Comparing the Slack-Variable Mixture Model With Other Alternatives

Abstract

There have been many linear regression models proposed to analyze mixture experiments including the Scheffé model, the slack-variable model, and the Kronecker model. The use of the slack-variable model is somewhat controversial within the mixture experiment research community. However, in situations that the slack-variable ingredient is used to fill in the formulation and the remaining ingredients have constraints such that they can be chosen independently of one another, the slack-variable model is extremely popular by practitioners mainly due to the ease of interpretation. In this article, we advocate that for some mixture experiments the slack-variable model has appealing properties including numerical stability and better prediction accuracy when model-term selection is performed. We also explain how the effects of the slack-variable model components should be interpreted and how easy it is for practitioners to understand the components effects. We also investigate how to choose the slack-variable component, what transformation should be used to reduce collinearity, and under what circumstances the slack-variable model should be preferred. Both simulation and practical examples are provided to support the conclusions.

• Condition number
• Mixture experiments
• Scheffé model
• Slack-variable model
• Variance inflation factor (VIF)
• Variable transformation

ACKNOWLEDGMENTS

This research was supported by the Procter & Gamble Statistics Innovation Grant from the Procter & Gamble Company. We thank the Editor, Associate Editor, and two referees for the valuable comments and suggestions. We also thank Robert Wells, retired Procter & Gamble Hair Care Principal Scientist, for motivating this work by being the first practitioner (many others followed) to question the need for the S-model based on many years of successfully using the SV-model when a large filler component is available.