Stat 226 - Section 11.1

Stat 226 - Section 11.1 - Introduction Chapter 11 Multiple...

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Chapter 11 Multiple Regression Chapter 11 1 Introduction We have investigated the relationship between a pair of quantitative variables with simple linear regression (Chapter 2) and inference for regression (Chapter 10). Multiple regression (Chapter 11) concerns the use of more than one explanatory variable to predict a response variable. Many of the exploratory tools remain the same. Chapter 11 2 Data for Multiple Regression In simple linear regression, the data consist of ( x i , y i ) pairs for each individual, with the subscript i identifying the individuals i = 1 , 2 ,..., n In multiple regression we have more than one explanatory variable, so the notation becomes slightly more complicated. It helps to think of a data matrix Chapter 11 3 Data Matrix If we have n individuals and p explanatory variables: Variables Individual y x 1 x 2 ... x p 1 y 1 x 11 x 12 ... x 1 p 2 y 2 x 21 x 22 ... x 2 p . . . . . . n y n x n 1 x n 2 ... x np Each explanatory variable has a different subscript in addition to each individual having a subscript. Chapter 11 4 Simple Linear Regression Recall the mathematical model for simple linear regression: μ y = β 0 + β 1 x For any fixed value of x , the response varies according to a Normal distribution with mean μ y and constant standard deviation σ . An alternative formulation is y i = β 0 + β 1 x i + ε i ε i N ( 0 , σ ) The parameters of the model are β 0 , β 1 and σ . Chapter 11 5 Multiple Regression Model The mathematical model for multiple regression is μ y = β 0 + β 1 x 1 + β 2 x 2 + ... + β p x p As in the simple case, the response varies according to a Normal distribution with mean μ y and constant standard deviation σ . An alternative formulation is y i = β 0 + β 1 x i 1 + β 2 x i 2 + ... + β p x ip + ε i ε i N ( 0 , σ ) The parameters of the model are β 0 , β 1 , β 2 ,..., β p and σ . Chapter 11 6
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Data Analysis Strategy When we encounter several variables, our analysis strategy consists of two exploratory steps. 1.
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