Three factors central composite designs

In a Box-Behnken design, the experimental points lie on a hypersphere equidistant from the center point as exemplified for a three-factor design in Figure 4.13 and Table 4.11. In contrast to the central composite design, the factor levels have only to be adjusted at three levels. In addition, if two replications are again performed in the center of the three-factor design, the total number of experiments is 15 compared to 17 with the central composite design. 

Replication is often included in central composite designs. If the response surface is thought to be reasonably homoscedastic, only one of the factor combinations (commonly the center point) need be replicated, usually three or four times to provide sufficient degrees of freedom for s. If the response surface is thought to be heteroscedastic, the replicates can be spread over the response surface to obtain an average purely experimental uncertainty. 

It has been remarked that a three-level two-factor factorial design can be the same as a two-factor central composite design. Comment. 

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... 

If a method of analysis is fast or can be fully automated and requires the testing of few factors (three or less) then the larger designs can be considered. Good choices are central composite designs, or if a linear factor response is expected a full factorial design at two levels. 

Blocking is done by running a set of experiments in balanced blocks and correcting the results for the differences (if any) between blocks prior to or in the process of analysing the results In a four-factor central composite design, the experiment may be split into three equal blocks as indicated. 

Figure 3.8. Possible configurations of nine sets of factor values (experiments) that could be used to discover information about the response of a system. Dotted ellipses indicate the (unknown) response surface, (a) A two-factor, three-level factorial design (b) a two-factor central composite design.

Figure 3.13. Central composite designs for (a) two and (b) three factors. The design in panel a is rotatable.

Table 3.7. Levels for a three-factor, rotatable, central composite design.

Typical models for two- and three-factor systems subjected to a central composite design are... 

The effects of different variables on a process can be determined using experimental design methodology, which employs a reduced, but meaningful, number of experiments (10). The statistical analysis of monolaurin molar fraction (Ymon) was made by means of a two-level-three-factors central composite design with six star points and six central points. The experimental data were analyzed using STATISTICA for Windows, release 5.5, produced by StatSoft. 

A central composite design for three factors was used to generate 20 combinations. The effects of independent variables—acid/glycerol molar ratio (R), temperature (T), and enzyme concentration (E)—on the response (i.e., the monolaurin molar fraction at 4 h) were investigated. The upper and lower limits of each variable were chosen based on published data and preliminary studies (12,13). Actual independent variables or factors and their corresponding coded levels are presented in Table 1. 

A three factor central composite design was used to study the volatiles formed as functions of the three independant variables temperature, rhamnose/proline ratio and pH. In this tjrpe of design the variables are changed both simultaneously and one-at-a-time. The specific experimental points are listed in Table I. 

Three-Factor Central Composite Design with Axial Values a and Four Center Points... 

Many designs for use in chemistry for modelling are based on die central composite design (sometimes called a response surface design), die main principles of which will be illustrated via a three factor example, in Figure 2.29 and Table 2.31. The first step,... 

Table 2.32 Three possible two factor central composite designs.

Three factors, namely (1) irradiation power as a percentage, (2) irradiation time in seconds and (3) number of cycles, are used to study the focused microwave assisted Soxhlet extraction of olive oil seeds, the response measuring the percentage recovery, which is to be optimised. A central composite design is set up to perform the experiments. The results are as follows, using coded values of the variables ... 

Fig. 4 Central composite design for three factors. The factorial points are shaded, the axial points unshaded, and the center point(s) filled.

Table 4 A central composite design for three factors...

Table 4 Central Composite Design Conditions for Three Factors...

Table 5 Sequential Approach for Conducting a Three-Factor Central Composite Design in Three Blocks...

Several studies have employed chemometric designs in CZE method development. In most cases, central composite designs were selected with background electrolyte pH and concentration as well as buffer additives such as methanol as experimental factors and separation selectivity or peak resolution of one or more critical analyte pairs as responses. For example, method development and optimization employing a three-factor central composite design was performed for the analysis of related compounds of the tetracychne antibiotics doxycycline (17) and metacychne (18). The separation selectivity between three critical pairs of analytes were selected as responses in the case of doxycycline while four critical pairs served as responses in the case of metacychne. In both studies, the data were htted to a partial least square (PLS) model. The factors buffer pH and methanol concentration proved to affect the separation selectivity of the respective critical pairs differently so that the overall optimized methods represented a compromise for each individual response. Both methods were subsequently validated and applied to commercial samples. 

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