Design-Expert® software, v12 offers formulators a simplified modeling option crafted to maximize essential mixture-process interaction information, while minimizing experimental costs. This new tool is nicknamed a “KCV model” after the initials of the developers – Scott Kowalski, John Cornell, and Geoff Vining. Below, Geoff reminisces on the development of these models.
To help learn this innovative methodology, first view a recorded webinar on the subject.
Next, sign up for the workshop "Mixture Design for Optimal Formulations".
The origin of the KCV designs goes back to a mixtures short-course that I taught for Doug Montgomery at an adhesives company in Ohio. One of the topics was mixture-process variables experiments, and Doug's notes for the course contained an example using ratios of mixture proportions with the process variables. Looking at the resulting designs, I recognized that ratios did not cover the mixture design space well. Scott Kowalski was beginning his dissertation at the time. Suddenly, he had a new chapter (actually two new chapters)!
The basic idea underlying the KCV designs is to start with a true second-order model in both the mixture and process variables and then to apply the mixture constraint. The mixture constraint is subtle and can produce several models, each with a different number of terms. A fundamental assumption underlying the KCV designs is that the mixture by process variable interactions are of serious interest, especially in production. Typically, corporate R&D develops the basic formulation, often under extremely pristine conditions. Too often, R&D makes the pronouncement that "Thou shall not play with the formula." However, there are situations where production is much smoother if we can take advantage of a mixture component by process variable interaction that improves yields or minimizes a major problem. Of course, that change requires changing the formula.
Cornell (2002) is the definitive text for all things dealing with mixture experiments. It covers every possible model for every situation. However, John in his research always tended to treat the process variables as nuisance. In fact, John's intuitions on Taguchi's combined array go back to the famous fish patty experiment in his book. The fish patty experiment looked at combining three different types of fish and involved three different processing conditions. John's intuition was how to create the best formulation robust to the processing conditions, recognizing that the use of these fish patties was in a major fast-food chain. The processing conditions in actual practice typically were in the hands of teenagers who may or may not follow the protocol precisely.
John's basic instincts followed the corporate R&D-production divide. He rarely, if ever, truly worried about the mixture by process variable interactions. In addition, his first instinct always was to cross a full mixture component experiment with a full factorial experiment in the process variables. If he needed to fractionate a mixture-process experiment, he always fractionated the process variable experiment, because it primarily provided the "noise".
The basic KCV approach reversed the focus. Why can we not fractionate the mixture experiment in such a way that if a process variable is not significant, the resulting design projects down to a standard full mixture experiment? In the process, we also can see the impact of possible mixture by process variable interactions.
I still vividly remember when Scott presented the basic idea in his very first dissertation committee meeting. Of course, John Cornell was on Scott's committee. In Scott's first committee meeting, he outlined the full second-order model in both the mixture and process variables and then proceeded to apply the mixture constraint in such a way as to preserve the mixture by process variable interactions. John, who was not the biggest fan of optimal designs when a standard mixture experiment would work well, immediately jumped up and ran to the board where Scott was presenting the basic idea. John was afraid that we were proposing to use this model and apply our favorite D-optimal algorithm, which may or may not look like a standard mixture design. John and I were very good friends. I simply told him to sit down and wait a few minutes. He reluctantly did. Five minutes later, Scott presented our design strategy for the basic KCV designs, illustrating the projection properties where if a process variable was unimportant the mixture design collapsed to a standard full mixture experiment. John saw that this approach addressed his basic concerns about the blind use of an optimal design algorithm. He immediately became a convert, hence the C in KCV. He saw that we were basically crossing a good design in the process variables, which itself could be a fraction, with a clever fraction of the mixture component experiment.
John's preference to cross a mixture experiment with a process variable design meant that it was very easy to extend these designs to split-plot structures. As a result, we had two very natural chapters for Scott's dissertation. The first paper (Kowalski, Cornell, and Vining 2000) appeared in Communications in Statistics in a special issue guest edited by Norman Draper. The second paper (Kowalski, Cornell, and Vining 2002) appeared in Technometrics.
There are several benefits to the KCV design strategy. First, these designs have very nice projection properties. Of course, they were constructed specifically to achieve this goal. Second, it can significantly reduce the overall design size while still preserving the ability to estimate highly informative models. Third, unlike the approach that I taught in Ohio, the KCV designs cover the mixture experimental design space much better while still providing the equivalent information. The underlying models for both approaches are equivalent.
It has been very gratifying to see Design-Expert incorporate the KCV designs. We hope that Design-Expert users find them valuable.
Cornell, J.A. (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, 3rd ed. NewYork: John Wiley and Sons.
Kowalski, S.M., Cornell, J.A., and Vining, G.G. (2000). “A New Model and Class of Designs for Mixture Experiments with Process Variables,” Communications in Statistics – Theory and Methods, 29, pp. 2255-2280.
Kowalski, S.M., Cornell, J.A., and Vining, G.G. (2002). “Split-Plot Designs and Estimation Methods for Mixture Experiments with Process Variables,” Technometrics, 44, pp. 72-79.
