5. Process Improvement
5.5. Advanced topics
5.5.4. What is a mixture design?

5.5.4.3.

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 2^q - 1. These points correspond to q permutations of (1, 0, 0, ..., 0) or q single component blends, the $\left(\begin{array}{c}q\\ 2\end{array}\right)$ permutations of (.5, .5, 0, ..., 0) or all binary mixtures, the $\left(\begin{array}{c}q\\ 3\end{array}\right)$ 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).