The central composite design

The upper left panel shows a surface of normalized uncertainty (defined in Equation 13.7 above) as a function of factors x and x. The normalized uncertainty is relatively small in the center (approximately 1.1) and relatively large at the comers (approximately 4.0). Note that this surface generally reflects the underlying design the uncertainty surface is relatively low in those regions where experiments have been carried out and is relatively high in those regions where experiments have not been carried out. 

It is important to note that the normalized uncertainty surface shown in the upper left panel is not the response surface generated by the FSOP model itself. Instead, this upper left panel is a measure of how much the response surface might flap around in different regions of the factor space. Experiments serve to anchor the underlying model, to pin it to the data, and thereby reduce the amount of uncertainty in the model at those points. The large amount of uncertainty at the comers of this upper left panel is a reflection of the freedom the model has to move up and down in those regions where experiments have not been performed. 

The lower right panel plots normalized information as a function of factor x, for Xj = -5, -4, -3, -2, -1, and 0. These lines show the left front edge (xj = -5) and parallel slices through the normalized information surface in the panel above. (For this design which is symmetric about the x, axis, the graph lines for Xj = 1, 2, 3, 4, and 5 are identical to lines that are already present.) 

One of the striking features of this central composite design is the flatness of the normalized uncertainty and normalized information surfaces near the center of the design. 

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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... 

A class of augmented designs, first proposed by Box and Wilson [6] and frequently applied in response surface work, is the central composite design. Composite designs consist of ... 

The use of the terms cube, star and center points is descriptive of the design pattern, as is clear when there are p = 3 variables. In that case the points of the central composite design, shown in Table 2.5, can be represented by the points in Figure 2.3. In Table 2.5, runs 1-8 are the cube portion, runs 9-14 are the star portion, and runs 15-17 are the center points. 

A possible disadvantage of the central composite design is that it requires five levels of each variable (0, 1, a). In some situations it might be necessary or preferable to have only three different settings of the variables. In this case a can be chosen to be 1 and the design is called a face-centered composite design. These designs are not rotatable. 

Some second-order designs, such as the uniform shell designs (Doehlert [28]), have been proposed which are not based on the central composite design. A more thorough treatment of additional second-order designs can be found in the texts mentioned earlier see Myers [11], Box and Draper [12], Khuri and Cornell [13]. 

To construct the central composite design to estimate the coefficients of the second-order model (equation (14)), usually a fractional factorial design of at least resolution V is used. In this case, if the model is valid, then all of the estimates of the main effect coefficients, p., and the interaction coefficients, p. are imbiased. An alternative to the central composite designs for estimating the coefficients of the second-order model are the Box-Behnken designs or the designs referenced in Section 2.2.5. 

Now if each of the design points in the central composite design is replicated five times, so that the complete design has 75 runs, then at each design point we can calculate the average response and the standard deviation of the response. The analysis techniques associated with response surface methodology can then be applied to fit separate models to... 

Box-Behnken provide efficient solutions for some k values compared with the central composite design for example, a design for A =7 with three levels uses 66 experiments compared to 92 for a similar central composite design. 

The Central Composite Designs Similar to the factorial designs, but with lines radiating from the centre of the figure, perpendicular to the faces and terminating outside. 

Many modifications of the central composite design are possible and may be used to fit specific situations. For example, one of the variables may only have two possible levels. In this case, the portion of the star pattern for this variable would be omitted and the star pattern for the other variables might be run at a selected level of the variable in question. In other cases, the outrigger points may be meaningless for a particular variable, and only the factorial and centre points used. 

The rotatable feature of the central composite designs makes it possible to complete a balanced portion of the design, evaluate the results and possibly shift the design to another area in some of the variables. The shift in area of interest pivots on some of the runs already obtained and these become part of the new experimental design. 

The central composite designs are very handy if it is necessary to run an experiment in several different blocks. There may be many reasons for doing this. For example, we may have only enough raw material to run a limited number of experiments Or a test method may be such that we can run a given number of samples at a time ... 

A fully central comp osite design was applied to optimize monolaurin synthesis. A three factorial design was proven effective to establish the influence of the variables on the monolaurin synthesis. The central composite design procedure was adopted to optimize variables affecting the monolaurin molar fraction. 

In the second optimization step, the exact values of the three variables that were identified to have significant effects on nisin and/or lactic acid production were determined using a central composite design (Table 2). The coded and actual values of each variable are given in Table 3. The fermentation media (pH 6.5) were composed of 50 g/L of whey, 5 g/L of polypeptone, 1 g/L of Tween-80, and 30 g/L of CaC03, and the predetermined amount of the three variables was assigned by the central composite design. The content of nisin and lactic acid after 24 h of fermentation at 30°C was measured and are presented as responses in Table 2. 

There is an important class of experimental design that largely avoids these kinds of problems and offers the experimenter the possibility of using the same data in the context of two different models. An example of these types of designs is the central composite design mentioned in Section 8.4.3. They have many useful features, but like all other symmetrical designs, we must perform all of the experiments in the list. If we fail to perform just one of them, the design will lose its desirable properties. 

The central composite design was often selected because of the limited number of experiments needed to sample the response surfaces. In the separation of As and Se species in tap water, the analysis of isoresponse curves allowed the determination of optimum chromatographic conditions and the robustness of the method [77]. The same design was also used to study the influence of an organic modifier and IPR concentration on retention of biogenic amines in wines. To obtain a compromise between resolution and chromatographic time, optimization through a multi-criteria approach was followed [78]. 

Figure 5.4. Response surface estimated from the central composite design obtained for the pair [HCIJ-US exposure time In the determination of tin in coal acidified slurries. (Reproduced with permission of Elsevier, Ref [6].)...

Occasionally c = 1 is selected which creates a so-called face-centred central composite design. An alternative to the central composite designs, which, as far as we know, has not been used in chromatography, is the Box-Behnken 52) design. 

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