A note comparing component-slope, Scheffé and Cox parameterizations of the linear mixture experiment model

Abstract

A mixture experiment involves combining two or more components in various proportions and collecting data on one or more responses. A linear mixture model may adequately represent the relationship between a response and mixture component proportions and be useful in screening the mixture components. The Scheffé and Cox parameterizations of the linear mixture model are commonly used for analyzing mixture experiment data. With the Scheffé parameterization, the fitted coefficient for a component is the predicted response at that pure component (i.e. single-component mixture). With the Cox parameterization, the fitted coefficient for a mixture component is the predicted difference in response at that pure component and at a pre-specified reference composition. This article presents a new component-slope parameterization, in which the fitted coefficient for a mixture component is the predicted slope of the linear response surface along the direction determined by that pure component and at a pre-specified reference composition. The component-slope, Scheffé, and Cox parameterizations of the linear mixture model are compared and their advantages and disadvantages are discussed.

Keywords:

• Mixture component effects
• Scheffé
• linear mixture model
• Cox linear mixture model
• component-slope linear mixture model

Acknowledgments

I gratefully acknowledge fruitful discussions several years ago with Pavel Hrma, a materials scientist at Pacific Northwest National Laboratory (PNNL), which led to the development of the component-slope linear mixture model and some related ideas. I also acknowledge contributions by Jason Loeppky as part of an internship at PNNL during 2001, when he was a graduate student at Simon Fraser University. Special thanks also go to PNNL co-worker Scott Cooley for related and follow-on contributions, as well as for an internal review of the initial manuscript. Finally, the comments of the JAS reviewer were instrumental in producing the final version of the article.