Unit – 3 (Response Surface Methodology) 
                    3.1  Introduction:  Response  surface  methodology  (RSM)  is  a  collection  of  mathematical  and 
                    statistical techniques useful for the modeling and analysis of problems in which a response of interest 
                    is  influenced  by  several  variables  and  the  objective  is  to  optimize  this  response.  For  example, 
                    suppose that a chemical engineer wishes to find the levels of temperature (x ) and pressure (x ) that 
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                    maximize the yield (y) of a process. The process yield is a function of the levels of temperature and 
                    pressure, say  
                           Y = f(x , x ) + error 
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                    where error represents the noise or error observed in the response y. If we denote the expected 
                    response by E(y) = f(x , x ) = ɳ, then the surface represented by 
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                           ɳ = f(x , x ) 
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                    is called a response surface. 
                    We usually represent the response surface graphically, such as in Figure 1, where ɳ is plotted versus 
                    the levels of x  and x . We have seen such response surface plots before, particularly in factorial 
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                    designs.  To  help  visualize  the  shape  of  a  response  surface,  we  often  plot  the  contours  of  the 
                    response surface as shown in Figure 2. In the contour plot, lines of constant response are drawn in 
                    the x , x  plane. Each contour corresponds to a particular height of the response surface. We have 
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                    also previously seen the utility of contour plots. 
                           Figure 1.                                     Figure 2. A contour plot of a response surface 
                                                                                                               
                    In  most  RSM problems, the form of the relationship between the response and the independent 
                    variables is unknown. Thus, the first step in RSM is to find a suitable approximation for the true 
                    functional  relationship  between  y  and  the  set  of  independent  variables.  Usually,  a  low-order  
                    polynomial  in  some  region  of  the  independent  variables  is  employed.  If the response is well 
                    modeled by a linear function of the independent variables, then the approximating function is the first-
                    order model 
                                                                                    
                    If there is curvature in the system, then a polynomial of higher degree must be used, such as the 
                    second-order model 
                                                                                       
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                         Designs for First-order Model                Designs for Second-order Model 
                             1.  2k full or fractional factorial design   1.  3k full or fractional factorial design 
                             2.  Plackett Burman design                   2.  Box-Behnken Design (BBD) 
                             3.  Simplex design                           3.  Central Composite Design (CCD 
                     3.2 Advantages of Regression Method: Regression methods are extremely useful when something 
                     “goes wrong” in a designed experiment. This is illustrated in the next two examples. 
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                     Example 1: A 2  Factorial Design with a Missing Observation  
                     A chemical engineer is investigating the yield of a process. Three process variables are of interest: 
                     temperature, pressure, and catalyst concentration. Each variable can be run at a low and a high 
                     level,  and  the  engineer  decides  to  run  a  23  design  with  four  center  points.  The  design  and  the 
                     resulting yields are shown in Figure 3, where we have shown both the natural levels of the design 
                     factor and the  1, +1 coded variable notation normally employed in 2k factorial designs to represent 
                     the factor levels. 
                      
                                                                 Figure 3                                          
                     The fitted regression model is  
                                                                                        
                     Suppose that when this experiment was performed, the run 8 (run with all variables at the high 
                     level) in Figure 3 was missing. This can happen for a variety of reasons; the measurement system can 
                     produce a faulty reading, the combination of factor levels may prove infeasible, the experimental 
                     unit may be damaged, and so forth. Therefore, the fitted model (using the remaining 11 runs) is 
                                                                                      
                     Compare this model to the one obtained where all 12 observations were used. The regression 
                     coefficients are very similar. Because the regression coefficients are closely related to the factor 
                     effects, our conclusions would not be seriously affected by the missing observation. However, note 
                     that the design with missing value is no more orthogonal for effect estimators. Furthermore, the 
                     variances of the regression coefficients are larger than they were in the original orthogonal design 
                     with no missing data. 
                      
                     Example 2. Inaccurate Levels in Design Factors 
                     When running a designed experiment, it is sometimes difficult to reach and hold the precise factor 
                     levels required by the design. Small discrepancies are not important, but large ones are potentially of 
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                    more concern. Regression methods are useful in the analysis of a designed experiment where the 
                    experimenter has been unable to obtain the required factor levels.  
                    To illustrate, the experiment presented in Table 1 shows a variation of the 23 design from Example 1, 
                    where many of the test combinations are not exactly the ones specified in the design. Most of the 
                    difficulty seems to have occurred with the temperature variable. 
                                        Table 1. Experimental design for the problem in Example 1 
                                                                                                          
                    The fitted regression model, with the coefficients reported to two decimal places, is 
                                                                                  
                    Comparing  this  to  the  original  model  in  Example 1,  where  the  factor  levels  were  exactly  those 
                    specified by the design, we note very little difference. The practical interpretation of the results of 
                    this experiment would not be seriously affected by the inability of the experimenter to achieve the 
                    desired factor levels exactly. 
                     
                    Example 3. De-aliasing Interactions in a Fractional Factorial (not discussed here) 
                     
                     
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                    3.3 Meaning of R  & Adjusted-R : The Table 1 reports the coefficient of multiple determination R , 
                    where 
                                                                            
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                    In designed experiments, R is a measure of the amount of reduction in the variability of y obtained by 
                    using the regressor variables x , x , . . . , x  in the model. However, it should be noted that a large 
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                    value of R does not necessarily imply that the regression model is a good one. Adding a variable to 
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                    the  model  will  always  increase  R ,  regardless  of  whether  the  additional  variable  is  statistically 
                    significant or not. Thus, it is possible for models that have large values of R2 to yield poor predictions 
                    of new observations or estimates of the mean response. 
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                    Because R always increases as we add terms to the model, some regression model builders prefer 
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                    to use an adjusted-R defined as 
                                                                                        
                            where (n – 1) and (n – p) are the total degree of freedoms (if n is the total number of runs) 
                            and the degree of freedom associated with error, respectively.  
                     
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                                   In general, the adjusted-R statistic will not always increase as variables are added to the model. In 
                                   fact, if unnecessary terms are added, the value of R2  will often decrease. For example, consider the 
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                                   viscosity  regression model.  The R  and the adjusted-R  for the model is shown in Table 1. It is 
                                   computed as 
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                                                R           = 1 – (3479 / 47636) = 0.927 or 92.7% 
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                                                R  = {1 – (15 / 13) } (1 – 0.927 ) = 0.916 or 91.6% 
                                                    adj
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                                   which is very close to the ordinary R . When R and R adj differ dramatically, there is a good chance 
                                   that non-significant terms have been included in the model. 
                                                                      Table 1. Minitab output for the viscosity regression model 
                                                                                                                                                                                     
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                                   PRESS (PRedicted Error Sum of Square) value and R                                                     : To calculate PRESS, we select an 
                                                                                                                              Prediction
                                   observation—for  example,  i.  Fitting  the  regression  model  to  the  observation  number  i  and  thus 
                                   denoting the predicted value as ŷ. We may find the prediction error for observation i as e  = y – ŷ. 
                                                                                        i                                                                                        (i)     i      i
                                   Similarly,  the  prediction  error  is  calculated  for  all  the  remaining  observations.  Then  the  PRESS 
                                   statistic is defined as  
                                                                                                        
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                                   Finally, we note that PRESS can be used to compute an approximate R  for prediction, say 
                                                                                    
                                   This  statistic  gives  some  indication  of  the  predictive  capability  of  the  regression  model.  For  the 
                                   viscosity regression model in Table 1, the value of the computed PRESS residuals is 5207.7. Then 
                                                                                                    
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