MANE 6313

Week 16, Module D

Student Learning Outcome

Select an appropriate experimental design with one or more factors,
Select an appropriate model with one or more factors,
Evaluate statistical analyses of experimental designs,
Assess the model adequacy of any experimental design, and
Interpret model results.

Module Learning Outcome

Explain mixture designs.

Introduction to Mixture Designs

Most experimental designs assume that the levels of each factor are independent of the levels of all the other factors
This assumption is not true in mixture experiments
Usually there is a restriction that the sum of some (or all) components must equal a value (typically 1)
Examine Figure 11.39

Figure 11.39

Simplex Lattice Designs

A \(\left\{p,m\right\}\) lattice design for \(p\) components consists of \(m+1\) equally spaced values from 0 to 1

\[ x_i=0,\frac{1}{m},\frac{2}{m},\ldots,1\;\;i=1,2,\ldots,p \]

Examine Figure 11-41
In general, a \({p,m}\) lattice design requires \(\(N=\frac{(p+m-1)!}{m!(p-1)!}\)\) points

Figure 11.41

Simplex Centroid Design

A design requiring \(2^p-1\) points corresponding to all permutations of the design points
See Figure 11.42
A Criticism of the simplex designs is that most experiments occur along the boundary of the region and not in the interior of the design

Figure 11.42

Mixture Models

Recall that \(\sum x_i=1\)
Slightly different standard forms
Linear model

\[ E(y)=\sum_{i=1}^p\beta_i x_i \]

Quadratic model

\[ E(y)=\sum_{i=1}^p\beta_i x_i + \sum\sum_{i<j}^p\beta_{ij}x_ix_j \]

Mixture Models, continued

Full cubic model

\[ \begin{aligned} E(y)&=&\sum_{i=1}^p\beta_i x_i + \sum\sum_{i<j}^p\beta_{ij}x_ix_j\\ &&+\sum\sum_{i<j}^p\delta_{ij}x_ix_j(x_i-x_j)\\ &&+\sum\sum_{i<j<k}\sum\beta_{ijk}x_ix_jx_k \end{aligned} \]

Special Cubic model

\[ \begin{aligned} E(y)&=&\sum_{i=1}^p\beta_i x_i + \sum\sum_{i<j}^p\beta_{ij}x_ix_j\\ &&+\sum\sum_{i<j<k}\sum\beta_{ijk}x_ix_jx_k \end{aligned} \]