1 / 16

Multiple regression analysis (MRA)

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!. MR equations.

mabyn
Download Presentation

Multiple regression analysis (MRA)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related