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Ch 7 Notes

Ch 7 Notes - Chapter 7 Multiple Linear Regression In the...

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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 explana- tory variables p . So then we’ll label our explanatory variables X 1 ,...,X p . 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 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.

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7.1 Concepts and Setup 131 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 sim- ple linear regression. We still have only one y -intercept α , but now we have multiple 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 num- ber (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. Visualizing the Data Unfortunately, it’s hard to make a single picture that gives a good visual rep- resentation of the data in multiple regression. If we have only two explanatory
7.1 Concepts and Setup 132 variables, X 1 and X 2 , we could try to make a sort of “3-D scatterplot” but this would probably be too confusing to be of much practical use.

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Ch 7 Notes - Chapter 7 Multiple Linear Regression In the...

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