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By Ron Schoenberg
rons@reflexnet.net
Response surface methods in DOE entail fitting a nonlinear model to an observed response, usually some form of a polynomial function. The coefficients of the terms of the fitted polynomial model are usually reported as well in summaries of results. What is to be made of these coefficients?
In my opinion nothing should be made of them. They should not even be reported. In fact they don’t even need to be computed. The response surface and its confidence limits can be computed using the singular value decomposition (SVD) without computing any coefficients. An SVD always exists for any set of independent variables, thus multicollinearity among the independent variables is irrelevant. The condition of the model doesn’t matter nor do VIF’s. Any polynomial model can be fit to the response data without concern for multicollinearity. Thus no special techniques are required for handling mixture models.
The STRATEGY program for DOE does in fact use the singular value decomposition. It reports standardized and unstandardized coefficients, standard errors, and VIF’s only because people have been taught that they need this information. But I argue that they have been taught by professors with little experience in the kinds of problems encountered in industrial applications of DOE. They are more interested in testing theories which evolve into questions about individual effects. Their theories might imply that one factor rather than another affects a response. Often, though, factors implied by multiple theories are statistically significant, and as a result the experimenter tries to reconfigure the question into one about the "relative importance" of factors. This usually means comparing "standardized" coefficients.
The issue of standardized coefficients and measuring "relative importance" was disposed of years ago, for example, O. D. Duncan, Notes on Social Measurement where he argued that the "standardization" has no theoretical meaning and that there is no substitute for first establishing the unit of measure. The most common standardized measure is the standard deviation. This can be rejected most quickly by pointing out that it is sample specific and cannot be derived from any theory. A reading of Duncan’s book and references therein should convince anyone that "relative importance" is an inappropriate approach to testing theories.
If one is testing theories, the measurement and testing of individual factors is important. Accomplishing this using response surface methods is not easy, however. The effects of a given factor on the response are nonlinear, considerably complicating the description of the effects – because the effects are now conditional on the levels of the other factors – and complicating the statistical inference as well. Moreover we would have to pay attention to the condition of the model, i.e., the matrix of independent variables must have full rank and we must remove linear dependencies, for example those found in mixture models.
Testing theories is not usually found in industrial DOE, however. The relevant factors are nearly always known in advance. There is no issue about the importance of the factors. Instead we want to know the optimal combination of factors, the sweet spot. The coefficients are irrelevant. The only issue is the observed and predicted response surfaces, and the confidence limits of the predicted response surface at any point. These quantities can all be computed without any regard to the coefficients. Moreover these quantities exist even when the moment matrix of the independent variables is singular. When this moment matrix is singular we don’t have a unique estimate of the coefficient vector. But this doesn’t matter for the calculation of the predicted response surface and its confidence limits when using the SVD method.
When the sole concern of the experiment is the determination of an optimal combination of factors, the measurement of individual factor effects is of no consequence. This has implications for the criterion used in calculating the experimental design. In particular it means that other criteria may be more efficient than the D-Optimal in this case. The I-Optimal design, for example, minimizes average prediction error over the response surface. This is the quantity that matters when the issue is the determination of an optimal combination of factors. The bottom line is that the I-Optimal design is always the best when the purpose of the experiment is to find a "sweet spot", and the D-Optimal will generally be best when individual factor effects are of interest.
[Ron Schoenberg is the head of Application Development at Aptech Systems (makers of Gauss) as well as the developer of the current versions of the math engines used in STRATEGY.]
Greg Piepel's response to this article.
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Dumb Questions by Al Corwin
with Smart Answers by Bill Kappele
One of the issues that is addressed more carefully in our new class than in our old is transformations. Here's a review for those of you that might have found them puzzling, as I did when I first encountered them.
A transformation is a mathematical calculation that is performed on each value in an Factor or Response column. (Transformations of responses are more common than transformations of factors.) Adding 1 to every value in a particular response column is a transformation; taking a square root is another.
You probably never WANT to make a transformation, but sometimes you should anyway. You should make a transformation when the s's for different trials are not estimates of the same s. How can this happen? Perhaps your instrument is not very precise when making measurements for a small response, but is very precise when measuring a large response. A trial with a small response will then have a large s and a trial with a large response will have a small s. You cannot combine, or "pool," these s's because they are different. Of course STRATEGY relies on pooling the s's. A transformation will make the s's consistent so that pooling is legitimate.
STRATEGY provides four indicators in the regression report that tell you when a transformation is necessary. They are,
If you check the box for "cues" before running the regression you will be provided with instructions in the printout for interpreting these tests.
Each test for consistency, or "uniformity," of the s's is imperfect. Each can give false alarms and false comfort. If two of the three plots indicate the need for a transformation, you should make one.
Yes, but this is less common. You also want to transform your factors BEFORE running an experiment if at all possible to insure that the trials will be well-spread-out. If you have a factor with levels spanning a few orders of magnitude, consider transforming the factor with a log transformation. STRATEGY provides plots at the bottom of the regression output to determine if a factor transformation is necessary. The cues provide complete instructions.
If you have run replicates of 5 to 8 trials, you can use the slope of the Box-Cox plot to determine an appropriate transformation. First, calculate the slope of the line from the plot. Next, subtract this slope from 1. Finally, raise Y to the power of the number just calculated. Precision is not necessary here - feel free to round. Some common transformations are square root (slope = 0.5), log (slope = 1), and reciprocal (slope = 2). STRATEGY also offers a wide range of alternative transformations. If you didn't collect many replicates, you can try several transformations to see if any of them helps. You can also go back to the lab and run more replicates to improve your Box-Cox plot.
