6: II 7: III
5.6 Response Surface and Definitive Screening Experi- ments
5.6 Response Surface and Definitive Screening Experi-
Committee 69 recognized the importance of response surface experimentation for process improvement studies, in addition to 2kfactorial and 2k−pfractional factorial designs (Johnson and Boulanger[46]).
FIGURE 5.19: Quadratic Response Surface
In order to fit a quadratic equation to the data obtained from experi- ments, there must be at least three levels of each factor rather than the two levels contained in the 2k factorial designs, 2k−p fractional factorial designs, or alternative screening designs.
One classical experimental design used in Response Surface Methodology is the Central Composite Design. It has five levels for each factor and is composed of a 2k design, plus a center point at the center-level of each factor, plus axial or star points that are outside the 2k region.Figure 5.20shows a visual representation of the experiments for a Central Composite Design in three factors.
FIGURE 5.20: Central Composite Design
Table 5.11 is a list of the runs in a Central Composite Design in three coded and scaled factors Xi, for i = 1, . . . ,3. In this table α = √
3, and is outside the coded low-high range of−1 to 1. In generalα=√
k, where k is the number of factors in the design.
TABLE 5.11: List of Runs for Central Composite Design in 3 Factors run X1 X2 X3
1 −1 −1 −1 Factorial
2 1 −1 −1 Points
3 −1 1 −1
4 1 1 −1
5 −1 −1 1
6 1 −1 1
7 −1 1 1
8 1 1 1
9 −α 0 0 axial
10 α 0 0 points
11 0 −α 0
12 0 α 0
13 0 0 −α
14 0 0 α
15 0 0 0 center
16 0 0 0 points
17 0 0 0
18 0 0 0
19 0 0 0
Table 5.12 shows the total number of runs or experiments required for a Central Composite design that includes 5 center points. The first number 2k is the number of factorial points, the second number 2×k is the number of axial points, and the third number is the number of center points. It can be seen that even for five factors, the number of runs or experiments for a Central Composite can be excessive. One way to reduce the number of runs is to substitute a resolution V 2k−p design for the factorial portion, but even doing that, the number of runs required for a typical process improvement study with 7 or more factors can be impractical.
In order to reduce the number of runs required, the traditional approach is to experiment sequentially. Start with a resolution III or IV 2k−p design, and identify which factors are significant. Next, augment the design with follow-up experiments to obtain unconfounded estimates of the significant main effects and two-factor interactions. Finally, add center points and axial points for the significant effects in order to fit a subset of the terms in Equation 5.12.Figure 5.21outlines this strategy.
TABLE 5.12: Number of runs for Central Composite Design Number of factorsk Number of runs for Central Composite Design
2 22+4+5=13
3 23+6+5=19
4 24+8+5=29
5 25+10+5=47
6 26+12+5=81
7 27+14+5=157
FIGURE 5.21: Sequential Experimentation Strategy
In 2011, Jones and Nachtsheim[50] proposed the use of Definitive Screening Designs that reduce the number of experiments required to fit a subset of Equation 5.12 in the significant factors. Their approach essentially combines an efficient screening design for identifying important factors and interactions, with center levels for the factors that allow fitting quadratic effects. These designs can be called super-saturated in that the number of runs in the design (2×k+ 1) is less than the number of coefficients
1 + 2×k+k×(k−2 1) in Equation 5.12.
However, if there are several proposed factors in a process improvement study, it is unlikely that each factor will have significant linear, quadratic, and interaction terms with all the other factors. This is again because of the effect sparsity principle. When using a Definitive Screening Design, like an alternative screening design, an appropriate model consisting of the important
subset of terms in Equation 5.12 can usually be obtained using a forward selection procedure.
When using a Definitive Screening Design, the selection procedure should take into account the special structure of the design (Jones and Nachtsheim[52]). The special structure of Definitive Screening Designs is that all linear main effects are completely unconfounded with all quadratic and two-factor interaction terms. Quadratic and two-factor interaction terms are partially confounded like they are in alternative screening designs. Therefore, when using a Definitive Screening Design, the important main effects can be determined first, and then, assuming strong effect heredity, a forward search is made with candidates being the quadratic terms and two-factor interactions involving the important main effects.
Jones and Nachtsheim’s[50] paper listed a catalog of Definitive Screening Designs. The DefScreen() function in the R package daewrcan be used to recall a design from this catalog. The example below illustrates this.
