11.1-11.3

11.1-11.3 - STAT3000 Chapter 11 Multiple Regression This...

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STAT3000 Chapter 11: Multiple Regression This chapter extends the basic concept of chapters 2 & 10 by modeling the mean value of y as a function of two or more independent variables. General Form of a Multiple Regression Model y = β 0 + β 1 x 1 + β 2 x 2 + + β p x p + ε Note// 1. β i determines the contribution of the independent variable x i . 2. x i may represent higher-order terms for quantitative predictors. 3. x i may also represent categorical predictors. In this case, x i is referred to as a dummy (indicator) variable. 224
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STAT3000 Section 11.1: Data Analysis for Multiple Regression The steps used to develop the multiple regression model are similar to those used for the simple linear regression model. Step 1: Determine appropriateness of linear model. a. Random sample (before fit model) b. Linear shape (before fit model) c. No pattern in residual plots (after fit model) d. Equal variance in residual plot (after fit) e. Histograms of residuals (after fit model) Step 2: Use the sample data to estimate the unknown model parameters, β 0 , β 1 , β 2 , , β p . But now, we have to choose which variables to include in the model. Step 3: Check that the assumptions about ε are valid. Statistically evaluate the usefulness of the model. R^2, s, inference procedures Step 4: Use the model for estimation and prediction. 225
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A First-Order Model* in Five Quantitative Independent Variables E(y) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5 where x 1 , x 2 , x 3 , x 4 , and x 5 are all quantitative variables that are not functions of other independent variables (*each independent variable is included, but no logs, no powers, no interactions). Interpreting β i : Each coefficient represents the slope of the line relating y to x i when all the other x’s are held fixed. In other words, β I represents the expected change (predicted average) in y for a unit increase in x i when all oth er x’s are held constant. The method of fitting first-order models and other multiple regression models is identical to that of fitting the SLR model, the method of least squares. We find the estimated model that minimizes the SSE. The primary difference is computational difficulty. Therefore, we will rely on computer output. 0 1 1 2 2 3 3 4 4 5 ˆ y = b b x b x b x b x b x 5 + + + + + 226
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STAT3000 Section 11.2: Inferences for Multiple Regression Model Assumptions For any given set of values of x 1 , x 2 , …, x p , 1. E( ε ) = 0 2. Var( ε ) = σ 2 3. ε ~ N 4. The values of ε associated with any two observed values of y are independent. MSE= 2 2 SSE s = and s = s n-p-1 = Square MSE n – p–1 = n–(p+1) = df of error = n – # estimated β ’s (variables + intercept) 227
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P= number of variables we have and then the -1 for intercept - For every variable we loose 1 degree of freedom b/c you have much bigger interval ( bad) s 2 = MSE or mean square for error and s = MSE Residual Analysis: Checking Model Assumptions Residual plots are useful for detecting whether or not the model assumptions hold.
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11.1-11.3 - STAT3000 Chapter 11 Multiple Regression This...

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