# Scheffé Mix Models

Scheffé models were specifically developed to handle the natural constraints of mixture designs.

Mixture models are only readily interpretable when the mixture components all go from 0 to the total for the design. Most mixture designs cover a more constrained space. Use the Model Graphs to better understand the models.

The Scheffé model forms are as follows:

## Linear

$\hat{y}= \sum_{i=1}^{q}\beta _{i} x_{i}$

### Example

$12A + 8B + 4C$

$\hat{y}=\sum_{i=1}^{q} \beta_{i}x_{i}\,+\sum_{i<j}^{q-1}\sum_{j}^{q} \beta_{ij} x_{i} x_{j}$

### Example

$12A + 8B +4C + 8AB - 8AC$

## Special Cubic

$\hat{y} \sum_{i=1}^{q} \beta_{i}x_{i}+\sum_{i<j}^{q-1}\sum_{j}^{q} \beta_{ij}x_{i}x_{j}+\sum_{i<j}^{q-2}\sum_{j<k}^{q-1}\sum_{k}^{q}\beta_{ijk} x_{i}x_{j}x_{k}$

### Example

$12A + 8B + 4C + 8AC - 8BC + 54ABC$

## Full Cubic

$\hat{y}=\sum_{i=1}^{q}\beta_{i}x_{i}+\sum_{i<j}^{q-1} \sum_{j}^{q}\beta_{ij}x_{i}x_{j}+\sum_{i<j}^{q-1}\sum_{j}^{q}\delta_{ij}x_{i} x_{j}(x_{i}-x_{j})+\sum_{i<j}^{q-2}\sum_{j<k}^{q-1}\sum_{k}^{q}\beta_{ijk}x_{i} x_{j}x_{k}$

### Example

$12A + 8B + 4C + 8AB - 8AC + 54ABC + 48AC(A - C)$

Standard Scheffé polynomials are available up to the fourth order. There are also partial quadratic mixture (PQM) models using a combination of linear, squared, and quadratic terms (see reference by Piepel, et al.).

References

• G. Piepel, J. Szychowski, and J. Loeppky. Augmenting scheffe linear mixture models with squared and/or crossproduct terms. Journal of Quality Technology, 2002.