A star design

Because the number of distinctly different factor combinations is seven (/ = 7), and because the number of experiments is eight (n = 8), there is only one degree of freedom for lack of fit f—p = 7 — 6 = 1) and only one degree of freedom for purely experimental uncertainty n-f-S-7= 1). 

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Can the following design be used to fit the model expressed by Equation 12.44 Can it be used to fit the model expressed by Equation 12.15 How is this design related to a star design To a comer design ... 

Figure 13.14 shows a star design that can be used to fit a two-factor FSOP model. The experimental design matrix is... 

Figure 13.14 A star design. (The replicates are Mizar and Alcor in Ursa Major.) DF = 1, DF...

Figure 5.7 A star design for three factors tested at three levels...

Central composite designs are constructed by a juxtaposition of a two level full factorial and a star design. They can be used to determine the effects of changing from the method value to the extremes and the effects of changing from extreme to extreme. 

This model is composite because it consists of the overlapping of a star design with 2k+ factor combinations, and a two level -factor design with 2 factor combinations to give a total of 2 "-i- -i-1 treatments. 

For a design with three variables we would require [2 + (2 x 3) + 1] = 15 experiments. In order to obtain repeatability information it is necessary to run an experiment several times. This is done by performing the centre point experiment twice. The total number of experiments would therefore be 16. The list of experiments is shown in Table 43.3 while Fig. 43.4 shows a diagrammatic representation of the CCD. The CCD is composed of a 3 factorial design superimposed with a star design (+a, —a). In order to minimize systematic eror (bias) it is necessary randomize the experimental run order. This is shown in Table 43.4. 

Central Composite Designs. CCDs are the most often used response surface designs (1,7,17).These designs are constructed by combining a two-level full factorial design (2 experiments), a star design (2/experiments), and a center point, which is often replicated a number of times. Thus, to examine / factors, at least = 2 + 2/ + 1 experiments are required. For more... 

There is an alternative, rather more economical, composite design for 3 factors, based on a (resolution III) 2 design, a star design and centre points (5). It allows determination of the second-order model, but its properties are inferior to those of the full design. In particular it is not rotatable. This design may be useful as it allows sequential experimentation with a small total number of experiments (about 12-14). Thus the exploratory 2 design may be completed by the axial points. It has been used for the optimization of a solid dispersion (6). 

One of the more important properties of composite designs is the possibility of carrying out the experiments in two or more blocks. This is already illustrated in the solubility example, where a 2 design plus centre points was carried out first and then augmented by a star design, also with centre points. 

This model has six parameters, and our design has only five levels , that is, five different combinations of stirring speed and concentration values. Because it is not possible to calculate coefficient estimates when there are more parameters than levels, we must augment our design. This can be done in several ways, the most common being to add a star design to the initial factorial design. 

Results of the central composite design obtained by augmenting the factorial design in Table 6.5 with a star design, and X2 are the values of the variables coded by the expressions in Table 6.5... 

The chemical volatilization of antimony(III) with bromide was favoured by the presence of iodide. The variables were studied and optimised by a central composite design 16 experiments (2 ) corresponding to a full two-level factorial design 8 more experiments (2 ) corresponding to a star design and a centre point that was replicated 3 times. This design allowed determination of the main effects and their interactions. 

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