.95.99-3-2-10123Normal Quantile Plot5101520253035Count-5 0 5 10 158.01.05.10.25.50.75.90.95.99-3-2-10123Normal Quantile Plot10203040Count-2 -1 0 1 2 3 4 5Standardized ResidualDistributions9Box Plot – Potential Outliers3.664812.573333.426746Volkswagen Jetta GLS TDI 4dr4.573515.690735.309351Toyota Prius4dr (gas/electric)2.17747.470135.529943Toyota Echo 2dr manual2.51298.621335.378744Honda Civic HX 2drStandardized Residual, zResidualPredicted MPGHighway MPGVehicle NameStat 401 B – Lecture 3110Bonferroni Correction Adjust what is a small P-value. If a P-value is less than 0.0005, then the standardized residual is statistically significant.0005.010005.0residuals of #05.0==11Standardized Residual0.000253.6648Volkswagen JettaGLS TDI 4dr0.000004.5735Toyota Prius 4dr (gas/electric)0.029452.1774Toyota Echo 2dr manual0.011972.5129Honda Civic HX 2drProb > |z|Standardized Residual, zVehicle Name12Outliers Both the Toyota Prius and the Volkswagon Jetta have standardized residuals so extreme that they are considered statistically significant (P-value < 0.0005).Stat 401 B – Lecture 3113Leverage Because we have multiple explanatory variables, there is not an easy formula for leverage, h. The leverage, h, value takes into account all of the explanatory variables.14Rule of Thumb High Leverage Value ifn = 100, p = 7, ⎟⎠⎞⎜⎝⎛+>nph1216.01008212 =⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛+np15Leverage There are 10 vehicles that have leverage, h, greater than 0.16. Of these, 2 have F-statistics large enough to produce P-values smaller than 0.0005.Stat 401 B – Lecture 3116Leverage0.000008.8520.4084Porsche 911 GT2 2 dr0.000394.2800.2532Chevrolet Corvette convertible 2 drProb > FFh17Leverage What makes the leverage so high? Have to look for extreme values for the explanatory variables.18Leverage Chevy Corvette Has the 2ndlargest engine of all the vehicles – 5.7 liter and the 2ndhighest horsepower – 350 horsepower. Porsche 911 Has the highest horsepower of all the vehicles – 477 horsepower.Stat 401 B – Lecture 3119Influence – Cook’s D None of the vehicles has a value of Cook’s D that is greater than 1. The largest value of Cook’s D is 0.16 for the Toyota Prius 4dr (gas/electric). The second largest is 0.12 for the VW Jetta.20Influence Just because there are no vehicles with Cook’s D greater than 1, you should still look at the Studentized residuals.2110203040Count-2 -1 0 1 2 3 4 5Studentized ResidualDistributionsStat 401 B – Lecture 3122Studentized Residual0.000273.7843Volkswagen JettaGLS TDI 4dr0.000014.5031Toyota Prius 4dr (gas/electric)0.027022.2471Toyota Echo 2dr manual0.011892.5663Honda Civic HX 2drProb > | rs|StudentizedResidual, rsVehicle Name23Outliers Both the Toyota Prius and the Volkswagon Jetta have studentized residuals so extreme that they are considered statistically significant (P-value < 0.0005).24Summary Toyota Prius and Volkswagon Jettaare statistically significant outliers. Chevy Corvette and Porsche 911 are statistically significant high leverage values. Toyota Prius and Volkswagon Jettaexert statistically significant influence.Stat 401 B – Lecture 3125RSquareRSquare AdjRoot Mean Square ErrorMean of ResponseObservations (or Sum Wgts)0.704860.6824043.43078627.7100Summary of FitModelErrorC. TotalSource79299DF2586.13281082.86723669.0000Sum ofSquares369.44811.770Mean Square31.3881F Ratio<.0001*Prob > FAnalysis of VarianceInterceptSUVMinivanAll WheelEngineHorsepowerWeightWheel BaseTerm30.735611-3.147224-3.283013-2.081883-1.654325-0.022587-0.0026880.1632806Estimate6.1906581.3856281.4367111.016240.7389660.0086840.0012210.075565Std Error4.96-2.27-2.29-2.05-2.24-2.60-2.202.16t Ratio<.0001*0.0255*0.0246*0.0433*0.0276*0.0108*0.0302*0.0333*Prob>|t|Parameter EstimatesResponse Highway MPG26Prediction Equation All 100 vehiclesPredicted Highway MPG = 30.74 – 3.15*SUV – 3.28*Minivan –2.08*All Wheel – 1.65*Engine –0.0226*Horsepower –0.0029*Weight + 0.163*Wheel Base27RSquareRSquare AdjRoot Mean Square ErrorMean of ResponseObservations (or Sum Wgts)0.7700870.7522052.66182527.2755198Summary of FitModelErrorC. TotalSource79097DF2135.8830637.67822773.5612Sum ofSquares305.1267.085Mean Square43.0646F Ratio<.0001*Prob > FAnalysis of VarianceInterceptSUVMinivanAll WheelEngineHorsepowerWeightWheel BaseTerm29.554037-2.543539-2.412591-1.649593-1.146582-0.016019-0.003680.1740468Estimate4.848561.078671.120140.7909450.578080.006820.0009590.059234Std Error6.10-2.36-2.15-2.09-1.98-2.35-3.842.94t Ratio<.0001*0.0205*0.0339*0.0398*0.05040.0210*0.0002*0.0042*Prob>|t|Parameter EstimatesResponse Highway MPGStat 401 B – Lecture 3128Prediction Equation Excluding Prius and JettaPredicted Highway MPG = 29.55 – 2.54*SUV – 2.41*Minivan –1.65*All Wheel – 1.15*Engine –0.0160*Horsepower –0.0037*Weight + 0.174*Wheel Base29Comment Note that Engine has a P-value of 0.0504 and so is not significant at the 0.05 level. This suggests that there is a different “best” model if we exclude Prius and Jetta. 30Comment A similar thing happens if you exclude the Porsche 911 and the Chevy Corvette.Stat 401 B – Lecture 3131Multicollinearity High correlation among explanatory variables is called multicollinearity. Multicollinearity causes standard errors of estimates to be larger than they should be. 32Variance Inflation Factor A general measure of the effect of multicollinearity is the variance inflation factor, VIF.211iiRVIF−=33Multiple R2 is the value of R2among the k – 1 explanatory variables excluding explanatory