Central composite design. See

If the F-test is significant then there is evidence of a quadratic effect due to at least one of the variables. With the present design, however, the investigator will not be able to determine which of the variables has a quadratic effect on the response. Additional experimentation, perhaps by augmenting the current design with some star points to construct a central composite design (see section on central composite designs below), will need to be conducted to fully explore the nature of the quadratic response surface. 

The most prominent design to fit a second-order polynomial is the so-called central-composite design, see e.g. Kleijnen (2007). Here, a resolution V design is enhanced by 1 + 2 -m configurations to be able to estimate quadratic effects, too. 

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]. 

The ability to detect Cu2+, Cd2+ and Pb2+ with different electrodes modified in the same way but with different peptides selective for different metals opens the door to fabricating an electrode array for more than one metal ion. To explore that possibility an electrode array with four elements was employed where each electrode was modified with thioctic acid. The four elements were electrodes modified with (a) no peptide (that is thioctic acid alone), (b) Gly-Gly-His, (c) GSH and (d) human angiotensin I (see Fig. 10.5). The electrode array was calibrated by preparing 16 different cocktails of Cu2+, Cd2+ and Pb2+ according to a central composite design [53]. OSWV for each electrode for one of the calibration solutions (0.100 pM Cu2+, 3.00 pM Cd2+ and 0.600 pM Pb2+ in 50 mM ammonium acetate at pH 7.0) are shown in Fig. 10.6. The two features apparent from Fig. 10.6 are that there are only two main peaks due to the overlap of the electrochemistry from... 

Cocaine has been extracted from coca leaves and the optimization procedure was investigated by means of a central composite design [17]. Pressure, temperature, nature, and percentage of polar modifier were studied. A rate of 2 mL/min CO2 modified by the addition of 29 % water in methanol at 20 M Pa for 10 min allowed the quantitative extraction of cocaine. The robustness of the method was evaluated by drawing response surfaces. The same compound has also been extracted by SEE from hair samples [18-20]. 

The extremely versatile full second-order polynomial model in Equation 3.32 can also be fitted when at least three-level factorial designs are used and 3 experiments are run. Alternatively, a central composite design may be used effectively (see Figure... 

As explained in Section 6.2 simple empirical models such as those of Eq. (6.1) and Eq. (6.2) are usually applied. They can be easily generalized to more than two variables. Usually not all possible terms are included. For instance, when including three variables one could include a ternary interaction (i.e. a term in. vi.vi.vy) in Eq. (6.1) or terms with different exponents in Eq. (6.2). such as. vi.v , but in practice this is very unusual. The models are nearly always restricted to the terms in the individual variables and binary interactions for the linear models of Eq. (6.1), and additionally include quadratic terms for individual variables for the quadratic models of Eq. (6.2). To obtain the actual model, the coefficients must be computed. In the case of the full factorial design, this can be done by using Eq. (6.5) and dividing by 2 (see Section 6.4.1). In many other applications such as those of Section 6.4.3 there are more experiments than coefficients in the model. For instance, for a three-variable central composite design, the model of Eq. (6.2)... 

Table 3.17 presents the results from an analysis of the standard errors of each effect. Dividing the coefficient value by the standard error for each effect gives the t ratio. The critical t ratio is given by f i, = VF ritlP/ h DPR) ratio. Here we see that 022 exceeds while flu does not. This method is not foolproof since fly and 022 are mutually biased. However, for central composite designs the degree of bias is often slight. 

We consider these designs to be highly recommendable as tools for pharmaceutical development. Although we will see that they may be inferior to the central composite designs according to some of our criteria, they are still of excellent quality and they have a number of interesting additional properties that make them particularly suitable for solving certain problems. 

Central composite designs (CCD) are the most frequently applied response surface designs. They consist of a two-level full factorial design (2 experiments), a star design (2/ experiments) and a central point. As a consequence the CCD requires 2 + 2/+ 1 experiments to examine / factors (9 experiments for two factors 15 experiments for three factors, etc.). The experiments of the full factorial are situated at levels -1 and +1, those of the star designs at levels -a and +a for each factor, and the central point has all levels at 0 (see Figure 3.19). The central point of the design (0, 0) is replicated commonly to estimate the experimental error. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

The so-called central composite (CC) designs are perhaps the most common ones used in RSM, p>erhaps due their simple structure (for other possible reasons, see section 5.3). As the name suggests they are composed of other designs, namely, of a factorial or fractional 2 ... 

The design can be written in two ways as a Taguchi style experiment (see, for example, Breyfogle, 2000) as shown, in Table 18.1 and as a modified Graeco Latin square (see, for example, Cochran and Cox, 1957) as shown in Table 18.2. Notice that each row and column in Table 18.2 has equal numbers of 2 h and 4 h lower dwell time and wet and dry environments. Other designs (see, for example Metcalfe, 1994, and Wu and Hamada, 2000), such as central composites and star designs exist but would require more trials to obtain the same balance (see Appendix III). 

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