Multiple regression analysis (MRA)

1. Multiple regression analysis (MRA) • We have seen how simple regression analysis can be used to model relationships – why then would we need to use multiple regression analysis (MRA)? • Because complex relationships may involve more than one independent variable!

2. MR equations • If we are modeling Y as a function of two independent variables X1 and X2, our MR equation is: Y=B0+B1X1+B2X2 • If Y is a function of 4 independent variables (X1 – X4), our equation: Y=B0+B1X1+B2X2+ B3X3+ B4X4

3. MRA Problem • Can we predict the mileage of an automobile(measured in MPG) based on its engine horsepower and weight? • How would horsepower affect mileage (would it be a positive or negative relationship)? • How would weight affect mileage (would it be a positive or negative relationship)?

4. Data To do the analysis please use the file Auto.xls that comes with the text book CD (or email me and I can send you the data file).

5. EXCEL output Regression Equation: MPG = 58.1571 -0.1175*Horsepower -0.0069*Weight

6. Interpreting MR coefficients: Horsepower • The coefficient for Horsepower is -0.1175; thus, holding constant Weight, MPG decrease by 0.1175 for every 1 unit increase in Horsepower • Stated another way, according to the regression equation, if 2 automobiles have the same weight, the auto with a higher horsepower will have lower MPG

7. Interpreting MR coefficients: Weight • The coefficient for Weight is -0.0069; thus, holding constant Horsepower, MPG decreases by 0.0069 for every additional lb of Weight • In other words, our regression model says that for 2 automobiles with the same horsepower, the automobile that weighs more will have lower MPG

8. Prediction using MR equation • Can we predict MPG for an automobile with a horsepower of 100 and weighs 2000 lbs? • Yes • Because the value for horsepower (100) is within the range of horsepower values used in developing the MR equation • And because the value for weight (2,000) is also within the range of weight values used in developing the MR equation

9. Prediction using MR equation MPG=58.1571-0. 1175*Horsepower-0.0069*Weight MPG=58.1571 - (0. 1175*100) – (0.0069*2000) MPG=32.66 Predicted MPG for an automobile with an engine horsepower of 100 and weighs 2000 lbs

10. Adjusted coefficient of determination (Adj.-r2) • It is meaningful to use adjusted R-squared (as opposed to R-squared) for the MR equation since this measure accounts for the number of independent variables and observations

11. Adjusted R-square from EXCEL Adj. R-square=0.74 tells us that about 74% of the variation in MPG is explained by Horsepower and Weight

12. Is the MR model statistically valid? • To assess validity of MR model, we need to use ANOVA (available in the EXCEL output). • The hypothesis we are testing is: H0: Slope (HP)=Slope (Weight)=0 H1: Not H0

13. Using ANOVA from EXCEL to test if MR model is valid F=70.28 and P-value=7.50524E-15. This P-value is (much) smaller than 0.01. Our rule is: if P-value is less than a reject null. Thus, at a=0.01 level, we reject null. Our regression model is statistically valid.

14. Assessing the contribution of Horsepower to the MR model t-Stat=-3.6003 and P-value=0.000763. Rule-- if P-value is smaller than a then reject null that Slope of horsepower=0. Because p-value is very small, we conclude that horsepower is a significant predictor of MPG in our model.

15. Assessing the contribution of Weight to the MR model t-Stat=-4.9035 and P-value=1.16E-05. Rule-- if P-value is smaller than a then reject null that Slope of weight=0. Because p-value is very small, we conclude that weight is a significant predictor of MPG in our model.

16. Is the analysis over? • As an academic example of multiple regression, yes • As a modeling exercise – perhaps not • The analyst can consider adding more variables if they are available, such as • Transmission type • Age of the automobile, etc.