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CHAPTER 12 MULTIPLE LINEAR REGRESSION
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Table 12-15 Analysis of Variance of Example 12-12 Source of Variation Regression SSR 1 1 0 0 2 SSR 1 2 0 1, Error Total Sum of Squares 1012.0595 130.6091 881.4504 7.7943 1019.8538 Degrees of Freedom 2 1 1 17 19 Mean Square 506.0297 130.6091 881.4504 0.4508 f0 1103.69 284.87 1922.52 P-value 1.02E-18 4.70E-12 6.24E-19
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The analysis of variance for this model is shown in Table 12-15. Note that the hypothesis H0: 1 0 (significance of regression) would be rejected at any reasonable level of 2 significance because the P-value is very small. This table also contains the sums of squares SSR SSR 1 SSR 1
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so a test of the hypothesis H0: 2 0 can be made. Since this hypothesis is also rejected, we conclude that tool type has an effect on surface finish. It is also possible to use indicator variables to investigate whether tool type affects both the slope and intercept. Let the model be Y
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where x2 is the indicator variable. Now if tool type 302 is used, x2 Y If tool type 416 is used, x2
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Note that 2 is the change in the intercept and that 3 is the change in slope produced by a change in tool type. Another method of analyzing these data is to fit separate regression models to the data for each tool type. However, the indicator variable approach has several advantages. First, only one regression model must be fit. Second, by pooling the data on both tool types, more degrees of freedom for error are obtained. Third, tests of both hypotheses on the parameters 2 and 3 are just special cases of the extra sum of squares method.
12-6.3
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Selection of Variables and Model Building
An important problem in many applications of regression analysis involves selecting the set of regressor variables to be used in the model. Sometimes previous experience or underlying theoretical considerations can help the analyst specify the set or regressor variables to use in a particular situation. Usually, however, the problem consists of selecting an appropriate set of
12-6 ASPECTS OF MULTIPLE REGRESSION MODELING
regressors from a set that quite likely includes all the important variables, but we are sure that not all these candidate regressors are necessary to adequately model the response Y. In such a situation, we are interested in variable selection; that is, screening the candidate variables to obtain a regression model that contains the best subset of regressor variables. We would like the final model to contain enough regressor variables so that in the intended use of the model (prediction, for example) it will perform satisfactorily. On the other hand, to keep model maintenance costs to a minimum and to make the model easy to use, we would like the model to use as few regressor variables as possible. The compromise between these conflicting objectives is often called finding the best regression equation. However, in most problems, no single regression model is best in terms of the various evaluation criteria that have been proposed. A great deal of judgment and experience with the system being modeled is usually necessary to select an appropriate set of regressor variables for a regression equation. No single algorithm will always produce a good solution to the variable selection problem. Most of the currently available procedures are search techniques, and to perform satisfactorily, they require interaction with judgment by the analyst. We now briefly discuss some of the more popular variable selection techniques. We assume that there are K candidate regressors, x1, x2, p , xK, and a single response variable y. All models will include an intercept term 0, so the model with all variables included would have K 1 terms. Furthermore, the functional form of each candidate variable (for example, x1 1 x, x2 ln x, etc.) is correct. All Possible Regressions This approach requires that the analyst fit all the regression equations involving one candidate variable, all regression equations involving two candidate variables, and so on. Then these equations are evaluated according to some suitable criteria to select the best regression model. If there are K candidate regressors, there are 2K total equations to be examined. For example, if K 4, there are 24 16 possible regression equations; while if K 10, there are 210 1024 possible regression equations. Hence, the number of equations to be examined increases rapidly as the number of candidate variables increases. However, there are some very efficient computing algorithms for all possible regressions available and they are widely implemented in statistical software, so it is a very practical procedure unless the number of candidate regressors is fairly large. Several criteria may be used for evaluating and comparing the different regression models obtained. A commonly used criterion is based on the value of R2 or the value of the 2 adjusted R2, Radj. Basically, the analyst continues to increase the number of variables in the 2 2 model until the increase in R2 or the adjusted Radj is small. Often, we will find that the Radj will stabilize and actually begin to decrease as the number of variables in the model increases. 2 Usually, the model that maximizes Radj is considered to be a good candidate for the best re2 gression equation. Because we can write Radj 1 {MSE [SSE (n 1)]} and SSE (n 1) 2 is a constant, the model that maximizes the Radj value also minimizes the mean square error, so this is a very attractive criterion. Another criterion used to evaluate regression models is the Cp statistic, which is a measure of the total mean square error for the regression model. We define the total standardized mean square error for the regression model as 1