Stat-Ease | The DOE FAQ Alert Vol. 12 No. 2

1: FAQ: Fold-over on resolution IV fractional two-level factorial did not de-alias interactions

Original Question:

From a Lean Six Sigma specialist:
“My original design was a 2^4-1 [8-run half-fraction on four factors at two levels each]. Confounded 2FIs could not be resolved [a resolution IV design]. We augmented the design with a full fold-over and ran the new runs as block 2 per Design-Expert® and entered the data. I’m wondering why the fold-over is still a res IV. Did I miss something?”

Answer:

From Stat-Ease Consultant Wayne Adams:
“Picture a cube to represent the possible design combinations for three factors. An orthogonal resolution IV design chooses opposite corners of the cube. A full fold-over merely reverses the signs of all the factor settings, which ends up just running the opposite corners thus replicating the design.”

Further comments:
Wayne helped this user get back on track. However, just so you know, the default choice presented by Design-Expert software for a res IV design augmentation is not the full fold-over, but rather a ‘semifold’ (see the screen shot below).

Recommended augmentation for a resolution IV design

The semifold will clear up the resolution problem. For more details, see our white paper showing How To Save Runs, Yet Reveal Breakthrough Interactions, By Doing Only A Semifoldover On Medium-Resolution Screening Designs. —Mark

(Learn more about two-level factorial design by attending the two-day computer-intensive workshop Experiment Design Made Easy. Click on the title for a description of this class and link from this page to the course outline and schedule. Then, if you like, enroll online.)

2: FAQ: Why the lack-of-fit test does not appear in my statistical analysis

Original Question:

From a Malaysian chemical-engineering graduate student:
“I just want to know why the lack-of-fit test does not appear in my statistical analysis from central composite designs (CCDs) that I perform for response surface method (RSM) optimizations. I've tried many times, but it seems helpless. Please guide me.”

Answer:

From Stat-Ease Consultant Shari Kraber:
“You do not have a lack-of-fit (LOF) test because you removed the replicates of the center points. The LOF computation requires an estimate of pure error. Therefore, replicates of some kind are required—at the center and/or other factor combinations in the experiment.

Replication of the center point, specifically, provides another advantage—it brings down the error associated with predictions throughout the middle of the design space for the CCD. Look at the standard error of prediction plot below. You can see the big ‘mountain’ in the middle which represents the standard error.

Prediction Plot

Standard error plot for CCD with center points removed

In future experiments, we strongly advise staying with the software defaults advised by Design-Expert, which in this case of three factors, calls for 6 center points.”

PS. At a post-conference workshop on RSM, I once watched DOE guru Doug Montgomery display the standard error plot on a CCD built according to the specification of its inventors (Box & Wilson, 1951). Doug expressed a great deal of enthusiasm (as only he would for such esoteric matters!) for the way this structure (with center points replicated!) created a flat-bottomed bowl on the standard error plot. Until then I had no idea how great this really was. —Mark

(Learn more about RSM designs by attending the two-day computer-intensive workshop Response Surface Methods for Process Optimization. Click on the title for a complete description. Link from this page to the course outline and schedule. Then, if you like, enroll online.)

3: Expert-FAQ: When results hit the upper limit (or lower limit) bound in the logit transformation what value for the correction “k” do you recommend I enter?

From a industrial statistician working on coatings:
“Some of our responses are scored on a scale which ranges between 0 and 10. In the analysis we treat it as a continuous response. When we analyze such a response using Design-Expert, we start off with no transformation, build the model, check diagnostics, etc. and then investigate how the factors affect the response under Model Graphs. Sometimes predictions go above the upper limit of our scale, which does not make sense, of course. However, if it is slightly above the upper limit, we do not consider it to be an issue. But when we get a prediction of 12 or higher, we then impose the upper (and lower) limit of our scale by applying the logit transformation.

This works very well and makes much more sense from a practical standpoint even though the model properties generally get slightly worse. However, it often occurs that the upper limit (or lower limit) of our scale is one of the scores in the experiment, and so we cannot enter those as upper limit (or lower limit) in the logit transformation. What I typically do is add 0.05 to the upper limit of our scale (or subtract 0.05 from our lower limit of the scale) and that works fine. One my colleagues showed that it can make quite a difference what value you add (the 0.05 or something else) in regards to R-squared and other model attributes. So the question is what should I add (or subtract) from theoretical bounds on the upper (or lower) for logit transformation—given these extenuating circumstances?

From Stat-Ease Consultant Pat Whitcomb:
“The idea behind the logit is that the data is bounded—the limits (relative to the minimum and maximum values) define the difficulty in approaching the boundaries. For an ordinal scale of 1 to 10 let’s look at the logit transformation using adjustment differences (Delta or 'd') of 0.01, 0.10 and 0.5 where the limits are 1-d and 10+d.

