1. © 2000 Prentice-Hall, Inc. Chap. 10 - 1 Multiple Regression Models

2. © 2000 Prentice-Hall, Inc. Chap. 10 - 2 The Multiple Regression Model The relationship between one dependent & two or more independent variables is a linear function Population Y-intercept Population slopes Dependent (Response) variable for sample Independent (Explanatory) variables for sample model Random Error

3. © 2000 Prentice-Hall, Inc. Chap. 10 - 3 Multiple Regression Model: Example ( 0 F) Develop a model for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.

4. © 2000 Prentice-Hall, Inc. Chap. 10 - 4 Sample Multiple Regression Model: Example Excel Output For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5.437 gallons, holding insulation constant. For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20.012 gallons, holding temperature constant.

5. © 2000 Prentice-Hall, Inc. Chap. 10 - 5 Slope (b i ) The average Y changes by b i each time X i is increased or decreased by 1 unit holding all other variables constant. For example: If b 1 = -2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X 1 ) given the inches of insulation (X 2 ). Interpretation of Estimated Coefficients

6. © 2000 Prentice-Hall, Inc. Chap. 10 - 6 Intercept (b 0 ) The intercept (b 0 ) is the estimated average value of Y when all X i = 0. Interpretation of Estimated Coefficients

7. © 2000 Prentice-Hall, Inc. Chap. 10 - 7 Using The Model to Make Predictions Predict the amount of heating oil used for a home if the average temperature is 30 0 and the insulation is 6 inches. The predicted heating oil used is 278.97 gallons

8. © 2000 Prentice-Hall, Inc. Chap. 10 - 8 Developing the Model Checking for problems. Being sure the model passes all tests for model quality.

9. © 2000 Prentice-Hall, Inc. Chap. 10 - 9 Identifying Problems Do all the residual tests listed for simple regression. Check for multicolinearity.

10. © 2000 Prentice-Hall, Inc. Chap. 10 - 10 Multicolinearity This occurs when there is a high correlation between the explanatory variables. This leads to unstable coefficients. The VIF used to measure colinearity (values exceeding 5 are not good and exceeding 10 are a big problem): = Coefficient of Multiple Determination of X j with all the others

11. © 2000 Prentice-Hall, Inc. Chap. 10 - 11 Is the fit to the data good?

12. © 2000 Prentice-Hall, Inc. Chap. 10 - 12 Coefficient of Multiple Determination Excel Output r 2 Adjusted r 2 The r 2 is adjusted downward to reflect small sample sizes.

13. © 2000 Prentice-Hall, Inc. Chap. 10 - 13 Do the variables collectively pass the test?

14. © 2000 Prentice-Hall, Inc. Chap. 10 - 14 Testing for Overall Significance Shows if there is a linear relationship between all of the X variables taken together and Y Hypothesis: H 0 :  1 =  2 = … =  p = 0 (No linear relationships) H 1 : At least one  i  0 (At least one independent variable effects Y)

15. © 2000 Prentice-Hall, Inc. Chap. 10 - 15 Test for Overall Significance Excel Output: Example p = 2, the number of explanatory variables n - 1 MSR MSE p value = F Test Statistic

16. © 2000 Prentice-Hall, Inc. Chap. 10 - 16 F 03.89 H 0 :  1 =  2 = … =  p = 0 H 1 : At least one  I  0  =.05 df = 2 and 12 Critical value(s): Test Statistic: Decision: Conclusion: Reject at  = 0.05 There is evidence that at least one independent variable affects Y.  = 0.05 F  Test for Overall Significance 168.47 (Excel Output)

17. © 2000 Prentice-Hall, Inc. Chap. 10 - 17 Test for Significance: Individual Variables Shows if there is a linear relationship between each variable X i and Y. Hypotheses: H 0 :  i = 0 (No linear relationship) H 1 :  i  0 (Linear relationship between X i and Y)

18. © 2000 Prentice-Hall, Inc. Chap. 10 - 18 T Test Statistic Excel Output: Example t Test Statistic for X 1 (Temperature) t Test Statistic for X 2 (Insulation)

19. © 2000 Prentice-Hall, Inc. Chap. 10 - 19 H 0 :  1 = 0 h 1 :  1  0 df = n-2 = 12 critical value(s): Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence of a significant effect of temperature on oil consumption. t 0 2.1788 -2.1788.025 Reject H 0 0.025 Does temperature have a significant effect on monthly consumption of heating oil? Test at  = 0.05. t Test : Example Solution t Test Statistic = -16.1699

20. © 2000 Prentice-Hall, Inc. Chap. 10 - 20 Confidence Interval Estimate For The Slope Provide the 95% confidence interval for the population slope  1 (the effect of temperature on oil consumption). -6.169   1  -4.704 The estimated average consumption of oil is reduced by between 4.7 gallons to 6.17 gallons per each increase of 1 0 F.

21. © 2000 Prentice-Hall, Inc. Chap. 10 - 21 Special Regression Topics

22. © 2000 Prentice-Hall, Inc. Chap. 10 - 22 Dummy-variable Models Create a categorical variable (dummy variable) with 2 levels:  For example, yes and no or male and female.  The date is coded as 0 or 1. The coding makes the intercepts different. This analysis assumes equal slopes. The regression model has same form:

23. © 2000 Prentice-Hall, Inc. Chap. 10 - 23 Dummy-variable Models Assumption Given: Y = Assessed Value of House X 1 = Square footage of House X 2 = Desirability of Neighborhood = Desirable (X 2 = 1) Undesirable (X 2 = 0) 0 if undesirable 1 if desirable Same slopes

24. © 2000 Prentice-Hall, Inc. Chap. 10 - 24 Dummy-variable Models Assumption X 1 (Square footage) Y (Assessed Value) Desirable Location Undesirable b 0 + b 2 b0b0 Same slopes Intercepts different

25. © 2000 Prentice-Hall, Inc. Chap. 10 - 25 Interpretation of the Dummy Variable Coefficient For example: : GPA 0 Female 1 Male : Annual salary of college graduate in thousand $ This 6 is interpreted as given the same GPA, the male college graduate is making an estimated 6 thousand dollars more than female on average. :