[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 text book for statistics. So please, take it as it is meant to be taken. Thanks!]
When I wrote my introduction to factorial design (Greg’s DOE Adventure - Factorial Design, Part 1), there were a couple of points that I left out. I’ll amend that post here to talk about making sure your experiment is planned out yet random.
You’ll see. Let me explain.
During the initial phase of an experiment, you should make sure that it is well planned out. First, think about the factors that affect the outcome of your experiment. You want to create a list that’s as all-encompassing as possible. Anything that may change the outcome, put on your list. Then pare it down into the ones that you know are going to be the biggest contributors.
Once you have done that, you can set the levels at which to run each factor. You want the low and high levels to be as far apart as possible. Not too low that you won’t see an effect (if your experiment is cooking something, don’t set the temperature so low that nothing happens). Not too high that it’s dangerous (as in cooking, you don’t want to burn your product).
Finally, you want to make sure your experiment is balanced when it comes to the factors in your experiment. Taking the cooking example above a little further, suppose you have three factors you are testing: time, temperature, and ingredient quality. Let’s also say that you are testing at two different levels: low and high (symbolized by minus and plus signs, respectively). We can write this out in a table:
This table contains all the possible combinations of the three factors. It’s called an ‘orthogonal array’ because it’s balanced. Each column has the same number of pluses and minuses (4 in this case). This balance in the array allows all factors to be uncorrelated and independent from each other.
With these steps, you have ensured that your experiment is well planned out and balanced when looking at your factors.
At the start of this post, I said that an experiment should be planned out, yet random. Well we have the planned-out part, now let’s get into the random part.
In any experimentation, influence from external sources (variables you are not studying) should be kept to a minimum. One way to do this is randomizing your runs.
As an example, let’s look at the table above with the cooking example. Let’s say that it represents the order of how the experiment was run. So, all the low temperature runs were made together and then all the high ones together. This makes sense, right? Perform all the runs at one temperature before adjusting up to the next setting.
The problem is, what if there is an issue with your oven that causes the temperature to fluctuate more, early in the experiment and less later. This time-related issue introduces variation (bias) into your results that you didn’t know about.
To reduce the influence of this variable, randomize your run order. It may take more time adjusting your oven for every run, but it will remove that unwanted variation.
Temperature is a popular example to illustrate randomization. But this can be said of any factor that may have time-related problems. It could be warm-up time on a machine or the physical tiring of an operator. Randomization is used to guard against bias as much as you can when running an experiment.
Hopefully, you see now why I said to keep your experiments planned but random. It sounds like an oxymoron, but it’s not. Not in the way I’m talking about it here!
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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 learn how to effectively use this innovative methodology, first view a recorded webinar on the subject.
Next, attend our Mixture Design for Optimal Formulations workshop, next offered July 15-16, 2020 in Minneapolis, MN.
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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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.
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Submission Deadline: February 15, 2020