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Unformatted text preview: 157 MULTIPLE REGRESSION AND THE GENERAL LINEAR MODEL We extend the simple linear regression case to the multi ple linear regression case to handle situations where the dependent variable y is assumed to be related to more than one independent variable, say, x 1 , x 2 , . . . , x k . The model we use is called a general linear model – its form is y = β + β 1 x 1 + β 2 x 2 + ··· + β k x k + Our assumptions about the ’s are the same, namely the ’s are independent with mean E ( ) zero and variance σ 2 , and they are normally distributed. Data consist of n cases of k + 1 values denoted by ( y i , x i 1 , x i 2 , . . . , x ik ) , i = 1 , 2 , . . . , n Using this full notation we can write the model as y i = β + β 1 x i 1 + β 2 x i 2 + ··· + β k x ik + i The form of the general linear model, as given, is under stood to permit polynomial terms like x 3 2 , crossproduct terms like x 2 x 3 – etc. – Thus the polynomial model in one variable y = β + β 1 x + β 2 x 2 + β 3 x 3 + fits the form of the general linear model 158 y = β + β 1 x 1 + β 2 x 2 + β 3 x 3 + with x 1 ≡ x, x 2 ≡ x 2 , x 3 ≡ x 3 . The multiple regression model y = β + β 1 x 1 + β 2 x 2 + β 3 x 2 1 + β 4 x 2 2 + β 5 x 3 1 + β 6 x 3 2 + β 7 x 1 x 2 + can also be written as y = β + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5 + β 6 x 6 + β 7 , x 7 + with x 3 = x 2 1 , x 4 = x 2 2 , x 5 = x 3 1 , x 6 = x 3 2 , x 7 = x 1 x 2 . Even terms like log( x 3 ), e x 2 , cos( x 4 ), etc., are permitted. The model is called a linear model because its expres sion is linear in the β ’s. The fact that it may involve nonlinear functions of the x ’s is irrelevant. The concept of interaction will be discussed later. Here let us just state that the presence (or absence) of interaction between two independent variables (say) x 1 and x 2 can be tested by including product terms such as x 1 x 2 in the model. Thus the model: y = β + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 allows us to test whether interaction between x 1 and x 2 exists. The interpretation of this model is postponed till later. 159 When a general linear model relates y to a set of quantitative independent variables ( x ’s) that may include squared terms, product terms, etc., we have a multiple regression model . When the indepen dent variables are dummy variables coded to 0 or 1 representing values of qualitative independent vari ables or levels of treatment factors, the resulting models are called analysis of variance models. Just as we did in Chapter 11 for the simple linear regres sion model y = β + β 1 x + , we will consider least squares estimation of the parameters β , β 1 , β 2 , . . . , β k in the gen eral linear model. These will be denoted as ˆ β , ˆ β 1 , . . . , ˆ β k ....
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This note was uploaded on 01/23/2010 for the course STAT 213 taught by Professor Hao during the Spring '10 term at Internet2.
 Spring '10
 hao
 Linear Regression

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