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Unformatted text preview: 1 Chapter 14: Multiple Regression We employ more than one independent (predictor) variable in the prediction equation. Section 14.1: The Multiple Regression Model and the Least Squares Estimates Example: Tasty Sub Shop Case y = yearly revenue (thousands of dollars) x 1 = population size (thousands of residents) x 2 = business rating (a measure of business activity in the area) This is measured on a scale from 1 to 10 1: limited business and shopping activity nearby 10: lots of business and shopping activity nearby Goal: Predict y using both x 1 and x 2 See Table 14.1 (on page 581 ) and Figures 14.1 and 14.2 ( on page 582) 2 Independent (predictor) Dependent v ariables variable Linear Relationship Linear Relationship y = β + β 1 x 1 + ε y = β + β 1 x 2 + ε where β 1 > 0 where β 1 > 3 Combined Model error term ( effects of all other factors on y ) effects of x 1 and x 2 on y are regression parameters relating y to and See Figure 14.3 on page 583 for a geometrical interpretation. 4 We will now use MINITAB to compute the least squares estimates of the regression parameters. See Figure 14.4(b) on page 584. Least squares estimates b , b 1 , and b 2 Prediction equation The least squares prediction equation : y ˆ = 125.29 + 14.1996 x 1 + 22.811 x 2 b b 1 b 2 5 The point prediction of y when and , which equals = 956.6 (that is, $956,600) We can understand the meaning of least squares by considering Table 14.2 on page 585. The least squares estimates minimize SSE = the sum of squared errors 6 Notation: Remember: β , β 1 , β 2 (betas) : are u nknown regression parameters b , b 1 , b 2 (bees) : are least squares estimates (aka the regression coefficients ) Interpretation of the parameters : μ = 2 2 1 1 x x β β β + + is the mean yearly revenue when population is x 1 and the business rating is x 2 . 1. β ’s interpretation Let x 1 = 0 and x 2 = 0 Then μ = β + 1 β (0) + 2 β (0) = β Therefore, β is the mean yearly revenue when x 1 = 0 and x 2 = 0. This does not make any sense – that’s ok, the y intercept often makes no sense on it’s own. 2. 1 β ’s interpretation Consider a site where x 1 = c and x 2 = d Then 7 μ = β + 1 β (c) + 2 β (d) Consider a different site where x 1 = (c + 1) and x 2 = d Then µ = β 0 + β 1 (c+1) + β 2 (d) The difference between the mean yearly revenue for the two sites is 1 β Therefore, 1 β = the change in the mean yearly revenue (mean y ) associated with a 1,000 resident increase in population size (that is, a 1 unit increase in x 1 ) when x 2 remains constant. 2 β is interpreted similarly The General Multiple Regression Model y = the dependent variable x 1 , x 2 , x 3 , … , x p = are p independent (predictor) variables The general multiple regression model says that y = ε β β β β β + + + + + + p p x x x x .......
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This note was uploaded on 02/09/2012 for the course DSC 203 taught by Professor Dr.weese during the Fall '11 term at Miami University.
 Fall '11
 Dr.Weese

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