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The RSREG Procedure

A Response Surface with a Simple Optimum

This example uses the three-factor quadratic model discussed in John (1971). Schneider and Stockett (1963) performed an experiment aimed at reducing the unpleasant odor of a chemical produced with several factors. The objective is to minimize the unpleasant odor of a chemical. The following statements read the data.

   title 'Response Surface with a Simple Optimum';
   data smell;
      input Odor T R H @@;
      label
         T = "Temperature"
         R = "Gas-Liquid Ratio"
         H = "Packing Height";
      datalines;
    66 40 .3 4     39 120 .3 4     43 40 .7 4     49 120 .7  4
    58 40 .5 2     17 120 .5 2     -5 40 .5 6    -40 120 .5  6
    65 80 .3 2      7  80 .7 2     43 80 .3 6    -22  80 .7  6
   -31 80 .5 4    -35  80 .5 4    -26 80 .5 4
   ;

The INPUT statement names the variables contained in the SAS data set smell; the variable Odor is the response, while the variables T, R, and H are the independent factors.

The following statements invoke PROC RSREG on the data set smell. Figure 56.1 through Figure 56.3 display the results of the analysis, including a lack-of-fit test requested with the LACKFIT option.

   proc rsreg data=smell;
      model Odor = T R H / lackfit;
   run;

Response Surface with a Simple Optimum

The RSREG Procedure

Coding Coefficients for the Independent
Variables
Factor Subtracted off Divided by
T 80.000000 40.000000
R 0.500000 0.200000
H 4.000000 2.000000

Response Surface for Variable Odor
Response Mean 15.200000
Root MSE 22.478508
R-Square 0.8820
Coefficient of Variation 147.8849

Regression DF Type I Sum of Squares R-Square F Value Pr > F
Linear 3 7143.250000 0.3337 4.71 0.0641
Quadratic 3 11445 0.5346 7.55 0.0264
Crossproduct 3 293.500000 0.0137 0.19 0.8965
Total Model 9 18882 0.8820 4.15 0.0657

Residual DF Sum of Squares Mean Square F Value Pr > F
Lack of Fit 3 2485.750000 828.583333 40.75 0.0240
Pure Error 2 40.666667 20.333333    
Total Error 5 2526.416667 505.283333    

Figure 56.1: Summary Statistics and Analysis of Variance

Figure 56.1 displays the coding coefficients for the transformation of the independent variables to lie between -1 and 1, simple statistics for the response variable, hypothesis tests for linear, quadratic, and crossproduct terms, and the lack-of-fit test. The hypothesis tests can be used to gain a rough idea of importance of the effects; here the crossproduct terms are not significant. However, the lack-of-fit for the model is significant, so more complicated modeling or further experimentation with additional variables should be performed before firm statements are made concerning the underlying process.

Response Surface with a Simple Optimum

The RSREG Procedure

Parameter DF Estimate Standard Error t Value Pr > |t| Parameter Estimate
from Coded Data
Intercept 1 568.958333 134.609816 4.23 0.0083 -30.666667
T 1 -4.102083 1.489024 -2.75 0.0401 -12.125000
R 1 -1345.833333 335.220685 -4.01 0.0102 -17.000000
H 1 -22.166667 29.780489 -0.74 0.4902 -21.375000
T*T 1 0.020052 0.007311 2.74 0.0407 32.083333
R*T 1 1.031250 1.404907 0.73 0.4959 8.250000
R*R 1 1195.833333 292.454665 4.09 0.0095 47.833333
H*T 1 0.018750 0.140491 0.13 0.8990 1.500000
H*R 1 -4.375000 28.098135 -0.16 0.8824 -1.750000
H*H 1 1.520833 2.924547 0.52 0.6252 6.083333

Factor DF Sum of Squares Mean Square F Value Pr > F Label
T 4 5258.016026 1314.504006 2.60 0.1613 Temperature
R 4 11045 2761.150641 5.46 0.0454 Gas-Liquid Ratio
H 4 3813.016026 953.254006 1.89 0.2510 Packing Height

