Central composite designs examples

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. 

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. 

One of the limitations to the Simplex design in additive experimentation is thet the desired total additive level is not always known. For example, we may be interested in obtaining a certain effect on a base stock using a combination of three additives. For cost reasons, we may limit the total concentration of additives to 6%. If we do this and fail to get the effect, we heve no information to estimate how much more additive we would need to get it. On the other hand, if we do get the effect at 6%, how do we know that we would not have obtained it less expensively at 4%. In situations of this kind, a central composite design can often be used, sat up in... 

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 ... 

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. 

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,... 

Example Optimization of a synthetic procedure by response surface modelling from a central composite design. Enamine synthesis by a modified TiCl -method... 

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. 

To estimate a model we need to carry out as many experiments as there are coefficients in the model. More experiments than this may be done, either replicating some of the experiments or from the addition of test points, or simply because of the nature of the experimental design. Certain statistical treatments then become possible. We will illustrate these using the example presented in section II.A, first treating the data of the factorial design plus centre point, then taking the full central composite design. 

The central composite design for 2 factors has already been discussed in section II. We now look in detail at a slightly more complex example of 3 factors. 

Sometimes known as Box-Wilson designs, these are the same as the central composite designs for the spherical domain, except for the positioning of the axial points. These are face-centred, with a = 1. An example is given for 3 factors in table A3.4. They may be derived easily by applying the rules given in chapter 5. 

As we have already seen, the design of Fig. 6.5 is an example of a central composite design for two factors. In general, a central composite design for k factors, coded as (xi,. .., x ), consists of three parts ... 

To build a central composite design, we need to specify each of these three parts. We have to decide how many cubic points to use and where they win be, what will be the value of a, and how many replicate runs should we conduct at the center point. For the design in Table 6.7, for example, k = 2. The cubic part is defined by the first four runs, the star design by the last four (with a = v ) and there are three replicate runs at the center point. The three-factor case is shown in Fig. 6.11, where the origin of the terms chosen to describe the three parts of the design is made clear. 

A rotatable central composite design is often used in this case. This design consists of 2 " factorial design combined with a number of repeated tests in the central (zero) point plus 2k star points. In the central composite factorial design, the independent variables are varied at five levels that are called a zero level, +1 and -1 levels, and two star levels. The +1 and -1 levels provide an increase or decrease in the level of the factors, from its zero level, Xq, by one increment, 5j. Star levels provide an increase or decrease in the level of the factor from its zero level by one increment multiplied by a coefficient, a, which depends on the number of variables in the design. The value of a can be calculated as a = 2 (Cochran and Cox, 1957). An example of the two-variable rotatable central composite design is shown in Table 1.2. 

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