Slack-variable models versus component-proportion models for mixture experiments: Literature review, evaluations, and recommendations

Abstract

A mixture experiment (ME) combines components in various proportions and observes the values of one or more responses for each mixture. The proportions of the q components in each mixture must sum to 1.0. Two statistical approaches for modeling MEs have been discussed in the literature. The slack-variable (SV) approach uses proportions of all but one (q − 1) of the components varied in a mixture (the SV). The component-proportion (CP) approach uses the relative proportions of all q components varied in a mixture. Several articles have claimed advantages and recommended the SV modeling approach over the CP modeling approach. In contrast, a 2009 article recommended the CP modeling approach for several reasons. Our article reviews the literature, evaluates the literature justifications for using the SV modeling approach, and uses literature examples to compare the CP and SV modeling approaches. Recommendations regarding when to use the CP and SV modeling approaches are provided.

Keywords:

• additive component
• collinearity
• component effect
• inactive component
• numerical instability
• partial quadratic mixture model
• slack-variable model
• variable selection

Acknowledgments

The work of Greg Piepel on this article was completed before he retired from the Pacific Northwest National Laboratory (PNNL). We thank Bryan Stanfill (PNNL employee at the time) for performing technical and editorial reviews of the manuscript, and John Vienna of PNNL for pointing out relevant discussion of the substitution and supplementation methods in Volf (1988). Also, the comments of two external reviewers and the two journal-assigned reviewers, through several rounds of revisions, led to many improvements in the article.

Additional information

Funding

Dayton Hoffmann was a student intern at PNNL during Summer 2018 with funding from the Science Undergraduate Laboratory Intern program sponsored by the U. S. Department of Energy. The authors gratefully acknowledge this funding source.

Notes on contributors

Greg F. Piepel

Greg F. Piepel is a retired Laboratory Fellow from Pacific Northwest National Laboratory. He is also the principal of MIXSOFT, through which he teaches mixture experiment short courses and makes available the MIXSOFT software for mixture experiments. Greg has developed and applied statistical experimental design and data analysis methods to mixture experiment and other physical, engineering, and environmental science problems for 42 years. He is a Fellow of the American Society for Quality, a Fellow of the American Statistical Association, and an Elected Member of the International Statistics Institute. He has won the Brumbaugh Award, the Statistics in Chemistry Award, the W.G. Hunter Award, and has given the Youden Memorial Address. He is on the Editorial Review Boards of the Journal of Quality Technology and Quality Engineering, served on the Management Committee for Technometrics, and reviews manuscripts for many journals. He has over 165 publications, including journal articles, book chapters, proceedings papers, software and manuals, and technical reports. He may be contacted by email at [email protected].

Dayton C. Hoffmann

Dayton C. Hoffmann was an undergraduate intern at the Pacific Northwest National Laboratory at the time of the initial research for this article during Summer 2018. He completed his degree and is employed as a Data Scientist with Circle K Stores, Inc. He may be contacted by email at [email protected].

Scott K. Cooley

Scott K. Cooley is a Senior Research Scientist in the Applied Statistics and Computational Modeling Group at the Pacific Northwest National Laboratory. He has worked on projects applying mixture experimental design and data analysis methods to nuclear waste glass for many years. He has also worked on a variety of tasks to support the Radiation Portal Monitoring Project, conducting data processing and statistical analyses to develop and test methods for detecting potential nuclear threats entering the country through border crossings and seaports. He has published papers in Quality Engineering, Journal of Quality Technology, other applied statistics journals, journals of other disciplines, and has coauthored numerous technical reports. He may be contacted by email at [email protected].