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Unformatted text preview: 354 Chapter 14: Introduction to Multiple Regression CHAPTER 14 OBJECTIVES • To learn how to develop a multiple regression model • To know how to interpret the regression coefficients • To learn how to determine which independent variables to include in the regression model • To know how to determine which independent variables are more important in predicting a dependent variable • To be able to use categorical variable in regression models OVERVIEW AND KEY CONCEPTS The Multiple Regression Model • The multiple regression model describes the relationship between one dependent variable and 2 or more independent variables in a linear function. The Simple Linear Regression and Multiple Regression Compared • Coefficients in a simple regression pick up the impact of that variable plus the impacts of other variables that are correlated with it and the dependent variable. • Coefficients in a multiple regression net out the impacts of other variables in the equation. Hence, they are called net regression coefficients . Interpretation of the Estimated Coefficients • The Y intercept ( ) b : The estimated average value of i Y when all i X = . • Slope ( ) i b : Estimated that the average value of Y changes by i b for each oneunit increase in i X holding constant the effect of all other independent variables. Predicting the Dependent Variable Y • Use the estimated sample regression equation (multiple linear regression equation): 1 1 ˆ i i k k i Y b b X b X = + + + L 1 1 2 2 i i i k k i i Y b b X b X b X e = + + + + + L Population Yintercept Population slopes Random Error Dependent (response) variable Independent (explanatory) variables 1 2 i i i k k i i Y X X X β β β β ε 1 2 = + + + + + L Residual Study Guide and Student’s Solutions Manual 355 The Venn Diagram and Explanatory Power of the Multiple Regression Model Coefficient of Multiple Determination • Coefficient of multiple determination measures the proportion of total variation in Y explained by all X variables taken together. • 2 12 Explained Variation Total Variation Y k SSR r SST • = = L • It never decreases when an additional X variable is added to the model, which is a disadvantage when comparing among models. Y X 1 2 r SSR SSR SSE = = + Y X 1 Variations in Y explained by X 1 or variations in X 1 used in explaining variation in Y Variations in Y not explained by X 1 Variations in X 1 not used in explaining variation in Y ( ) SSE ( ) SSR Y X 1 X 2 Overlapping Overlapping variation in both X 1 and X 2 are used in explaining the variation variation in Y but NOT NOT in the estimation estimation of nor 1 β 2 β Variation NOT NOT explained by X 1 nor X 2 ( ) SSE 356 Chapter 14: Introduction to Multiple Regression Adjusted Coefficient of Multiple Determination • It measures the proportion of variation in Y explained by all X variables adjusted for the number of X variables used....
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This note was uploaded on 03/26/2012 for the course MATH 104 taught by Professor Green during the Spring '10 term at Golden Gate.
 Spring '10
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