notes Ch 13(II) - CHAPTER 13(II Multiple Regression Butler...

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CHAPTER 13(II) Multiple Regression Butler Trucking Company provides deliveries throughout its local area. To develop better work schedules, the managers want to estimate the total daily travel time for their drivers. Initially the managers believed that the total daily travel time would be closely related to the number of miles traveled in making the daily deliveries. In attempting to identify another independent variable, the managers felt that the number of deliveries could also contribute to the total travel time. The data of 10 observations is provided below. Assignment Miles Deliveries Time 1 100 4 9.3 2 50 3 4.8 3 100 4 8.9 4 100 2 6.5 5 50 2 4.2 6 80 2 6.2 7 75 3 7.4 8 65 4 6 9 90 3 7.6 10 90 2 6.1 10.1 Model The multiple regression model is given by ε β β β β + + + + + = p p x x x y ... 2 2 1 1 0 Where p is the number of independent variables x, y is the dependent variable, and ε is a random error. Assuming ε is normally, the multiple regression equation is reduced to p p x x x y E β β β β + + + + = ... ) ( 2 2 1 1 0 Where E(y) is the expected value of y. The sample statistics provide an estimate of the above equation with the following: p p x b x b x b b y + + + + = ... ˆ 2 2 1 1 0 Where b 0 , b 1 , b 2 , …, b p are the estimates of β 0 , β 1 , β 2 , …, β p y ˆ = estimated value of the dependent variable 1
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13.2 Least Squares Method Similar to Chapter 12, the least squares criterion is: ( 29 2 ˆ min - i i y y Where: i y = observed value of the dependent variable for the i th observation i y ˆ = estimated value of the dependent variable for the i th observation The coefficients b 0 , b 1 , b 2 , …b p are obtained in Excel (see Appendix: Regression in Excel) 13.3 Multiple Coefficient of Determination The sum of squares due to error is 2 ) ( - = i i y y SSE The total sum of squares is given by 2 ) ( - = y y SST i The sum of squares due to regression is 2 ) ˆ ( - = y y SSR i The relationship among SST, SSR, and SSE is SST = SSR + SSE The multiple coefficient of determination is SST SSR r = 2 r 2 measures the percentage of variation of the dependent variable explained by changes in the independent variables.
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