R>library(daewr)
R>Design<-DefScreen(m=8,c=0) R>Design
A B C D E F G H
1 0 -1 1 1 -1 1 1 1 2 0 1 -1 -1 1 -1 -1 -1 3 -1 0 -1 1 1 1 1 -1 4 1 0 1 -1 -1 -1 -1 1 5 -1 -1 0 1 1 -1 -1 1 6 1 1 0 -1 -1 1 1 -1 7 1 -1 1 0 1 1 -1 -1 8 -1 1 -1 0 -1 -1 1 1 9 -1 -1 1 -1 0 -1 1 -1 10 1 1 -1 1 0 1 -1 1 11 1 -1 -1 -1 1 0 1 1 12 -1 1 1 1 -1 0 -1 -1 13 -1 1 1 -1 1 1 0 1 14 1 -1 -1 1 -1 -1 0 -1 15 1 1 1 1 1 -1 1 0 16 -1 -1 -1 -1 -1 1 -1 0
17 0 0 0 0 0 0 0 0
The argumentm=8specifies that there are 8 quantitative 3-level factors in the design. The number of runs is 2×k+ 1 = 17, whenk = 8. This design is listed in standard order since the default optionrandomize=FALSEwas not overridden in the function call. When planning a Definitive Screening Design, the option randomize=TRUE should be used to recall the runs in a random order, and thus prevent biases from changes in unknown factors.
To illustrate the analysis of data from an Definitive Screening Design, con- sider the following example. Libbrecht et. al. [65] completed an 11-run Defini- tive Screening Design for optimizing the process of synthesizing soft template mesoporus carbons. Mesoporus carbons can be used for various applications such as electrode materials, absorbents, gas storage hosts, and support mate- rial for catalysts. The effect of five synthesis factors on the material properties of the carbons were considered. Because the relationship between the mate- rial properties, or responses, and the synthesis parameters, or factors, was expected to be nonlinear, it was desirable to use an experimental design that would allow fitting a subset of the terms in Equation 5.11. The five factors studied were: A (Carbonation temperature [◦C]), B (Ratio of precursor/sur- factant), C (EISA-Time [h]), D (Curing time [h]), and E (EISA surface area [cm2]).Table 5.13shows the range of each of the five factors.
TABLE 5.13: Factors and Uncoded Low and High Levels in Mesoporus Car- bon Experiment
Factors Level (−) Level (+)
A–Carbonation temperature 600(◦C) 1000(◦C) B–Ratio of precursor to surfactant 0.7 1.3
C–EISA-Time 6(h) 24(h)
D–Curing time 12(h) 24(h)
E–EISA surface area 65(cm2) 306(cm2)
The experimental outputs or properties of the mesoporus carbon were mi- croporus and mesoporus surface area, pore volumes, and the weight % carbon.
Among the purposes of the experimentation was to understand the effects of the acid catalyzed EISA synthesis parameters on the material properties and to find the optimal set of synthesis parameters to maximize mesopore surface area.
A traditional central composite design for five factors would entail 47 ex- periments. Since it was doubtful that all terms in a Equation 5.11 fork= five factors would be significant, a Definitive Screening Design using less than one fourth that number of experiments (11) could be used.
The R code below shows the command to create the Definitive Screening Design using the DefScreen() function. The randomize=FALSE option was used for this experiment to create the design in standard order. The actual experiments were conducted in random order, but by creating the design in standard order, it was easier to pair the responses shown in the article to the experimental runs. The factor levels are in coded and scaled units. All Definitive Screening Designs in Jones and Nachtsheim’s[50] catalog as recalled by theDefScreenfunction contain one center point where all factors are set at their mid levels. In this experiment, there were three center points, including additional center points allows for a lack of fit test to be performed. The argumentcenter=2tells the function to add two additional center points.
R>library(daewr)
R>design<-DefScreen(m=5,c=0,center=2, randomize=FALSE)
R>design
A B C D E
1 0 1 1 -1 -1 2 0 -1 -1 1 1 3 1 0 -1 -1 1 4 -1 0 1 1 -1 5 1 -1 0 1 -1 6 -1 1 0 -1 1 7 1 -1 1 0 1 8 -1 1 -1 0 -1
9 1 1 1 1 0
10 -1 -1 -1 -1 0 11 0 0 0 0 0 12 0 0 0 0 0 13 0 0 0 0 0
In the design listing shown above, the three center points (runs 11, 12, and 13) can be seen to have the center level (in coded and scaled units) for each factor.
The code below shows the responses recorded from the synthesis experi- ments, and the call of the functionFitDefSc()used to fit a model to the data.
The response, Smeso, is the mesoporus surface area. The option alpha=.05 controls the number of second-order terms that will be allowed in the model.
A smaller the value ofalphais more stringent and allows fewer second-order terms to enter. A larger value ofalphais less stringent and allows more second- order terms to enter. Using theFitDefSc() values of alpha=.05within the range .05 - .20 should be used.
