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chapter07_mine

# chapter07_mine - Chapter 7 Multiple Linear Regression In...

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Chapter 7 Multiple Linear Regression In the last chapter, we developed a set of techniques called simple linear regression for analyzing data with a quantitative explanatory variable and a quantitative response variable. In this chapter, we’ll adapt and extend those ideas to cover situations with more than one explanatory variable. We call these techniques multiple linear regression, or just multiple regression for short. 7.1 Concepts and Setup Again we’ll have a quantitative response variable Y , but now we’ll have more than one quantitative explanatory variable. Let’s call the number of explanatory variables p . So then we’ll label our explanatory variables X 1 ,...,X p . Note: We will be consistent about this definition of p, but be warned that other people may not be. Some textbooks use p to refer to the total number of unknown coefficients that our regression equation will estimate, including the y -intercept, so in those textbooks, p will equal the number of explanatory variables plus one. Population Each individual in the population has a value of Y and a value of each of X 1 ,...,X p . In the last chapter, we introduced the notation μ Y X to indicate the population mean of Y for just those individuals with a certain

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7.1 Concepts and Setup 126 X value. In this chapter, we’ll need to talk about μ Y X 1 ,...,X p , the population mean of Y for just those individuals with a certain combination of explanatory variable values. Linear Relationship What we’re going to assume about the population is that the relationship between μ Y X 1 ,...,X p and each of the explanatory variables X 1 ,...,X p is linear, so that we can write μ Y X 1 ,...,X p α β 1 X 1 β p X p . Our population relationship includes more parameters than it did in simple linear regression. We still have only one y -intercept α , but now we have multiple slope-like parameters β 1 ,...,β p , one for each explanatory variable. Sample In practice, we have to try to learn about the population by taking a sample. Each observation in our sample will have a value of Y and a value of each of the explanatory variables X 1 ,...,X p . Figure 7.1 shows a what a sample might look like with three explanatory variables. Notice that labeling the explanatory variable values is a little more complicated now. The first number (from 1 to n ) indicates the observation, and the second number (from 1 to p ) indicates the explanatory variable. Note: Occasionally people will reverse these labels on the X variables. It doesn’t really matter what you do, as long as you’re consistent. X 1 X 2 X 3 Y 38 84 7 19 39 81 14 48 48 95 18 62 X 1 X 2 X 3 Y x 11 x 12 x 13 y 1 x 21 x 22 x 23 y 2 x n 1 x n 2 x n 3 y n Figure 7.1: Sample, in a multiple regression context. Here there are three explanatory variables.
7.1 Concepts and Setup 127 Visualizing the Data Unfortunately, it’s hard to make a single picture that gives a good visual representation of the data in multiple regression. If we have only two explanatory variables, X 1 and X 2 , we could try to make a sort of “3-

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