You can always use our free design check service. We will help you determine the need for a transformation
[Bill Kappele is the President of Math Options and the author of most of the courses that we sponsor. Visit his web site at <http://www.mathoptions.com/>, and sign up for his wonderful newsletter, E-Math News.]
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Time to get your registration in for the introductory class in Boulder. As requested, we have added an advanced class in Boulder for early September, but you must have completed one of the approved basic classes to get into that class.
Week of June 28, 29, 30
A Modern DOE Workshop
Boulder, CO
July 20, 21, 22
Performing Objective Experiments
Union City, CA
August 27
Basic Statistics for Industry
Anacortes, WA
More complete information available next month.
Sep 9, 10
Advanced DOE Workshop
Boulder, CO
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A new version of STRATEGY (11.3) is now available on the Process Builder web site.
Note that we have received a major upgrade of the underlying Microsoft components. This will offer a new degree of compatibility with Excel. A version of STRATEGY incorporating these new components is complete and currently being tested. Expect that version to appear on our web site during June unless we encounter unexpected problems.
In addition, we have completed a release version of the basic Excel utility to save a selected range in STRATEGY format. This is a simple utility, provided free of charge to all STRATEGY users. Source code and documentation are included with the relatively small download (less than 1 MB).
All of these are available through the following web page:
http://www.processbuilder.com/doe/Software/strategy/downloads.htm
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You can go from any displayed plot in STRATEGY back to the list of plots simply by pushing the "L" key.
STRATEGY screens that are not designed for text entry have many of these one-key commands. Usually, these are as simple as pressing the key associated with the underlined letter in the command prompt. Try it!
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July 5 -- 9
Edinburgh, Scotland
For additional information, see the ICIAM 99 Web site at
http://www.ma.hw.ac.uk/iciam99/
---
"How modern technology is affecting the teaching and learning of statistics, and how this technology can best be incorporated into the classroom."
August 5, 1999; 8:30 AM to 4:30 PM and 6:30 PM to 9:00 PM
August 6, 1999; 8:30 AM and ends at 3:00 PM
Monroe Community College, Rochester, New York
Robin Lock, St. Lawrence University:
Opening Keynote: "How has technology changed the teaching of statistics?"
Closing Keynote: "Where do we go from here?"
Roxy Peck, Cal Poly:
"An Eclectic Approach to Teaching Introductory Statistics -- Integrating Multiple
Pedagogies"
George Cobb, Mount Holyoke:
"Course content: The clothes are there, but is the emperor wearing them?"
Gordon Black, Harris Black:
"The Advent of Internet Research: A Replacement Technology"
Patty Cyr, Eastman Kodak:
"What should you know about your data to draw the proper conclusions?"
Rick Cleary, Cornell University:
"Perceptions of Randomness"
Kunita Cooper, Minitab Inc.:
"Basic's of MINITAB"
"Using MINITAB in the classroom"
Fred Djang, Choate Rosemary Hall, Connecticut:
"Introduction to TI-83"
"Utilizing the TI-83"
Robert Heckard, Penn State:
"Visualizing Statistics"
"Combining in-class activities and technology in the classroom"
Robin Lock, St. Lawrence U:
"WWW Resources for Teaching Statistics"
"Some Tasks for Evaluating Statistical Software"
David Mathiason, RIT:
"Web-based delivery: The good, the bad, and the ugly"
"Web as a Data Resource"
Gary McClelland, Univ. of Colorado:
"Seeing Statistics: A Webbook of Interactive Graphics for Visualizing
Statistics."
"Breaking the 'Virtual' Metaphor: Teaching on the Web in Ways We Never Could
Before"
Roxy Peck
"Visualizing Statistical Concepts: Using Statistics Conceptual Software to
Enhance Student Understanding"
J. Laurie Snell, Dartmouth College:
"Chance News and Chance Videos"
Lori Thombs, U of South Carolina:
"Elementary Statistics Laboratory Course"
"Statistics Labs: Possible Formats and Some Examples"
Registration Fee: $125, which includes 5 meals for participant.
Limited seating, first-come basis.
For complete information (conference, abstracts of presentations, registration,
accommodations, travel) and updates, see our webpage:
http://www.monroecc.edu/depts/math/beyond1.htm
or contact Robert Johnson by E-mail:
RJohnson@MonroeCC.edu
Phone: Dept. Office: 716-292-2930
Fax: Dept. Office: 716-292-3874
Post:
Dept. of Mathematics
Monroe Community College
Rochester, NY 14623
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--------------------------
We hope you like Ron Schoenberg's article. Whether you do or not, we hope that it inspires some spirited feedback.
By the way, Ron brought a wonderful article by Veronica Czitrom in the most recent issue of The American Statistician to our attention. The article in titled "One-Factor-at-a-Time Versus Designed Experiments". This article should be on-line soon at the web site of The American Statistician:
http://www.amstat.org/publications/tas/index.html
I went to Bill Kappele's new class this week. Not only did I learn a lot, it was a rare opportunity for me to see some of our customers in the flesh. Since it was the advanced class, these were mostly people with whom I had a previous relationship by email. (Sign of age -- everybody is younger than I expect them to be these days.)
It was a great class Bill did his usual stellar job, the labs were wonderful, and we had exceptional students. (Part of putting on great classes is getting the right people to come.)
Thanks for your attention.
Sincerely yours,
Al Corwin
President
Process Builder
<http://www.processbuilder.com/doe/>
abcorwin@processbuilder.com