Design central-composite

A central composite design consists of three parts  

The distributions of experimental points for designs in two and three variables are shown in Fig. 6. 

A response surface model can be used for predictions and simulations by entering values of variable settings in test points to the model. However, the model 

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  

A factorial (or cubic) part, containing a total of rifact points with coordinates Xi — —1 or +1, for i — 

An axial (or star) part, formed by nax = 2A points with all their coordinates null except for one that is set equal to a certain value a (or —a). 

A total of /Zc runs performed at the center point, where, of course, xi = 

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. 

One of the most useful models for approximating a region of a multifactor response surface is the full second-order polynomial model. For two factors, the model is of the form 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Efficiency of full second-order polynomial models fit to data from central composite designs without replication. 

Central composite designs are relatively efficient for small numbers of factors. Efficiency in this case means obtaining the required parameter estimates with little wasted effort. One measure of efficiency is the efficiency value, E 

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. 

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After the preceding considerations have been taken into account, a test plan is developed to best meet the goals of the program. This might involve one of the standard plans developed by statisticians. Such plans are described in various texts (Table 1) and are considered only briefly here. Sometimes, combinations of plans are encountered, such as a factorial experiment conducted in blocks or a central composite design using a fractional factorial base. 

The experimental designs discussed in Chapters 24-26 for optimization can be used also for finding the product composition or processing condition that is optimal in terms of sensory properties. In particular, central composite designs and mixture designs are much used. The analysis of the sensory response is usually in the form of a fully quadratic function of the experimental factors. The sensory response itself may be the mean score of a panel of trained panellists. One may consider such a trained panel as a sensitive instrument to measure the perceived intensity useful in describing the sensory characteristics of a food product. 

This model is capable of estimating both linear and non-linear effects observed experimentally. Hence, it can also be used for optimization of the desired response with respect to the variables of the system. Two popular response surface designs are central composite designs and Box-Behnken designs. Box-Behnken designs were not employed in the experimental research described here and will therefore not be discussed further, but more information on Box-Behnken designs can be obtained from reference [15]. 

Note that, again, three different types of variables were combined chain length, component ratio, and absolute component level. Thus, a "standard" constrained mixture design was not appropriate. In this case a full factorial, central composite design was used, with a total of 20 data points. The star points were... 

Table VII. Design IIIi 23 FULL FACTORIAL CENTRAL COMPOSITE DESIGN...

Central composite design, commercial experimental design software compared, 8 398t... 

An augmented central composite design was used in obtaining reaction-rate data in a flow differential reactor the reaction occurring was the isomerization of normal pentane to isopentane in the presence of hydrogen (Cl). Using the subscripts 1, 2, and 3 for hydrogen, normal pentane, and isopentane respectively, an empirical rate equation can be written... 

Sketch a 3-factor non-central composite design for which the center of the star coincides with one of the factorial points. 

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

Figure 13.1 Sums of squares and degrees of freedom tree for a two-factor full second-order polynomial model fitted to a central composite design with a total of four center point replicates.

Figure 13.2 Central composite design. Square points 2, star points 4, = 3, DF = 3.

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... 

One of the striking features of this central composite design is the flatness of the normalized uncertainty and normalized information surfaces near the center of the design. 

Figure 13.3 shows a similar set of four panels for a slightly different central composite design. The lower left panel shows the placement of experiments in factor space (i.e., it shows the experimental design). The upper left panel shows the normalized uncertainty as a function of factors x, and x. The upper right panel shows the normalized information as a function of factors x, and Xj. The lower right panel plots normalized information as a function of factor x, for X2 = -5, -4, -3, -2, -1, and 0. The experimental design matrix is... 

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