Example 56.1: A Saddle-Surface Response Using Ridge Analysis
Frankel (1961) reports an experiment aimed at maximizing
the yield of mercaptobenzothiazole (MBT) by varying
processing time and temperature.
Myers (1976) uses a two-factor 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 read the data and invoke PROC RSREG.
These statements produce Output 56.1.1 through Output 56.1.5:
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
;
proc sort;
by Time Temp;
run;
proc rsreg;
model MBT=Time Temp / lackfit;
ridge max;
run;
Output 56.1.1: 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 |
| R-Square |
0.8003 |
| Coefficient of Variation |
5.7760 |
|
Output 56.1.2: Analyses of Variance
|
| Regression |
DF |
Type I Sum of Squares |
R-Square |
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) |
|
Output 56.1.2 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 56.1.3: Canonical Analysis
|
| The RSREG Procedure |
| Canonical Analysis of Response Surface Based on Coded Data |
| 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. |
|
The canonical analysis (Output 56.1.3) 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 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 56.1.4: 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 |
|
However, the ridge analysis in Output 56.1.4 indicates that
maximum yields will result from relatively
high reaction times and low temperatures. A contour plot of the
predicted response surface, shown in Output 56.1.5, confirms
this conclusion.
Output 56.1.5: Contour Plot of Predicted Response Surface
The statements that produce this plot follow. Note that contour and
three-dimensional plots can be created interactively using SAS/INSIGHT
software or the ADX menu system in SAS/QC software. Initial DATA
steps create a grid over Time and Temp and combine this grid
with the
original data, using a variable flag to indicate the grid. Then,
PROC RSREG is used to create predictions for the combined data. Finally, a
series of statements subsets the predictions over just the grid and
uses PROC GCONTOUR to display a contour plot. An ANNOTATE data set
adds level values to the contours.
data b;
set d;
flag=1;
MBT=.;
do Time=0 to 20 by 1;
do Temp=220 to 280 by 5;
output;
end;
end;
data c;
set d b;
run;
proc rsreg data=c out=e noprint;
model MBT=Time Temp / predict;
id flag;
run;
data f;
set e;
if flag=1;
data annote;
length function color style $8 text $8;
retain hsys ysys xsys '2' size 1 function 'label'
color 'black' style 'swissl' position '5';
x=255; y=10 ; text='80.3'; output;
x=245; y=11 ; text='82.9'; output;
x=227; y= 7 ; text='80.3'; output;
x=235; y= 8 ; text='82.9'; output;
x=235; y=14.5; text='85.5'; output;
x=230; y=18 ; text='88.1'; output;
x=250; y= 3 ; text='85.5'; output;
run;
axis1 label=(angle=90) minor=none;
axis2 order=(220 to 280 by 20) minor=none;
proc gcontour data=f annotate=annote;
plot Time*Temp=MBT
/ nlevels=12 vaxis=axis1 haxis=axis2 nolegend
llevels=2 2 2 1 1 1 1 1 1 1 1 1 ;
run;
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
