Centered experimental designs

A two levels of full factorial experimental design with three independent variables were generated with one center point, which was repeated[3]. In this design, F/P molar ratio, Oh/P wt%, and reaction temperature were defined as independent variables, all receiving two values, a high and a low value. A cube like model was formed, with eight comers. One center point (repeated twice) was added to improve accuracy of the design. Every analysis results were treated as a dependent result in the statistical study. 

Considerable discussion of reparameterization and examples of its usefulness have been published (B3, B8, B12, Gl, G2, M7). Although several specific techniques are useful, one reparametrization of kinetic models often necessary is a redefinition of the independent variables so that the center of the new coordinate system is near that of the experimental design. In particular, the exponential parameter... 

First of all, such approach would require too many experiments when the number of factors becomes too large. To vary factors independently between —1 and -Fl with the OVAT approach, at least 2/+1 experiments are required. Eor example, to vary 10 factors with the OVAT approach, at least 21 experiments are needed, i.e., 20 experiments with one factor varying (once at —1 and once at -Fl) and 1 experiment with all factors at nominal level (center point). With an experimental design approach, on the other hand, these 10 factors can be examined in 12 experiments. 

It is to be stressed, however, that the geometric interpretation of the parameter estimates obtained using coded factor levels is usually different from the interpretation of those parameter estimates obtained using uncoded factor levels. As an illustration, pj (the intercept in the coded system) represents the response at the center of the experimental design, whereas Pq (the intercept in the uncoded system) represents the response at the origin of the original coordinate system the two estimates (Pq and Pq) are usually quite different numerically. This difficulty will not be important in the remainder of this book, and we will feel equally free to use either coded or uncoded factor levels as the examples require. Later, in Section 11.5, we will show how to translate coded parameter estimates back into uncoded parameter estimates. 

In this as in all other problems of experimental design, prior information is helpful. For example, if the enzyme we are dealing with is naturally found in a neutral environment, then it would probably be most active at a neutral pH, somewhere near pH = 7. If it were found in an acidic environment, say in the stomach, it would be expected to exhibit its optimal activity at a low (acidic) pH. When information such as this is available, it is appropriate to center the experimental design about the best guess of where the desired region might be. In the absence of prior information, factor combinations might be centered about the midpoint of the factor domain. 

Given these five levels of pH, how can the five replicate experiments be allocated One way is to place all of the replicates at the center factor level. Doing so would give a good estimate of at the center of the experimental design, but it would give... 

Let us try instead an experimental design in which two replicates are carried out at the center point and three replicates are carried out at each of the extreme points (-2 and +2). Then... 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is... 

The rotatable central composite design in Figure 13.7 is related to the rotatable central composite design in Figure 13.3 through expansion by a factor of V2 the square points expand from 2 to 2 2 from the center the star points expand from 2 2 to 4 from the center. The experimental design matrix is... 

Replicates don t all have to be carried out at the center point. Using the experimental design of Figure 13.2 as a basis. Figures 13.9 and 13.10 show the effects of different distributions of replicates. 

In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. The experimental design matrix is... 

Figure 12.4 The spatial representations of four different types of experimental designs that are useful for process analyzer calibration (A) full-factorial, (B) Box-Behnken, (C) face-centered cube, and (D) central composite.

For theexample discussed here, the calibration sets for classes A and B are selected gs hically, and for class C are selected as the extremes and centers of each ofdie three levels in the experimental design. The selection results in 15 samples in each of the calibration sets and 12 in each of the validation sets. A score pS>t of all samples in class A is shown in Figure 4.69 with the calibration set samples indicated by X and the validation samples indicated by O. Similarly, SCO plots of clas.es B and C with calibration and validation samples identifiedsre shown in Figures 4.70 and 4.71, respectively. 

The complete design is seen in the score space with replicate center points clearly visible. Note that the interpretation of scores plots is not always as straightforward as in this example. The experimental design is not seen if the experiment is not well designed or if the problem is high dimensional. The level of impEcidy modeled components (e.g., component O also has an effect on the relative position of the samples in score space. For this example, the effect of C on the relative placement of the samples in score space is small. 

Scores Plot (Sample Diagnostic) Figure 5-132 displays the Factor 2 versus Factor l scores for the MCB model (showing 98.41% of the spectral variance). The experimental design for this data set is not readily discernible because this plot shows only two dimensions. However, samples 1-4 define the extremes, which makes sense because these are the pure spectra. As expected, the center-point replicates lie very near each other and are in the middle of the plot. 

Figure 7 The first four loading vectors of the NIR PLS model (MSC and mean centering) generated on the NIR data collected on the 10% APAP surrogate tablets prepared according to the Latin square experimental design.

The LFMM FF for the oxidized Cu(II) centers was designed around suitable homoleptic species, viz., [Cu(imidazole)4]2+, [Cu(SCH3)4]2-, [Cu(S(CH3)2)4]2+, and [Cu(0=CH2)4]2+ (37). These complexes represent models for Cu-histidine, Cu-cysteine, Cu-methionine, and Cu-glutamine O/peptide respectively. Only the first of these species is known experimentally. However, it is amply documented that DFT gives excellent structures for metal complexes (64,65) so we can access the remaining species computationally (Fig. 20). 

---