 5. Process Improvement
5.5.4. What is a mixture design?

## Simplex-centroid designs

Definition of simplex-
centroid designs
A second type of mixture design is the simplex-centroid design. In the q-component simplex-centroid design, the number of distinct points is 2q - 1. These points correspond to q permutations of (1, 0, 0, ..., 0) or q single component blends, the $$\small \left( \begin{array}{c} q \\ 2 \end{array} \right)$$ permutations of (.5, .5, 0, ..., 0) or all binary mixtures, the $$\small \left( \begin{array}{c} q \\ 3 \end{array} \right)$$ permutations of (1/3, 1/3, 1/3, 0, ..., 0), ..., and so on, with finally the overall centroid point (1/q, 1/q, ..., 1/q) or q-nary mixture.
Model supported by simplex-
centroid designs
The design points in the Simplex-Centroid design will support the polynomial

$\begin{array}{lcl} E(Y) & = & \sum_{i=1}^{q}{\beta_{i}x_{i}} + \sum_{i=1}^{q}{\sum_{i < j}^{q}{\beta_{ij}x_{i}x_{j}}} + \\ & & \sum_{k=1}^{q}{\sum_{j < k}^{q}{\sum_{i < j}^{q} {\beta_{ijk}x_{i}x_{j}x_{k}}}} + \cdots + \\ & & \beta_{12 \dots q}x_{i}x_{j} \dots x_{q} \end{array}$

which is the qth-order mixture polynomial. For q = 2, this is the quadratic model. For q = 3, this is the special cubic model.

Example of runs for three and four components For example, the fifteen runs for a four component (q = 4) simplex-centroid design are:
(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (.5,.5,0,0), (.5,0,.5,0) ..., (0,0,.5,.5), (1/3,1/3,1/3,0), ...,(0,1/3,1/3,1/3), (1/4,1/4,1/4,1/4).
The runs for a three component simplex-centroid design of degree 2 are
(1,0,0), (0,1,0), (0,0,1), (.5,.5,0), (.5,0,.5), (0,.5,.5), (1/3, 1/3, 1/3).
However, in order to fit a first-order model with q =4, only the five runs with a "1" and all "1/4's" would be needed. To fit a second-order model, add the six runs with a ".5" (this also fits a saturated third-order model, with no degrees of freedom left for error). 