Fitting Mixture Designs

When fitting a model for mixture designs, one must take into account that the factors sum to a constant. A traditional full linear model is not fully estimable.

An appropriate response surface model for mixture responses is the Scheffé polynomial (Scheffé 1958). See the discussion of Cox Mixtures and the Scheffé Cubic macro in Cox Mixtures in the Fitting Linear Models book. The Scheffé polynomial model does the following:

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suppresses the intercept

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includes all the linear main-effect terms

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excludes all the square terms (such as X1*X1)

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includes all the cross terms (such as X1*X2)

In this model, the parameters are easy to interpret (Cornell 1990). The coefficients on the linear terms are the fitted response at the extreme points where the mixture consists of a single factor. The coefficients on the cross terms indicate the curvature across each edge of the factor space.

For a mixture model with third-degree polynomial terms, the Scheffé Cubic model can be used. The Scheffé Cubic model includes terms of the form X1*X2*(X1-X2).

To fit a Scheffé polynomial model use the Mixture Response Surface macro in the Fit Model platform. To fit the Scheffé cubic model use the Sheffe Cubic macro in the Fit Model platform. If you chose to fit a different mixture model the Fit Model platform includes options for a No Intercept model and Mixture Effects as an effect attribute.

Tip: The custom design model outline has an option for adding the Scheffé Cubic model terms. The generated design table will include a script for the Scheffé Cubic model.