George Box and Response Surface Methods

News of George Box’s death escaped me while I was on vacation earlier this year and I thought it about time I commented on a huge statistical career. There are plenty of thorough obituaries on the web and I’m sure that the RSS will do a splendid job in due course. It is sad that there was no obituary in the Fleet Street press for somebody who has made such an eminent contribution to science and technology. Box’s particular talents were formed through his English training and learning on the job. Perhaps his neglect on the national stage is a measure of the extent to which the biggest ideas work gradually and organically, away from the grandstanding of the celebrity culture.

The word statistician feels inadequate to describe Box’s work. He was a man actively engaged in seeking novel methodologies for solving practical problems. Many of his solutions embraced what we conventionally think of as statistics. However, his work always seems that of somebody who looked for methodological solutions and sometimes found them in statistics, rather than a statistician looking to sell his product. Box described himself as “an accidental statistician” and I have a soft spot for people who arrive at their destinations by unconventional routes.

I was spurred on to reflect on Box’s work by Tim Davis’ worthwhile advocacy of dimensional analysis in experimental design;. Box himself lamented that engineers are often hypnotised when adopting statistical tools and discard their engineering knowledge in the process with little regret. There is a gap between the engineer incubating a, possibly ill-formed, problem and a statistically inspired structured investigation. Sometimes it’s a hazardous leap between the two. From the far side it’s sometimes difficult to look back and see what motivated the investigation. The more bridges we can find across that gap the better. I think few have approached the effectiveness with which Box pontificated (in the exact sense of the word).

One of Box’s greatest contributions was his advocacy of Response Surface Methods (“RSMs”). I think some of my most enjoyable statistical experiences were back in my automotive industry days when we were using RSMs with computer models to optimise design details on mechanical components. We were looking to improve durability and reduce warranty costs. I recall one situation where we exploited an elastic-plastic model of a feature that took 16 hours to run on the company’s CRAY supercomputer, a situation where even computer experiments needed a structured investigation.

As I said, Tim had got me thinking and I returned to a frustrating book that I have put down years ago: Walter G Vinenti’s What Engineers Know and How They Know It (1990, Johns Hopkins UP). Vincenti was an eminent aerospace engineer and the book is a fascinating history of a number of notable events in aerospace design. I do have a problem with this book. Vincenti seems rather dismissive of statistics. There are no statisticians in the index! There is however a compelling chapter on W F Durand and E P Lesley’s First World War propeller experiments. These were executed through quite a nice little factorial design. Durand’s trials and tribulations in managing the experimentation show that really the statistics is the easy bit. You can find the full report here. It is well worth reading.

Vincenti is rather dismissive of Durand’s statistical skill and relegates it to a footnote. He doesn’t really acknowledge Durand’s methodological sophistication. The truly frustrating thing about the book is the difficulty in drawing generalised conclusions that answer the question in the title. However, Vincenti does come up with the suggestion that “parameter variation” in the broadest sense is a key part of the engineering learning process. I think it’s a disappointing takeaway as his descriptive part of the book is much richer than the conclusion suggests. Perhaps I will come back to this.

One of Box’s key insights was engineers’ need for immediacy and sequentiality in the parameter variation process (“Statistics as a Catalyst to Learning by Scientific Method Part II – A Discussion”, Journal of Quality Technology, 31(1), 1999, pp16-29).

Psychologist Daniel Kahneman has described two ways of thinking that typify human decision making. System 1 is instinctive, fluent, heuristic and integrated with the experience base. System 1 is over confident and often leads us astray. System 2 employs reflective considered analysis. It can, when properly guided by statistical theory, guard against the hazards of System 1. Problems such as “What factors determine this process output?” are difficult. Kahneman observes that often, when confronted with difficult problems, System 1 substitutes a simpler problem such as “What factors are we currently relying on to control this process?”. Experts think they are answering the first question when they are in fact answering the second. Box’s requirement for immediacy allows engineers to exploit their, sometimes misleading, experience base while subjecting it to a rigorous experimental test in a rapid and efficient manner.

Experimental results feed into System 2 thinking. However, the human mind is still much too confident in adopting explanations that are in reality merely plausible rather than probable. The requirement of sequentiality allows analysis of those explanations in a rich and diverse context that puts them to a rigorous test.

One of the fascinations of engineering research is exploring the partially known. Jon Schmidt made the following remark about structural engineering but I think it applies to engineering in general. I t certainly applied to mechanical engineering in my automotive days.

Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot precisely analyse so as to withstand forces we cannot properly assess in such a way that the public at large has no reason to suspect the extent of our ignorance.

The application of statistics, and in particular RSMs, to engineering is one of the great tools we have for decision making under uncertainty. Modern psychology has tended to confirm Box’s instincts, learned on the job, about the tools that best support human decision making and guard against its inadequacies. Box remains a role model in developing strategies for operational excellence.

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