Building from our successful 2019 analytics conference, co-hosted by Stat-Ease and Camo Analytics, we are once again partnering to bring you a full week of technical skills education! Start with the 2-day Experiment Design Made Easy workshop on March 9-10, focusing on factorial design of experiments (DOE). These highly efficient designs help you discover those key factors and interactions that bring about breakthrough achievements in this new year! Follow this up with the 3-day Multivariate Data Analysis – Level 1 workshop on March 11-13, where you will learn how to use methods such as Principle Components Analysis (PCA) and Partial Least Squares (PLS) to identify important relationships between variables. In both courses, learn to use top-notch graphical visualization to effectively communicate your results!
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Click on the links above for more information about either workshop. Register for either one, or both for additional savings. Early Bird and multi-student discounts may be available. Contact email@example.com if you have questions. We look forward to seeing you!
Don't forget to check out our 8th European DOE Meeting, held in Groningen, Netherlands in 2020.
European DOE Meeting
Do you have a story to tell? Share your design of experiments (DOE) success with us. We invite you to submit an abstract to speak at the 2020 European DOE Meeting.
Stat-Ease and Science Plus Group are partnering together to host the premier conference on the practical applications of design of experiments. Topics feature using the tools of factorial design, response surface methods, and mixture design to solve business problems.
The 2020 European DOE Meeting is the perfect venue to learn how others are using these statistical techniques to dramatically impact the bottom line. Industries include, but are not limited to: pharmaceutical, medical device, electronics, food science, oil and gas, chemical processes, aerospace, etc. Presentation length: approximately 25 minutes.
The program committee is looking for presentations that:
Speakers (one per presentation) will be given discounted registration rates.
The conference will include a pre-conference short-course, keynote speakers, and some fun evening events.
Abstracts must include:
Submit abstracts (up to 1 page) to the program committee at firstname.lastname@example.org
Submission Deadline: February 15, 2020
Disclaimer: I’m not a statistician. Nor do I want you to think that I am. I am a marketing guy (with a few years of biochemistry lab experience) learning the basics of statistics, design of experiments (DOE) in particular. This series of blog posts is meant to be a light-hearted chronicle of my travels in the land of DOE, not be a textbook for statistics. So please, take it as it is meant to be taken. Thanks!
As I learn about design of experiments, it’s natural to start with simple concepts; such as an experiment where one of the inputs is changed to see if the output changes. That seems simple enough.
For example, let’s say you know from historical data that if 100 children brush their teeth with a certain toothpaste for six months, 10 will have cavities. What happens when you change the toothpaste? Does the number with cavities go up, down, or stay the same? That is a simple comparative experiment.
“Well then,” I say, “if you change the toothpaste and 6 months later 9 children have cavities, then that’s an improvement.”
Not so fast, I’m told. I’ve already forgotten about that thing called variability that I defined in my last post. Great.
In that first example, where 10 kids got cavities. That result comes from that particular sample of 100 kids. A different sample of 100 kids may produce an outcome of 9, other times it’s 11. There is some variability in there. It’s not 10 every time.
[Note: You can and should remove as much variability as you can. Make sure the children brush their teeth twice a day. Make sure it’s for exactly 2 minutes each time. But there is still going to be some variation in the experiment. Some kids are just more prone to cavities than others.]
How do you know when your observed differences are due to the changes to the inputs, and not from the variation?
It’s called the F-Test.
I’ve seen it written as:
s = standard deviation
s2 = variance
n = sample size
y = response
ӯ (“y bar”) = average response
In essence, this is the amount of variance for individual observations in the new experiment (multiplied by the number of observations) divided by the total variation in the experiment.
Now that, by itself, does not mean much to me (see disclaimer above!). But I was told to think of it as the ratio of signal to noise. The top part of that equation is the amount of signal you are getting from the new condition; it’s the amount of change you are seeing from the established mean with the change you made (new toothpaste). The bottom part is the total variation you see in all your data. So, the way I’m interpreting this F-Test is (again, see disclaimer above): measuring the amount of change you see versus the amount of change that is naturally there.
If that ratio is equal to 1, more than likely there is no difference between the two. In our example, changing the toothpaste probably makes no difference in the number of cavities.
As the F-value goes up, then we start to see differences that can likely be credited to the new toothpaste. The higher the value of the F-test, the less likely it is that we are seeing that difference by chance and the more likely it is due to the change in the input (toothpaste).
Question: Why is this thing called an F-Test?
Answer: It is named after Sir Ronald Fisher. He was a geneticist who developed this test while working on some agricultural experiments. “Gee Greg, what kind of experiments?”. He was looking at how different kinds of manure effected the growth of potatoes. Yup. How “Peculiar Poop Promotes Potato Plants”. At least that would have been my title for the research.
Graduate students are frequently expected to use design of experiments (DOE) in their thesis project, often without much DOE background or support. This results in some classic mistakes.
The number one thing to remember is this: Using previous student’s theses as a basis for yours, means that you may be repeating their mistakes and propagating poor practices! Don’t be afraid to forge a new path and showcase your talent for using state-of-the-art statistical designs and best practices.