Myers (1976) analyzes an experiment reported by Frankel (1961) aimed at maximizing the yield of mercaptobenzothiazole (MBT) by varying processing time and temperature. Myers (1976) uses a twofactor model in which the estimated surface does not have a unique optimum. A ridge analysis is used to determine the region in which the optimum lies. The objective is to find the settings of time and temperature in the processing of a chemical that maximize the yield. The following statements produce Output 81.1.1 through Output 81.1.6:
data d; input Time Temp MBT; label Time = "Reaction Time (Hours)" Temp = "Temperature (Degrees Centigrade)" MBT = "Percent Yield Mercaptobenzothiazole"; datalines; 4.0 250 83.8 20.0 250 81.7 12.0 250 82.4 12.0 250 82.9 12.0 220 84.7 12.0 280 57.9 12.0 250 81.2 6.3 229 81.3 6.3 271 83.1 17.7 229 85.3 17.7 271 72.7 4.0 250 82.0 ;
ods graphics on; proc rsreg data=d plots=(ridge surface); model MBT=Time Temp / lackfit; ridge max; run; ods graphics off;
Output 81.1.1 displays the coding coefficients for the transformation of the independent variables to lie between –1 and 1 and some simple statistics for the response variable.
Output 81.1.1: Coding and Response Variable Information
Coding Coefficients for the Independent Variables 


Factor  Subtracted off  Divided by 
Time  12.000000  8.000000 
Temp  250.000000  30.000000 
Response Surface for Variable MBT: Percent Yield Mercaptobenzothiazole 


Response Mean  79.916667 
Root MSE  4.615964 
RSquare  0.8003 
Coefficient of Variation  5.7760 
Output 81.1.2 shows that the lack of fit for the model is highly significant. Since the quadratic model does not fit the data very well,
firm statements about the underlying process should not be based only on the current analysis. Note from the analysis of variance
for the model that the test for the time factor is not significant. If further experimentation is undertaken, it might be
best to fix Time
at a moderate to high value and to concentrate on the effect of temperature. In the actual experiment discussed here, extra
runs were made that confirmed the results of the following analysis.
Output 81.1.2: Analyses of Variance
Regression  DF  Type I Sum of Squares  RSquare  F Value  Pr > F 

Linear  2  313.585803  0.4899  7.36  0.0243 
Quadratic  2  146.768144  0.2293  3.44  0.1009 
Crossproduct  1  51.840000  0.0810  2.43  0.1698 
Total Model  5  512.193947  0.8003  4.81  0.0410 
Residual  DF  Sum of Squares  Mean Square  F Value  Pr > F 

Lack of Fit  3  124.696053  41.565351  39.63  0.0065 
Pure Error  3  3.146667  1.048889  
Total Error  6  127.842720  21.307120 
Parameter  DF  Estimate  Standard Error  t Value  Pr > t  Parameter Estimate from Coded Data 

Intercept  1  545.867976  277.145373  1.97  0.0964  82.173110 
Time  1  6.872863  5.004928  1.37  0.2188  1.014287 
Temp  1  4.989743  2.165839  2.30  0.0608  8.676768 
Time*Time  1  0.021631  0.056784  0.38  0.7164  1.384394 
Temp*Time  1  0.030075  0.019281  1.56  0.1698  7.218045 
Temp*Temp  1  0.009836  0.004304  2.29  0.0623  8.852519 
Factor  DF  Sum of Squares  Mean Square  F Value  Pr > F  Label 

Time  3  61.290957  20.430319  0.96  0.4704  Reaction Time (Hours) 
Temp  3  461.250925  153.750308  7.22  0.0205  Temperature (Degrees Centigrade) 
The canonical analysis (Output 81.1.3) indicates that the predicted response surface is shaped like a saddle. The eigenvalue of 2.5 shows that the valley orientation
of the saddle is less curved than the hill orientation, with an eigenvalue of –9.99. The coefficients of the associated eigenvectors
show that the valley is more aligned with Time
and the hill with Temp
. Because the canonical analysis resulted in a saddle point, the estimated surface does not have a unique optimum.
Output 81.1.3: Canonical Analysis
Factor  Critical Value  Label  

Coded  Uncoded  
Time  0.441758  8.465935  Reaction Time (Hours) 
Temp  0.309976  240.700718  Temperature (Degrees Centigrade) 
Predicted value at stationary point: 83.741940 
Eigenvalues  Eigenvectors  

Time  Temp  
2.528816  0.953223  0.302267 
9.996940  0.302267  0.953223 
Stationary point is a saddle point. 
However, the ridge analysis in Output 81.1.4 and the ridge plot in Output 81.1.5 indicate that maximum yields result from relatively high reaction times and low temperatures. A contour plot of the predicted response surface, shown in Output 81.1.6, confirms this conclusion.
Output 81.1.4: Ridge Analysis
Estimated Ridge of Maximum Response for Variable MBT: Percent Yield Mercaptobenzothiazole  

Coded Radius  Estimated Response  Standard Error  Uncoded Factor Values  
Time  Temp  
0.0  82.173110  2.665023  12.000000  250.000000 
0.1  82.952909  2.648671  11.964493  247.002956 
0.2  83.558260  2.602270  12.142790  244.023941 
0.3  84.037098  2.533296  12.704153  241.396084 
0.4  84.470454  2.457836  13.517555  239.435227 
0.5  84.914099  2.404616  14.370977  237.919138 
0.6  85.390012  2.410981  15.212247  236.624811 
0.7  85.906767  2.516619  16.037822  235.449230 
0.8  86.468277  2.752355  16.850813  234.344204 
0.9  87.076587  3.130961  17.654321  233.284652 
1.0  87.732874  3.648568  18.450682  232.256238 
Output 81.1.5: Ridge Plot of Predicted Response Surface
Output 81.1.6: Contour Plot of Predicted Response Surface