General Example: Gas Mileage (Stat 5044 Schabenberger & J.P.Morgen)

Transcription

1 General Example: Gas Mileage (Stat 5044 Schabenberger & J.P.Morgen) From Motor Trend magazine data were obtained for n=32 cars on the following variables: Y= Gas Mileage (miles per gallon, MPG) X1= Engine size in cubic inches (SIZE). X2= Horsepower (HP). X3= Weight in pounds (Weight). X4= Engine shape (1=straight, 0= V, ShAPE). X5= Number of cylinders (CYLIN). X6= Transmission type (1= manual, 0= automatic, TRANS). X7= Number of forward gears (GEARS). (1) If we were to perform a SLR of MPG on TRANS, the slope of the fitted Regression line estimates the difference in the mean mileage of cars with (TRANS=1) and without (TRANS=0) manual transmission. Model <.0001 Error Root MSE R-Square Dependent Adj R-Sq Coeff Var Parameter Estimates Variable DF Estimate Error t Value Pr > t Intercept <.0001 trans <.0001 A manual transmission increases the gas mileage by mile per gallon on average. This increase is significant. However, we know that the cars in the two groups are different with respect to other characteristics, not only the transmission type. Does having a manual transmission still improve the gas mileage if, e.g., the car engine size and horsepower are taken into account?? To answer this question, we need to fit a model with three variables simultaneously, A multiple Regression Model.

2 Model <.0001 Error Root MSE R-Square Dependent Adj R-Sq Coeff Var Parameter Estimates Variable DF Estimate Error t Value Pr > t Intercept <.0001 size hp trans The Correlation between the response and the seven regressors in our example is as follow: The SAS System 1 The CORR Procedure 1 With Variables: mpg 7 Variables: size hp weight shape cylin trans gears Pearson Correlation Coefficients, N = 32 Prob > r under H0: Rho=0 size hp weight shape cylin trans gears mpg <.0001 The Correlation among the seven regressors in our example is as follow: The CORR Procedure 7 Variables: size hp weight shape cylin trans gears Pearson Correlation Coefficients, N = 32 Prob > r under H0: Rho=0 size hp weight shape cylin trans gears size hp weight shape cylin trans

3 gears This indicator for A Multicolinearity problem Correlations among predictors resulting in an increase in variance Reduces the significance value of the variable Occurs when several predictor variables are used in the model Model <.0001 Error Root MSE R-Square Dependent Adj R-Sq Coeff Var Parameter Estimates Variable DF Estimate Error t Value Pr > t Intercept size hp weight shape cylin trans gears Model Selection 1- Forward Selection Forward Selection: Step 1 Variable size Entered: R-Square = and C(p) = Model <.0001 Error Parameter Standard

4 Intercept <.0001 size <.0001 Bounds on condition number: 1, 1 Forward Selection: Step 2 Variable trans Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size <.0001 trans Bounds on condition number: , Forward Selection: Step 3 The SAS System 11 Forward Selection: Step 3 Variable weight Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size weight trans Bounds on condition number: , Forward Selection: Step 4 Variable cylin Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size weight cylin trans

5 Bounds on condition number: , Forward Selection: Step 5 The SAS System 12 Forward Selection: Step 5 Variable hp Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size hp weight cylin trans Bounds on condition number: , No other variable met the significance level for entry into the model. Summary of Forward Selection Variable Number Partial Model Step Entered Vars In R-Square R-Square C(p) F Value Pr > F 1 size < trans weight cylin hp Backward Elimination Backward Elimination: Step 0 All Variables Entered: R-Square = and C(p) = Model <.0001 Error Intercept size hp weight

6 shape cylin trans gears Bounds on condition number: , Backward Elimination: Step 1 Variable trans Removed: R-Square = and C(p) = Model <.0001 Error The SAS System 14 Backward Elimination: Step 1 Intercept size hp weight shape cylin gears Bounds on condition number: , Backward Elimination: Step 2 Variable shape Removed: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size hp weight cylin gears Bounds on condition number: , All variables left in the model are significant at the level. The SAS System 15 Summary of Backward Elimination

7 Variable Number Partial Model Step Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 trans shape Stepwise Selection Stepwise Selection: Step 1 Variable size Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size <.0001 Bounds on condition number: 1, 1 Stepwise Selection: Step 2 Variable trans Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 size <.0001 trans Bounds on condition number: , Stepwise Selection: Step 3 The SAS System 17 Stepwise Selection: Step 3 Variable weight Entered: R-Square = and C(p) = Model <.0001 Error

8 Intercept <.0001 size weight trans Bounds on condition number: , Stepwise Selection: Step 4 Variable size Removed: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 weight <.0001 trans Bounds on condition number: , Stepwise Selection: Step 5 The SAS System 18 Stepwise Selection: Step 5 Variable shape Entered: R-Square = and C(p) = Model <.0001 Error Intercept <.0001 weight shape trans Bounds on condition number: , All variables left in the model are significant at the level. No other variable met the significance level for entry into the model. Summary of Stepwise Selection Variable Variable Number Partial Model Step Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 size < trans weight size shape

9 Model Comparison Reduced Model Model <.0001 Error Root MSE R-Square Dependent Adj R-Sq Coeff Var Parameter Estimates Variable DF Estimate Error t Value Pr > t Intercept <.0001 size <.0001 trans Full Model Model <.0001 Error Root MSE R-Square Dependent Adj R-Sq Coeff Var Parameter Estimates Variable DF Estimate Error t Value Pr > t Intercept size hp weight shape cylin trans gears

10 F-TEST Test 1 Results for Dependent Variable mpg Source DF Square F Value Pr > F Numerator Denominator SAS Code data Gasmileage; input mpg size hp weight shape cylin trans gears; datalines; ;;; model mpg=trans; model mpg=size hp trans; proc corr data=gasmileage nosimple; var size hp weight shape cylin trans gears; with mpg; proc corr data=gasmileage nosimple; var size hp weight shape cylin trans gears; model mpg=size hp weight shape cylin trans gears; model mpg=size hp weight shape cylin trans gears/ ss1 ss2; /* model mpg=size hp weight shape cylin trans gears/ selection=rsquare cp; */ model mpg=size hp weight shape cylin trans gears/ selection=forward slentry=.25; model mpg=size hp weight shape cylin trans gears/ selection=backward slstay=.25; model mpg=size hp weight shape cylin trans gears/ selection=stepwise slstay=.25 slentry=.25;

11 model mpg=size trans; model mpg=size hp weight shape cylin trans gears; test weight=0, shape=0, cylin=0, hp=0, gears=0;