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Unformatted text preview: 06.05.1 Chapter 06.05 Adequacy of Models for Regression Quality of Fitted Model In the application of regression models, one objective is to obtain an equation ) ( x f y = that best describes the n response data points ) , ( ),......., , ( ), , ( 2 2 1 1 n n y x y x y x . Consequently, we are faced with answering two basic questions. 1. Does the model ) ( x f y = describe the data adequately, that is, is there an adequate fit? 2. How well does the model predict the response variable (predictability)? To answer these questions, let us limit our discussion to straight line models as nonlinear models require a different approach. Some authors [1] claim that nonlinear model parameters are not unbiased. To exemplify our discussion, we will take example data to go through the process of model evaluation. Given below is the data for the coefficient of thermal expansion vs. temperature for steel. We assume a linear relationship between the data as T a a T 1 ) ( + = α Table 1 Values of coefficient of thermal expansion vs. temperature. F) ( T F) μin/in/ ( α340 260 180 100 20 60 2.45 3.58 4.52 5.28 5.86 6.36 Following the procedure for conducting linear regression as given in Chapter 06.03, we get T T 0096964 . 0325 . 6 ) ( + = α Let us now look at how we can evaluate the adequacy of a linear regression model. 1. Plot the data and the regression model. Figure 1 shows the data and the regression model. From a visual check, it looks like the model explains the data adequately. 06.05.2 Chapter 06.05 Figure 1 Plot of coefficient of thermal expansion vs. temperature data points and regression line. 2. Calculate the standard error of estimate . The standard error of estimate is defined as 2 / − = n S s r T α where ∑ = − − = n i i i r T a a S 1 2 1 ) ( α Table 2 Residuals for data. i T i α i T a a 1 + i i T a a 1 − − α340 260 180 100 20 60 2.45 3.58 4.52 5.28 5.86 6.36 2.7357 3.5114 4.2871 5.0629 5.8386 6.6143 0.28571 0.068571 0.23286 0.21714 0.021429 0.25429 Adequacy of Regression Model 06.05.3 Table 2 shows the residuals of the data to calculate the sum of the square of residuals as 2 2 2 2 2 2 ) 25429 . ( ) 021429 . ( ) 21714 . ( ) 23286 . ( ) 068571 . ( ) 28571 . ( − + + + + + − = r S 25283 . = The standard error of estimate 2 / − = n S s r T α 2 6 25283 . − = 25141 . = The units of T s / α are same as the units of α . How is the value of the standard error of estimate interpreted? We may say that on average the difference between the observed and predicted values is F μin/in/ 0.25141 . Also, we can look at the value as follows. About 95% of the observed α values are between T s / 2 α ± of the predicted value (see Figure 2). This would lead us to believe that the value of...
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida  Tampa.
 Spring '08
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