R>library(daewr)
R>Smeso<-c(241,295,260,338,320,265,275,248,92.5, 383,313,305,304)
R>FitDefSc(Smeso,design,alpha=.05) Call:
lm(formula = y ~ (.), data = ndesign) Residuals:
Min 1Q Median 3Q Max
-13.667 -3.653 0.000 3.550 12.373
Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 304.000 5.012 60.656 1.35e-09 ***
B -76.030 4.362 -17.429 2.29e-06 ***
D -26.609 4.533 -5.870 0.001082 **
B:D -21.023 4.779 -4.399 0.004572 **
I(B^2) -44.318 6.500 -6.819 0.000488 ***
A -42.803 4.362 -9.812 6.45e-05 ***
E -21.459 4.533 -4.734 0.003213 **
---
Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 11.21 on 6 degrees of freedom
Multiple R-squared: 0.9866, Adjusted R-squared: 0.9733 F-statistic: 73.9 on 6 and 6 DF, p-value: 2.333e-05
Sums of Squares df F-value P-value Lack of Fit 704.89318 4 7.24205 0.04059
Pure Error 48.66667 2
The lack of fit F-test shown at the bottom of the output is only produced when there is more than one center point in the design. The F-ratio compares the variability among the replicated center point responses, and the lack of fit variability. The lack of fit variability is the variability among the residuals (i.e., actual responses−fitted values) minus the variability among replicate center points. In this example, the F-test is significant. This means the difference in predictions from the fitted model and new experiments at the prediction conditions will be larger than replicate experiments at those conditions.
To improve the model,alphawas increased as shown in the code below to expand the model.
R>library(daewr)
R>FitDefSc(Smeso,design,alpha=.1) Call:
lm(formula = y ~ (.), data = ndesign) Residuals:
1 2 3 4 5 6 7
-0.39670 -2.32094 0.46480 3.15873 -0.12433 1.03021 3.54358
8 9 10 11 12 13
0.07995 -0.71346 -1.09831 4.45882 -3.54118 -4.54118
Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 308.541 2.386 129.329 2.14e-08 ***
I(D^2) -11.353 3.046 -3.727 0.020353 *
D -26.609 1.855 -14.343 0.000137 ***
B -76.030 1.785 -42.589 1.82e-06 ***
E -21.459 1.855 -11.567 0.000319 ***
B:E 11.344 2.125 5.338 0.005933 **
B:D -21.024 1.974 -10.649 0.000440 ***
I(B^2) -37.509 2.974 -12.611 0.000228 ***
A -42.803 1.785 -23.976 1.79e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 4.586 on 4 degrees of freedom
Multiple R-squared: 0.9985,Adjusted R-squared: 0.9955 F-statistic: 334.9 on 8 and 4 DF, p-value: 2.216e-05
Sums of Squares df F-value P-value Lack of Fit 35.46992 2 0.72883 0.57842
Pure Error 48.66667 2
In the output it can be seen that two additional terms ((I(D^2), and B:E)) were added to the model and the lack of fit test is no longer significant.
This means that predictions from this fitted model will be just as accurate as running additional experiments at the factor levels where the predictions are made.
Sometimes theFitDefSc()may include some terms in the model that are not significant at theα=.05 level of significance. If that is the case, thelm() function can be used to refit the model eliminating the insignificant term with the largest p-value, as shown in the example in Section 5.5.1. This can be repeated until all remaining terms in the model are significant. After using thelm()function to fit a model, the assumptions of the least-squares fit can be checked by looking at the diagnostic plots produced by the plot.lm() functions as shown in the block of R code beforeFigure 5.7
Predictions from the fitted model can be explored graphically to identify the optimal set of synthesis parameters to maximize mesopore surface area, Smeso. Figure 5.22 shows contour plots of the predicted mesoporus surface area as a function of curing time and ratio of precursor/surfactant. The plot on the left is made when EISA surface area is at its low level of 65, and the plot on right is when EISA surface area is at its high level of 306. The code for producing these plots is in the R code forChapter 5.
FIGURE 5.22: Contour Plots of Fitted Surface
It can be seen that the maximum predicted mesopore surface area is just over 400(m2/g) when carbonation temperature = 600, Ratio ≈ .80, EISA time = 15, curing time ≈ 15, and EISA surface area = 65. These conclu- sions were reached after conducting only 13 experiments, far less than would be required using the traditional approach illustrated inFigure 5.21. The R function ConstrOptim() can also be used to get the optimal combination synthesis parameters numerically. For an example see Lawson[59].