Here are the results (table below):

To simplify the comparison I centered the ordinal scale of 1 to 10 on zero by subtracting 5.5 from each value (series 1). Series 2, 3 and 4 are the logit transform with ds of 0.01, 0.1 and 0.5:

As can be seen in the above graph smaller values of d translate into the boundaries being more difficult to approach. One way to choose d is to base it on the expected response behavior: If it is exponentially difficult to approach the boundaries, then use a small d (e.g. 0.01). If the approach is more linear, then use a larger value (e.g. 0.5).”

PS. Here's what I get from this intriguing answer by Pat:

4: Info Alert: White paper on QbD, tutorial on mixture-process experiments, inspirational article on using DOE for coatings development

Check out the white Paper posted on “Using DOE with Tolerance Intervals to Verify Specifications” posted here. It’s based on a talk given by Pat Whitcomb to the 11th annual meeting of the European Network for Business and Industrial Statistics (ENBIS) at the University of Coimbra (Portugal) last September. The tools are geared to the Quality by Design (QbD) initiative by the U.S. Food and Drug Administration (FDA). However, they can just as well be applied to transform how any products are discovered, developed, and manufactured.

If you formulate products, take a look at this “Tutorial for mixture-process experiments with an industrial application” by Luiz Bello and Antonio Vieira published in Pesquisa Operacional (Operations Research), Vol.31 No.3 Rio de Janeiro Sept./Dec. 2011. They present a case study of a three-component mixture used in the delay mechanism for starting a rocket engine. Two process factors are considered as well.

5: Reader Testimonial: DOE Simplified used to teach experiment design in high school

“My daughter’s high school teaches statistics. Most schools cannot afford the full license to your software. I passed along the DOE Simplified flyer to the stats teacher. He was ecstatic to see this economical book including a 180-day license to Design-Ease software. Teachers can get such budgeting through their administration and thus use the program in lessons.”
—Dan Doe, Senior Research Chemist, ITW Plexus, Product Development Group, Performance Polymers Div., Danvers, MA

(Fitting coming from a fellow named “Doe”! —Mark)

6: Webinar Alert: Overview of Robust Design, Propagation of Error, and Tolerance Analysis

Response Surface Methods (RSM) can lead you to the peak of process performance. In this advanced-level webinar presented on Wednesday, May 23 at 2:00 PM CT,* Stat-Ease Consultant Pat Whitcomb will discuss robust design, propagation of error, and tolerance analysis. Propagation of error (POE) accounts for variation transmitted from deviations in factor levels. It finds the flats—high plateaus or broad valleys of response, whichever direction one wants to go—maximum or minimum; respectively. Tolerance analysis drills down to the variation of individual units, thus facilitating improvement of process capability.

If you are working to make your system more robust, this webinar is for you! It will be especially valuable to those involved in Design for Six Sigma (DFSS).

Stat-Ease webinars vary somewhat in length depending on the presenter and the particular session—mainly due to breaks for questions: Plan for 45 minutes to 1.5 hours, with 1 hour being the target median. When developing these one-hour educational sessions, our presenters often draw valuable material from Stat-Ease DOE workshops.

Attendance may be limited, so sign up soon by contacting our Communications Specialist, Karen Dulski, via [email protected]. If you can be accommodated, she will provide immediate confirmation and, in timely fashion, the link with instructions from our web-conferencing vendor GotoWebinar.

*(To determine the time in your zone of the world, try using this link. We are based in Minneapolis, which appears on the city list that you must manipulate to calculate the time correctly. Evidently, correlating the clock on international communications is even more complicated than statistics! Good luck!)

You are invited to participate in our Fourth European Design of Experiments (DOE) User Meeting in Vienna, Austria. Find all the details here.

Pat Whitcomb jets halfway around the world to present a one-day workshop on DOE for QbD to pharma researchers in India. See all the details here.

See Stat-Ease also at the 29th Quality and Productivity Research Conference in Long Beach, CA, June 4-7. Wayne Adams will provide a demo of Design-Expert software on June 6th, 3-4 pm.

Click here for a list of upcoming appearances by Stat-Ease professionals. We hope to see you sometime in the near future!

PS. Do you need a speaker on DOE for a learning session within your company or technical society at regional, national, or even international levels? If so, contact me. It may not cost you anything if Stat-Ease has a consultant close by, or if a web conference will be suitable. However, for presentations involving travel, we appreciate reimbursement for travel expenses. In any case, it never hurts to ask Stat-Ease for a speaker on this topic.