ࡱ> vxu{ ,:bjbjzz ;0m*****>>>8vR>V.&(&&&'''Շׇׇׇׇׇׇ$66*'''''**&&4p,p,p,'^*&*&Շp,'Շp,p,{9&dG^D(^&0V- l+l99l*Mt''p,'''''+'''V''''l''''''''' :  Chapter 06.05 Adequacy of Models for Regression After reading this chapter, you should be able to determine if a linear regression model is adequate determine how well the linear regression model predicts the response variable. Quality of Fitted Model In the application of regression models, one objective is to obtain an equation  EMBED Equation.3  that best describes the  EMBED Equation.3  response data points  EMBED Equation.3 . Consequently, we are faced with answering two basic questions. Does the model  EMBED Equation.3  describe the data adequately, that is, is there an adequate fit? 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  EMBED Equation.3  Table 1 Values of coefficient of thermal expansion vs. temperature.  EMBED Equation.3  EMBED Equation.3 -340 -260 -180 -100 -20 602.45 3.58 4.52 5.28 5.86 6.36Following the procedure for conducting linear regression as given in Chapter 06.03, we get  EMBED Equation.3  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.  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  EMBED Equation.3  where  EMBED Equation.3  Table 2 Residuals for data.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 -340 -260 -180 -100 -20 602.45 3.58 4.52 5.28 5.86 6.362.7357 3.5114 4.2871 5.0629 5.8386 6.6143-0.28571 0.068571 0.23286 0.21714 0.021429 -0.25429 Table 2 shows the residuals of the data to calculate the sum of the square of residuals as  EMBED Equation.3   EMBED Equation.3  The standard error of estimate  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The units of  EMBED Equation.3  are same as the units of  EMBED Equation.3 . 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  EMBED Equation.3 . Also, we can look at the value as follows. About 95% of the observed  EMBED Equation.3  values are between  EMBED Equation.3  of the predicted value (see Figure 2). This would lead us to believe that the value of  EMBED Equation.3  in the example is expected to be accurate within  EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3 .  Figure 2 Plotting the linear regression line and showing the regression standard error. One can also look at this criterion as finding if 95% of the scaled residuals for the model are in the domain [-2,2], that is Scaled residual EMBED Equation.3  For the example,  EMBED Equation.3  Table 4 Residuals and scaled residuals for data.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Scaled Residuals-340 -260 -180 -100 -20 602.45 3.58 4.52 5.28 5.86 6.36-0.28571 0.068571 0.23286 0.21714 0.021429 -0.25429-1.1364 0.27275 0.92622 0.86369 0.085235 -1.0115and the scaled residuals are calculated in Table 4. All the scaled residuals are in the [-2,2] domain. 3. Calculate the coefficient of determination. Denoted by EMBED Equation.3 , the coefficient of determination is another criterion to use for checking the adequacy of the model. To answer the above questions, let us start from the examination of some measures of discrepancies between the whole data and some key central tendency. Look at the two equations given below.  EMBED Equation.3  (1)  EMBED Equation.3  (2) where  EMBED Equation.3  For the example data  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Table 5 Difference between observed and average value.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 -340 -260 -180 -100 -20 602.45 3.58 4.52 5.28 5.86 6.36-2.2250 -1.0950 -0.15500 0.60500 1.1850 1.6850where  EMBED Equation.3  is the sum of the square of the residuals (residual is the difference between the observed value and the predicted value), and  EMBED Equation.3  is the sum of the square of the difference between the observed value and the average value. What inferences can we make about the two equations? Equation (2) measures the discrepancy between the data and the mean. Recall that the mean of the data is a measure of a single point that measures the central tendency of the whole data. Equation (2) contrasts with Equation (1) as Equation (1) measures the discrepancy between the vertical distance of the point from the regression line (another measure of central tendency). This line obtained by the least squares method gives the best estimate of a line with least sum of deviation.  EMBED Equation.3  as calculated quantifies the spread around the regression line. The objective of least squares method is to obtain a compact equation that best describes all the data points. The mean can also be used to describe all the data points. The magnitude of the sum of squares of deviation from the mean or from the least squares line is therefore a good indicator of how well the mean or least squares characterizes the whole data. We can liken the sum of squares deviation around the mean,  EMBED Equation.3  as the error or variability in  EMBED Equation.3  without considering the regression variable  EMBED Equation.3 , while  EMBED Equation.3 , the sum of squares deviation around the least square regression line is error or variability in  EMBED Equation.3  remaining after the dependent variable EMBED Equation.3  has been considered. The difference between these two parameters measures the error due to describing or characterizing the data in one form instead of the other. A relative comparison of this difference EMBED Equation.3 , with the sum of squares deviation associated with the mean  EMBED Equation.3  describes a quantity called coefficient of determination,  EMBED Equation.3   EMBED Equation.3  (5)  EMBED Equation.3   EMBED Equation.3  Based on the value obtained above, we can claim that 97.7% of the original uncertainty in the value of  EMBED Equation.3  can be explained by the straight-line regression model of  EMBED Equation.3 . Going back to the definition of the coefficient of determination, one can see that  EMBED Equation.3  is the variation without any relationship of  EMBED Equation.3  vs.  