1. INTRODUCTION TO MULTIPLE REGRESSION

2. 11.1 MULTIPLE REGRESSION MODEL 11.2 MULTIPLE COEFFICIENT OF DETERMINATION 11.3 MODEL ASSUMPTIONS 11.4 TEST OF SIGNIFICANCE 11.5 ANALYSIS OF VARIANCE (ANOVA)

3. 11.1 MULTIPLE REGRESSION MODEL  In multiple regression, there are several independent variables (X)and one dependent variable (Y).  The multiple regression model:  This equation that describes how the dependent variable y is related to these independent variables x 1, x 2,... x p. where: are the parameters, and e is a random variable called the error term are the independent variables.

4. 11.1 MULTIPLE REGRESSION MODEL  Multiple regression analysis is use when a statistician thinks there are several independent variables contributing to the variation of the dependent variable.  This analysis then can be used to increase the accuracy of predictions for the dependent variable over one independent variable alone.

5. Estimated Multiple Regression Equation Estimated Multiple Regression Equation Estimated Multiple Regression Equation  In multiple regression analysis, we interpret each regression coefficient as follows: regression coefficient as follows: β i represents an estimate of the change in y β i represents an estimate of the change in y corresponding to a 1-unit increase in x i when all corresponding to a 1-unit increase in x i when all other independent variables are held constant. other independent variables are held constant.

6. 11.2 MULTIPLE COEFFICIENT OF DETERMINATION (R 2 )  As with simple regression, R 2 is the coefficient of multiple determination, and it is the amount of variation explained by the regression model.  Formula:  In multiple regression, as in simple regression, the strength of the relationship between the independent variable and the dependent variable is measured by correlation coefficient, R. MULTIPLE CORRELATION COEFFICIENT (R)

7. 11.3 MODEL ASSUMPTIONS The errors are normally distributed with mean and variance Var =. The errors are statistically independent. Thus the error for any value of Y is unaffected by the error for any other Y-value. The X-variables are linear additive (i.e., can be summed).

8. 11.5 ANALYSIS OF VARIANCE (ANOVA) General form of ANOVA table: SourceDegrees of Freedom Sum of Squares Mean SquaresValue of the Test Statistic RegressionpSSRMSR=SSR pF=MSR MSE Errorn-p-1SSEMSE=SSE n-p-1 Totaln-1SST n Excel’s ANOVA Output SSR SST

9. 11.4 TEST OF SIGNIFICANCE  In simple linear regression, the F and t tests provide the same conclusion. the same conclusion.  In multiple regression, the F and t tests have different purposes. purposes. The F test is used to determine whether a significant relationship exists between the dependent variable and the set of all the independent variables. The F test is referred to as the test for overall significance. The F test is used to determine whether a significant relationship exists between the dependent variable and the set of all the independent variables. The F test is referred to as the test for overall significance. The t test is used to determine whether each of the individual independent variables is significant. independent variables is significant. A separate t test is conducted for each of the independent variables in the model. independent variables in the model. We refer to each of these t tests as a test for individual We refer to each of these t tests as a test for individual significance. significance. The t test is used to determine whether each of the individual independent variables is significant. independent variables is significant. A separate t test is conducted for each of the independent variables in the model. independent variables in the model. We refer to each of these t tests as a test for individual We refer to each of these t tests as a test for individual significance. significance.

10. Testing for Significance: F Test - Overall Significance Hypotheses Rejection Rule Test Statistics H 0 : β 1 = β 2 =... = β p = 0 H 0 : β 1 = β 2 =... = β p = 0 H 1 : One or more of the parameters H 1 : One or more of the parameters is not equal to zero. is not equal to zero. F = MSR/MSE Reject H 0 if p -value F α where : F  is based on an F distribution With p d.f. in the numerator and n - p - 1 d.f. in the denominator.

11. Testing for Significance: t Test- Individual Parameters Hypotheses Rejection Rule Test Statistics Reject H 0 if p -value < α or t t α/2 t t α/2Where: t α/2 is based on a t distribution t α/2 is based on a t distribution with n - p - 1 degrees of freedom.

12. Example: An independent trucking company, The Butler Trucking Company involves deliveries throughout southern California. The managers want to estimate the total daily travel time for their drivers. He believes the total daily travel time would be closely related to the number of miles traveled in making the deliveries. a)Determine whether there is a relationship among the variables using b) Use the t-test to determine the significance of each independent variable. What is your conclusion at the 0.05 level of significance?

13. Solution: a)Hypothesis Statement: Test Statistics: Rejection Region: Since 32.88>4.74, we Reject H 0 and conclude that there is a significance relationship between travel time (Y) and two independent variables, miles traveled and number of deliveries.

14. Solution: b) Hypothesis Statement: Test Statistics: Rejection Region: Since 6.18>2.365, we Reject H 0 and conclude that there is a significance relationship between travel time (Y) and miles traveled (X1).

15. Solution: b) Hypothesis Statement: Test Statistics: Rejection Region: Since 4.18>2.365, we Reject H 0 and conclude that there is a significance relationship between travel time (Y) and number of deliveries (X2).

16. End of Chapter 11