ch12 - 157 MULTIPLE REGRESSION AND THE GENERAL LINEAR MODEL...

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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 , cross-product 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.

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ch12 - 157 MULTIPLE REGRESSION AND THE GENERAL LINEAR MODEL...

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