Chapter 6 - F-2011 STAT 410/510 Chapter 6. Multiple...

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F-2011 STAT 410/510 1 Chapter 6. Multiple Regression I 6.1 Multiple regression models: First-order model with two predictor variables: i i i i X X Y 2 2 1 1 0 Assuming that , 0 } { i E 2 2 1 1 0 } { X X Y E Then, the regression function is a plane. See Figure 6.1. The parameter 0 indicates the Y intercept of the regression plane. If the scope of the model includes , 0 , 0 2 1 X X then 0 indicates the mean response } { Y E at . 0 , 0 2 1 X X Otherwise, it does not have any particular meaning. The parameter 1 indicates the change in the mean response } { Y E per unit increase in 1 X when 2 X is held constant. Likewise, 2 indicates the change in the mean response per unit increase in 2 X when 1 X is held constant. ( additive effects ). The parameters 1 and 2 are sometimes called partial regression coefficients because they reflect the partial effect of one predictor variable when the other predictor variable is included in the model and is held constant.
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F-2011 STAT 410/510 2 Qualitative predictor variables: Consider a regression analysis to predict the length of hospital stay ( Y ) based on the age ( 1 X ) and gender ( 2 X ) of the patient. Then, we define 2 X as follows: patient male if 0 patient female if 1 2 X Then, the first-order regression model is i i i i X X Y 2 2 1 1 0 . For male patient , 0 2 X and response function becomes 1 1 0 } { X Y E and for female patient , , 1 2 X 1 1 2 0 ) ( } { X Y E These two response functions represent parallel straight lines with different intercepts. In general, we represent a qualitative variable with c classes by means of 1 c indicator variables. Polynomial regression: i i i i X X Y 2 2 1 0 Despite the curvilinear nature of the response function, it is a special case of general linear regression model. If we let i i X X 1 and , 2 2 i i X X i i i i X X Y 2 2 1 1 0 Transformed variable: i i i i i X X X Y 3 3 2 2 1 1 0 ) log( If we let ) log( i i Y Y , i i i i i X X X Y 3 3 2 2 1 1 0 Another example would be i i i i X X Y 2 2 1 1 0 1 By letting , / 1 i i Y Y i i i i X X Y 2 2 1 1 0 Interaction effects:
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F-2011 STAT 410/510 3 i i i i i i X X X X Y 2 1 3 2 2 1 1 0 Let 2 1 3 i i i X X X , then i i i i i X X X Y 3 3 2 2 1 1 0 Meaning of linear in general linear regression model: The term linear model refers to the fact that regression model is linear in the parameters; it does not refer to the shape of the response surface. An example of a nonlinear regression model is the following:
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Chapter 6 - F-2011 STAT 410/510 Chapter 6. Multiple...

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