EMBED Equation.3 , while  EMBED Equation.3  is the variation with the straight-line relationship. The limits of the values of  EMBED Equation.3  are between 0 and 1. What do these limiting values of  EMBED Equation.3  mean? If  EMBED Equation.3 , then  EMBED Equation.3 , which means that regressing the data to a straight line does nothing to explain the data any further. If  EMBED Equation.3 , then  EMBED Equation.3 , which means that the straight line is passing through all the data points and is a perfect fit. Caution in the use of  EMBED Equation.3  The coefficient of determination,  EMBED Equation.3  can be made larger (assumes no collinear points) by adding more terms to the model. For instance,  EMBED Equation.3  terms in a regression equation for which  EMBED Equation.3  data points are used will give an  EMBED Equation.3  value of 1 if there are no collinear points. The magnitude of  EMBED Equation.3  also depends on the range of variability of the regressor  EMBED Equation.3  variable. Increase in the spread of  EMBED Equation.3  increases  EMBED Equation.3  while a decrease in the spread of  EMBED Equation.3  decreases EMBED Equation.3 . Large regression slope will also yield artificially high  EMBED Equation.3 . The coefficient of determination,  EMBED Equation.3  does not measure the appropriateness of the linear model.  EMBED Equation.3  may be large for nonlinearly related  EMBED Equation.3  and  EMBED Equation.3 values. Large coefficient of determination  EMBED Equation.3  value does not necessarily imply the regression will predict accurately. The coefficient of determination , EMBED Equation.3 does not measure the magnitude of the regression slope. These statements above imply that one should not choose a regression model solely based on consideration of  EMBED Equation.3 . 4. Find if the model meets the assumptions of random errors. These assumptions include that the residuals are negative as well as positive to give a mean of zero, the variation of the residuals as a function of the independent variable is random, the residuals follow a normal distribution, and that there is no auto correlation between the data points. To illustrate this better, we have an extended data set for the example that we took. Instead of 6 data points, this set has 22 data points (Table 6). Drawing conclusions from small or large data sets for checking assumption of random error is not recommended. Table 6 Instantaneous thermal expansion coefficient as a function of temperature. TemperatureInstantaneous Thermal Expansion EMBED Equation.3  EMBED Equation.3 806.47606.36406.24206.1206.00-205.86-405.72-605.58-805.43-1005.28-1205.09-1404.91-1604.72-1804.52-2004.30-2204.08-2403.83-2603.58-2803.33-3003.07-3202.76-3402.45 Figure 3 Plot of thermal expansion coefficient vs. temperature data points and regression line for more data points. Regressing the data from Table 2 to the straight line regression line  EMBED Equation.3  and following the procedure for conducting linear regression as given in Chapter 06.03, we get (Figure 3)  EMBED Equation.3   Figure 4 Plot of residuals. Figure 4 shows the residuals for the example as a function of temperature. Although the residuals seem to average to zero, but within a range, they do not exhibit this zero mean. For an initial value of  EMBED Equation.3 , the averages are below zero. For the middle values of  EMBED Equation.3 , the averages are below zero, and again for the final values of  EMBED Equation.3 , the averages are below zero. This may be considered a violation of the model assumption. Figure 4 also shows the residuals for the example are following a nonlinear variance. This is a clear violation of the model assumption of constant variance.  Figure 5 Histogram of residuals. Figure 5 shows the histogram of the residuals. Clearly, the histogram is not showing a normal distribution, and hence violates the model assumption of normality. To check that there is no autocorrelation between observed values, the following rule of thumb can be used. If  EMBED Equation.3  is the number of data points, and  EMBED Equation.3  is the number of times the sign of the residual changes, then if  EMBED Equation.3 , you most likely do not have an autocorrelation. For the example,  EMBED Equation.3 , then  EMBED Equation.3   EMBED Equation.3  is not satisfied as  EMBED Equation.3 . So this model assumption is violated. References ADEQUACY OF REGRESSION MODELSTopicAdequacy of Regression ModelsSummaryTextbook notes of Adequacy of Regression ModelsMajorGeneral EngineeringAuthorsAutar Kaw, Egwu KaluDate DATE \@ "MMMM d, yyyy" \* MERGEFORMAT May 31, 2013Web Site HYPERLINK "http://numericalmethods.eng.usf.edu" http://numericalmethods.eng.usf.edu     06.05. PAGE 2 Chapter 06.05 Adequacy of Regression Model 06.05. 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