Figure 56.2: Parameter Estimates and Hypothesis Tests

Parameter estimates and the factor ANOVA are shown in Figure 56.2. Looking at the parameter estimates, you can see that the crossproduct terms are not significantly different from zero, as noted previously. The "Estimate" column contains estimates based on the raw data, and the "Parameter Estimate from Coded Data" column contains those based on the coded data. The factor ANOVA table displays tests for all four parameters corresponding to each factor -the parameters corresponding to the linear effect, the quadratic effect, and the effects of the cross products with each of the other two factors. The only factor with a significant over-all effect is R, indicating that the level of noise left unexplained by the model is still too high to estimate the effects of T and H accurately. This may be due to the lack of fit.

Response Surface with a Simple Optimum

The RSREG Procedure
Canonical Analysis of Response Surface Based on Coded Data

Factor Critical Value Label
Coded Uncoded
T 0.121913 84.876502 Temperature
R 0.199575 0.539915 Gas-Liquid Ratio
H 1.770525 7.541050 Packing Height
Predicted value at stationary point: -52.024631

Eigenvalues Eigenvectors
T R H
48.858807 0.238091 0.971116 -0.015690
31.103461 0.970696 -0.237384 0.037399
6.037732 -0.032594 0.024135 0.999177
Stationary point is a minimum.

Figure 56.3: Canonical Analysis and Eigenvectors

Figure 56.3 contains the canonical analysis and eigenvectors. The canonical analysis indicates that the directions of principle orientation for the predicted response surface are along the axes associated with the three factors, confirming the small interaction effect in the Regression ANOVA. The largest eigenvalue (48.8588) corresponds to the eigenvector {0.238091, 0.971116, -0.015690}, the largest component of which (0.971116) is associated with R; similarly, the second largest eigenvalue (31.1035) is associated with T. The third eigenvalue (6.0377), associated with H, is quite a bit smaller than the other two, indicating that the response surface is relatively insensitive to changes in this factor. The coded form of the canonical analysis indicates that the estimated response surface is at a minimum when T and R are both near the middle of their respective ranges and H is relatively high; in uncoded, terms, the model predicts that the unpleasant odor will be minimized when T = 84.876502, R = 9.539915, and H = 7.541050.

To plot the response surface with respect to two of the factor variables, first fix H, the least significant factor variable, at its estimated optimum value and generate a grid of points for T and R. To ensure that the grid data do not affect parameter estimates, the response variable (Odor) is set to missing. (See the "Missing Values" section.) The following statements produce and graph the necessary data. Initial data steps creates a grid over T and R, with H set to a constant value, and combine this grid with the original data. Then, PROC RSREG is used to create predictions for the combined data. Finally, PROC G3D is used to create a surface plot of the predictions.

   data grid;
      do;
         Odor =  .  ;
         H    = 7.541;
         do T = 20 to 140 by 5;
            do R = .1 to .9 by .05;
               output;
            end;
         end;
      end;
   data grid;
      set smell grid;
   run;

   proc rsreg data=grid out=predict noprint;
      model Odor = T R H / predict;
   run;

   data plot;
      set predict;
      if H = 7.541;
   proc g3d data=plot;
      plot T*R=Odor / rotate=38 tilt=75 xticknum=3 yticknum=3
      zmax=300 zmin=-60 ctop=red cbottom=blue caxis=black;
   run;

The first DATA step creates grid points for T and R at H=7.541 and sets Odor to missing, and the second DATA step concatenates these grid points with the original data. Predicted values are created in the SAS data set predict by invoking the RSREG procedure with the PREDICT option in the MODEL statement. The analysis is not displayed due to the NOPRINT option. The third DATA step subsets the predicted values over just the grid points (excluding the predictions at the original points). PROC G3D is then used to create the three-dimensional plot shown in Figure 56.4.

rsrg4.gif (5838 bytes)

Figure 56.4: The Response Surface Obtained from the PREDICT Option

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