Simplex centroid designs

Scheffe s simplex centroid designs contain 2q-l points, q of which fall on straight components, Cq2 on binary mixtures, Cq3 on ternary mixtures, and so forth, and one observation on a q-component mixture. Simplex centroid designs, consist of the points whose coordinates are (1,0.0), (1/2, 1/2,0.0).(l/q,l/q.l/q), and of all the points that can be obtained from these by permutations of coordinates. Thus, the design contains a point at the center (centroid) of the simplex and the centroids of all the component simplexes of lesser dimension, its proper faces. 

Polynomials obtained from simplex-centroid designs contain as many coefficients as there are points in the design, and for the q-component mixture they have the form  

Making recourse to the saturation property of the design and substituting in succession the coordinates of experimental points 1 through 15 into the polynomial of Eq. (3.67), we determine the polynomial coefficients  

In a similar manner, for the polynomial of Eq. (3.66), for the q-component mixture, regression coefficients are calculated as follows  

In the general case, the formula for coefficients of regression equation obtained from the simplex-centroid design, takes the form [6]  

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Scheffe, H. The simplex-centroid design for experiments with mixtilrtffe, Statist. Soc., B25, 235,... 

Table 3.26 Matrix of simplex centroid design for a quaternary system q=4...

Adequacy of a regression equation derived by the simplex-centroid design is tested and the confidence intervals of property values, predicted by the equation, are assigned in much the same way as in the case of the simplex-lattice method. 

The simplex centroid design for q=3 is applied. The design matrix and experimental results are arrayed in Table 3.27. 

Figure 3.33a Simplex-centroid design in each point of a 2 full factorial experiment...

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

The common structure of the regression model applicable to all simplex-centroid designs is shown in Equation 8.20. 

Another standard mixture experiment strategy is the so-called simplex centroid design, where data are collected at the extremes of the experimental region and for every equal-parts two-component mixture, every equal-parts three-component mixture, and so on. Figure 5.22 identifies the blends included in a p = 3 simplex centroid design. 

Table 2.34 Three factor simplex centroid design.

A general simplex centroid design for k factors consists of 2k — 1 experiments, of which there are... 

Another class of designs called simplex lattice designs have been developed and are often preferable to the reduced simplex centroid design when it is required to reduce the number of interactions. They span the mixture space more evenly. 

How many combinations are required in a full five factor simplex centroid design Construct this design. 

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