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Design-Expert » General Sequence of Analysis » Model Selection » Scheffé Mix Models

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

\[12A + 8B + 4C\]
../../../_images/scheffe_linear.png

Note that for a linear model the edges of the graphs are straight. In the unconstrained simplex the coefficient value is the prediction when the mixture is a pure component.

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

\[12A + 8B +4C + 8AB - 8AC\]
../../../_images/scheffe_quadratic.png

When there are two component blending effects the edges curve away from the linear model by one-fourth of the coefficient at the 50/50% blend (1/2 squared).

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

\[12A + 8B + 4C + 8AC - 8BC + 54ABC\]
../../../_images/scheffe_special_cubic.png

When there is three component blending, the curve away happens in the middle of the simplex and is one-twenty-seventh (1/3 cubed) of the coefficient.

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

\[12A + 8B + 4C + 8AB - 8AC + 54ABC + 48AC(A - C)\]
../../../_images/scheffe_cubic.png

Higher-order terms are used to model wavy surfaces.

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.

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