Stat 112 Notes 8 Today: –Chapters 4.3 (Assessing the Fit of a Regression Model) –Chapter 4.4 (Comparing Two Regression Models) –Chapter 4.5 (Prediction with a Multiple Regression Equation)

Gas Mileage Regression

R-Squared (Coefficient of Determination) The coefficient of determination for multiple regression is defined as for simple linear regression: Represents percentage of variation in y that is explained by the multiple regression line. is between 0 and 1. The closer to 1, the better the fit of the regression equation to the data. For the gas mileage regression, RSquare Summary of Fit

Comparing Two Regression Models Multiple Regression Model for automobile data: We use t test to test if one variable, for example, cargo is useful after putting the rest of the three variables into the model. How to test whether cargo and/or seating are useful predictors once weight and hp are taken into account, i.e., test

Full vs. Reduced Model General setup for testing whether any of the variables are useful for predicting y after taking into account variables Full model: Reduced model: Is the full model better than the reduced model?

Partial F test Test statistic: Under H 0, F has an distribution. Round both degrees of freedom down when using Table B.4. Decision rule for test with significance level –Reject H 0 if –Accept H 0 if p-value = Prob (F (K-L, n-K-1) >F)

Automobile Example Test whether seating and length are useful predictors once weight and hp are taken into account. From Table B.4, F(.05; 2,120)=3.07 [rounding down to nearest denominator degrees of freedom] Because 60.59>3.07, we reject H 0. There is evidence that seating and/or length are useful predictors once weight and hp are taken into account.

Test of Usefulness of Model Are any of the variables useful for predicting y? Multiple Linear Regression model:

F Test of Usefulness of Model Under, F has F(K,n-K-1) distribution. Decision rule: Reject if [see Appendix B.3-B.5] F test in JMP in Analysis of Variance table. Prob>F is the p-value for the F test.

Test of Usefulness of Model for Gas Mileage Data

Prediction in Gas Mileage Data The design team is planning a new car with the following characteristics: weight=4000 lbs, horsepower = 200, seating = 5 adults, length=200 inches. What is a 95% prediction interval for the GPM1000 of this car?

Prediction with Multiple Regression Equation Prediction interval for individual with x 1,…,x K : For a large number of observations (say n>30+number of explanatory variables *10), the 95% prediction interval is approximately

Finding Prediction Interval in JMP Enter a line with the independent variables x 1,…,x K for the new individual. Do not enter a y for the new individual. Fit the model. Because the new individual does not have a y, JMP will not include the new individual when calculating the least squares fit. Click red triangle next to response, click Save Columns: –To find, click Predicted Values. Creates column with –To find 95% PI, click Indiv Confid Interval. Creates column with lower and upper endpoints of 95% PI.

Prediction in Automobile Example The design team is planning a new car with the following characteristics: weight = 4000 lbs, horsepower =200, seating =5, length =200 inches From JMP, – –95% prediction interval: (33.36, 45.63)