Response surfaces, blocking, and split plots: A predictive distribution case study

Abstract

This article presents a case study re-analysis of a complex response-factor data set involving a split-plot design with blocking for two quality responses. The analysis presented herein makes use of multivariate predictive distributions to both optimize and quantify the risk of meeting specifications. This article shows how a modern approach using predictive distributions can provide deeper insight and improved process optimization over the use of classical response surface methodology tools such as “overlapping means” plots and (mean-based) desirability functions. It is shown how the R and the Stan programming languages are used to facilitate the analysis.

Keywords:

• Bayesian
• central composite design
• DoE
• MCMC
• multiple response optimization
• posterior prediction
• predictive distribution
• quality
• random effects
• reliability

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Correction

Acknowledgments

I would like to thank Jensen and Kowalski (2012) for making their interesting and rich data set publicly available for analysis and learning. I would also like to thank the referees for their careful and helpful reviews. In addition, I’d like to express my appreciation to Enrique del Castillo for helpful discussions and his feedback on the manuscript.

Notes

1 My understanding is the Crosby meant that specifications were a proper subset of quality requirements, which met customer needs.

2 Strictly speaking, if $\nu \le 2,$ then a moment-based “correlation” does not exist. Even so, if the off-diagonal term in Σ is non-zero, then the associated random variables are not stochastically independent.

Additional information

Notes on contributors

John J. Peterson

John J. Peterson is a principal consultant and holds a Ph.D. in Statistics from The Pennsylvania State University. He is on the editorial board of the Journal of Quality Technology. His research interests include process optimization, Bayesian modeling, and pharmaceutical manufacturing applications of statistics. He is the only statistician to have won the American Institute of Chemical Engineers’ Pharmaceutical Discovery, Development, and Manufacturing Award for his contributions to Quality by Design. He is a Senior Member of the American Society for Quality and a Fellow of the American Statistical Association.