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Unformatted text preview: CHAPTER 12: MULTIPLE LINEAR REGRESSION & CORRELATION 12.1 MULTIPLE LINEAR REGRESSION AND CORRELATION ANALYSIS As we mentioned in Chapter 7 , we can use more than one independent variable to estimate the value of the dependent variable and, in this way, attempt to increase the accuracy of the estimate. This process is called multiple linear regression and correlation analysis . In fact, multiple linear regression analysis is merely an extension of simple linear regression analysis and is used for testing hypotheses about the relationship between a dependent variable ( response variable ) and two or more independent variables ( predictors ) and for prediction. For example, the selling price of a home may be modeled as a function of the number of rooms, the size of the surrounding lot, and the total square footage of a house. Also, we can use the customers age, gender, income level, type of residence, etc. to predict how much they will spend on an automobile. In general, the dependent variable is designated by Y, while the k quantitative independent variables are designated sequentially by X 1 , X 2 , , and X k . The general descriptive form of a multiple linear regression equation for the population is then given by the following formula: Y j = + 1 X j1 + 2 X j2 + 3 X j3 + + k X jk + j where: Y j = The value of the j th observation of the variable Y , with j = 1 , 2 , , n . = Populations regression constant h = Populations regression coefficient for each independent variable X h ; where h = 1, 2, , k . k = Number of independent variables = (Greek letter epsilon) Random error term or residual. Based on the sample data, the least-squares multiple linear regression equation is written as = b + b 1 x 1 + b 2 x 2 + b 3 x 3 + + b k x k Constants b 1 , b 2 , b 3 , , and b k are called partial regression coefficients . They indicate the change in the estimated value of the dependent variable for a unit change in one of the independent variable, when the other independent variables are held constant. The constant b is the y-intercept , the value of Y when all the x h s are zero; and x 1 , x 2 , x 3 , etc. are respectively the values of the independent variables X 1 , X 2 , X 3 , etc. Dr. LOHAKA QBA 2305 CHAPT. 12: MULTIPLE REGRESSION & CORRELATION Page 118 The principal advantage of multiple linear regression over simple regression is that it allows us to use more of the information available to us to estimate or predict the value of the dependent variable. Sometimes the correlation between two variables may be insufficient to determine a reliable estimating equation. Yet, if we add the data from more independent variables, we may be able to determine an estimating equation that describes the relationship with greater accuracy....
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- Spring '08