Models, the terms that they fit, and the type of blending that they model

The model terms that are available depend on the type of mixture design. You can fit a model to a simple mixture design (components only), a mixture-process variable design (components and process variables), or a mixture-amounts design (components and amounts).

The order of the model you choose determines which terms are fit and whether or not you can model linear or curvilinear aspects of the response surface.

In Stat > DOE > Mixture > Analyze Mixture Design > Terms, you can choose a linear, quadratic, special cubic, full cubic, special quartic, or full quartic model. Or, you can fit a model that is a subset of these terms. The following table summarizes these models. For a discussion of the various blending effects you can model, see [1].

Model type Terms Type of blending

linear (first-order)

linear

additive

quadratic (second-order)

linear and quadratic

additive

nonlinear synergistic binary

or

additive

nonlinear antagonistic binary

special cubic (third-order)

linear, quadratic, and special cubic

additive

nonlinear synergistic binary

nonlinear antagonistic binary

full cubic (third-order)

linear, quadratic, special cubic, and full cubic

additive

nonlinear synergistic binary

nonlinear antagonistic binary

nonlinear synergistic ternary

nonlinear antagonistic ternary

special quartic (fourth-order)

linear, quadratic, special cubic, full cubic, and special quartic

additive

nonlinear synergistic binary

nonlinear antagonistic binary

nonlinear synergistic ternary

nonlinear antagonistic ternary

nonlinear synergistic quaternary

nonlinear antagonistic quaternary

full quartic (fourth-order)

linear, quadratic, special cubic, full cubic, special quartic, and full quartic

additive

nonlinear synergistic binary

nonlinear antagonistic binary

nonlinear synergistic ternary

nonlinear antagonistic ternary

nonlinear synergistic quaternary

nonlinear antagonistic quaternary

You can fit inverse terms with any of the previous models if the lower bound for any component is not zero and you choose to analyze the design in proportions. Inverse terms lets you model extreme changes in the response as the proportion of one or more components nears its boundary. Suppose you are formulating lemonade and you are interested in the acceptance rating for flavor. An extreme change in the acceptance of lemonade occurs when the proportion of sweetener goes to zero. That is, the flavor becomes too sour.

Analyze Mixture Design fits a model without a constant term. For example, a quadratic in three components is as follows:

Y = b1A + b2B + b3C + b12AB + b13AC + b23BC

To open Analyze Mixture Design, choose Stat > DOE > Mixture > Analyze Mixture Design.

[1] J.A. Cornell (2002). Experiments With Mixtures: Designs, Models, and the Analysis of Mixture Data, Third Edition, John Wiley & Sons.

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