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

$$12A + 8B + 4C$$

Quadratic

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

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.

References

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