Centre point design

A five-factor central composite design consists of the five-factor, two-level factorial, with the centre point and with the star pattern in all five variables. This would ordinarily call for running 32 4- 10 + 4-1-43 conditions, with some replication at the centre. Usually, the half replicate of the factorial plus the star points and centre is enough to give an adequate picture of the relationships. 

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 three-factor central composite may be represented as a cube, with a centre point, experiments at each vertex and at the ends of axes radiating out from the centre through the middle of each face. The vertices will be designated as +1 or -1 for each factor. It is suggested that six runs be made at the centre, This design can be represented as indicated below the diagram,... 

Block l Half replicate of factorial design and centre point. 

Fig. 22. Energy levels for the compensated semiconductor of the n-type. Charge transfer is carried out by means of tunneling from an occupied donor level to a vacant donor level. The presence or the absence of the point designated "e indicates the presence or absence of the electron on the impurity centre.

Fig. 6. Distribution of experimental points in central composite designs factorial points, O centre point, x axial points...

Besides the experiments required by a given design, frequently additional experiments are performed. For instance, replication of experimental points allows to have an idea of the experimental error (most frequently the centre point is replicated) and/or to validate the model (see also Section 6.4.4). 

The most evident design would appear to be a three-level factorial design. An example of a three-level factorial design is shown in Fig. 6.1.. A full three-level factorial design, 3, can be used to obtain quadratic models. However, unless / is small (/ = 2) the design requires a number of experiments (= V) that is not often feasible. An example of the use of a 3" design can be found in Ref. 51]. The two factors, the pH and the percentage acetonitrile were examined to evaluate their influence on the retention and resolution of three isoxazolyl penicillin antibiotics. The centre point experiment was triplicated to evaluate experimental error. 

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. 

A small design which varies just two or three of the factors expected to be most influential can be carried out. An example might be the 2 factorial design described in chapter 3, section IV.E, requiring 4 runs. Replicated centre points may be included. In this way, interpretable results can be obtained without sacrificing too many resources. 

It is better to replicate experiments within the design, thus estimating the experimental repeatability. If all factors are quantitative it is the centre point that is selected. This has the advantages that if the experimental standard deviation changes within the domain one could reasonably hope that the centre point would represent a mean value and also that if the response within the domain is curved, this curvature may be detected. 

Estimates of the statistical significance of the coefficients can and frequently should be obtained by other means - in particular by replicated experiments (usually centre points) followed by multi-linear regression of the data, and analysis of variance, as developed in chapter 4. The methods we have described above are complementary to those statistical methods and are especially useful for saturated designs of 12 to 16 or more experiments. For designs of only 8 experiments, the results of these analyses should be examined with caution. 

Analysis of variance for the replicated 2 full factorial design with centre points... 

Factorial Design with Centre Point and Each Experiment Repeated... 

The structure of the information matrix of a 2-level factorial design, with or without centre-points, is simple, and is easily inverted to give (X X) . Inversion for most other designs is by no means trivial, but it can be done rapidly by computer. 

B. Analysis of Variance for the Replicated 2 Full Factorial Design with Centre Points... 

Replicated experiments within a design allow for estimates of the repeatability. For quantitative factors, it is usually the centre points which are added and replicated. The design may then be analysed by ANOVA instead of using the saturated design methods. Further examples are given in chapters 5 and 9. 

Formulations of the kind described above have been used quite widely in solubilising lipophilic drugs and vitamins, for oral, parenteral, and transdermal drug delivery. We consider the 2 design plus centre point tested in an animal model for the pharmacokinetic profile or pharmacological effect of the solubilised active substance. If each of the formulation was tested in two animals, with no testing of different formations on the same animal, the analysis would be identical to that of the solubility experiment above. The major part of the variance consists of that between animals and within the same animal o/, where cF = + a/. It is... 

The 2 design with centre point can be extended by adding experiments along each of the axes at values a of the other coded variable (A, = a, Xj = 0 and X, = 0, 2 = a). These are called axial points. The result is the central composite design for 2 factors. If a is set equal to 1, as in figure 5.1b, the design is also a full factorial design at 3 levels (3 ), quite often used for studies on 2 factors. 

Although all the design points (other than the centre point) here lie on the circumference of the circle limiting the experimental domain, this is not the case for all such composite designs. In particular, for 3 and 5 or more variables and a spherical (or more accurately for more than 3 variables, hyperspherical) domain, the star points usually lie just inside the boundary. 

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 previous treatment was based on mean data, but in fact each solubility experiment was duplicated. The full results for the factorial design and axial points, plus centre points in each case were given in tables 4.2 and 5.2 respectively. 

The columns X, Xi and X2 can be seen to be identical for the factorial experiments. Following the same reasoning as in chapter 3, we conclude that the estimator bo for the constant term, obtained from the factorial points, is biased by any quadratic effects that exist. On the other hand the estimate b o obtained only from the centre point experiments is unbiased. Whatever the polynomial model, the values at the centre of the domain are direct measurements of Pq. So the difference between the estimates, bo - b o, (which we can write as U+22 using the same notation as in chaptw 3) is a measure of the curvature P, + P22- The standard deviation o o is s/v 8 (as it is the mean of 8 data of the factorial design) and that of b o is s/ /2. (being the mean of 2 centre points). We define a function t as ... 

ANOVA of the regression on the 2 design with 2 centre points... 

Table 5.4 ANOVA of the Regression 2 Design Plus Centre Points...

The individual results for the duplicated star points and the extra centre point were listed in table 5.2. These, when added to the factorial design of table 4.2, make up the duplicated central composite design. Analysis of variance for regression may be carried out on the complete data set of the composite design for the second degree model (20 data). The results are summarised in table 5.5. 

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