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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.

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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