Issue: Volume 3, Number 12
Date: December 2003
From: Mark J. Anderson, Stat-Ease, Inc. (

Dear Experimenter,

Here's another set of frequently asked questions (FAQs) about doing design of experiments (DOE), plus alerts to timely information and free software updates. If you missed previous DOE FAQ Alerts, please click on the links at the bottom of this page. Feel free to forward this newsletter to your colleagues. They can subscribe by going to If this newsletter prompts you ask to your own questions about DOE, please address them to

I am fighting off the lassitude created by an overdose of tryptophan from the traditional Thanksgiving turkey and the long-lasting leftovers (maybe it is not the tryptophan according to Thank goodness that Stat-Ease, like many companies here in the USA, gives its employees an extra day off to deal with the after-effects of the holiday feasting.  Even though you may not share in these fowl traditions, it might do you some good to think positive and reflect on things to be thankful for.  While you're at it, how about a wish for peace on earth.  Wouldn't that be nice?

Here's an appetizer (zero calories!) to get this Alert off to a good start:  This web site displays a plaque, simply labeled "Experiments," that commemorates the first flight by Orville Wright 100 years ago this coming December 17.  For a far richer collection of information on this pioneering aviator and his brother Wilbur, see  Click the "Lift and Drift" link for the story of how these two developed state-of-the-art wind tunnels and their calculations on lift coefficients that proved to be the key for success.  Let your imagination soar as you contemplate what it must have been like to be among the first who broke free from being earth-bound.

Now return to the 21st century to view a fanciful flight that takes you from the tiniest nooks of nanospace to the broadest expanse of the universe.  See all this via the fascinating applet at the Molecular Expressions web site.  It will give you a whole new perspective on things.

Here's what I cover in the body text of this DOE FAQ Alert (topics that delve into statistical detail are designated "Expert"):  

1. FAQ: How center points are fitted in two-level factorials  
2. Expert-FAQ: How to make proper use of the logit transformation  
3. Challenge to readers: A puzzle from my daughter's math teacher  
4. Info alert: Links to a case study on mixture design for optimal formulation; plus a primer on how to detect outliers  
5. Events alert: Link to a schedule of appearances by Stat-Ease, now including 2004 conferences and shows  
6. Workshop alert: The 2004 schedule is now posted to our web site

PS. Quote for the month -- Why pioneering pilot (and native son of Minnesota) Charles Lindbergh loved aviation [and a link to breaking news about his other love -- a German hat maker]


1. FAQ: How center points are fitted in two-level factorials

-----Original Question-----
From: Switzerland

"Recently, I had a three-day course on DOE based on your book "DOE Simplified" [available at].  In the book there was a CD-ROM containing Design-Ease® software.  Here is a brief description of my first design: 5 factors (4 numeric, 1 categorical) in 16 runs plus 2 center points for a total of 18 runs.  In the ANOVA analysis I find the equation of the model and the table "actual value" vs. "predicted  value."  I noticed at the bottom of the table the following sentence: "Predicted values of center points include center point coefficient."  I do not understand the meaning of this coefficient.  The analysis performed with other DOE software gives me, more or less, the same predicted values, except the center points, which are not corrected with any coefficient."


Stat-Ease software, unlike some others, provides a test for overall curvature in analysis of variance (ANOVA) for two-level factorial designs with center points.  Then to compare 'apples-to-apples' for measurement of residuals, we fit center points separately from the factorial points.  Thus, even though curvature may be significant according to the test in ANOVA, center-point residuals can be diagnosed for individual outliers, rather than rejected as a whole for being so far off from the average of the outer (factorial) runs.

To recap: Our software computes the mean of center points only. Then it measures residuals of each center point from this mean. Not only does this make outlier-detection more accurate, but it also facilitates other diagnostics of residuals such as normality and constant variance.  Nevertheless, we do provide a 'work-around' for these features for those who desire output more like other software for regression modeling: Select View, Effects List from the Effects screen and turn off the modeling ("M") for Curvature. Then you will not get the special treatment of residuals for center points.

(Learn more about center points and curvature by attending the three-day computer-intensive workshop "Experiment Design Made Easy."  See for a complete description. Link from this page to the course outline and schedule.  Then, if you like, enroll online.)


2. Expert-FAQ: How to make proper use of the logit transformation

-----Original Question-----
From: Indiana

"I have a question regarding the use of the logit transformation.  In my scenario, I will be using Design-Expert® version 6 to analyze data reported in percentages that varies from 0.00 to  0.14.  Now, the data are, of course, skewed to the right and are not normally distributed, so I would like to use the logit transformation to make the response more conducive to data analysis.  However, a quick glance at the User Guide suggests that I need to decide on upper and lower limits first.  Since I have never used the logit transformation before and my minimum data value is zero, my question is can my Lower Limit for the transformation be zero?"


As you can infer from the function shown below:
    Logit(Y) = Ln[(Y-LL)/(UL-Y)]
the logit becomes mathematically impossible if any of your actual responses (Y) equal either the lower limit (LL) or upper limit (UL).  In such cases Design-Expert suggests slightly wider boundaries.  In your case it will see zero and insert a lower limit of -1 by default, but you might try putting in lesser absolute values, such as -0.1, or even lower.  Then see whether the residual diagnostics improve.

I like logit for bounded responses, such as molecular yield, because it prevents nonsensical predictions -- for example 101 percent conversion.  Otherwise, even though a model may be very useful for identifying an optimal configuration, it will be discredited due to making impossible predictions.

For more details on the logit transformation see FAQ 2 at (DOE FAQ Alert, Volume 1, Number 1, March 1, 2001).


3. Challenge to readers: A puzzle from my daughter's math teacher

Here's a puzzle that mathematics teachers probably have posed for many centuries.  My daughter brought it home the other day.  I do not recall seeing it before so I thought it would be worth passing along.

Diophantus was an Alexandrian whom many consider to be the 'father of algebra.'  He is best known for his text Arithmetica, which presented the solution for quadratic equations -- the primary model for response surface methods (RSM).  Historians know very little about his life or even when he lived. The Greek author Metrodorus around 500 AD published a famous puzzle about Diophantus which says:

"His boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/12th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son."

A more florid version of this puzzle (JR Newman (ed.), "The World of Mathematics," New York 1956) is said to be his epitaph:

"This tomb holds Diophantus.  Ah, what a marvel!  And the tomb tells scientifically the measure of his life.  God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son.  Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him.  After consoling his grief by this science of numbers for four years, he reached the end of his life.

Can you figure out how long Diophantus lived?  If you cannot (or care not to), look up the answer at:


4. Info alert: Links to a case study on mixture design for optimal formulation; plus a primer on how to detect outliers

Click to see a write up on how mixture design, aided by use of Design-Expert software, provided "An Efficient Approach to Cell Culture Medium Optimization."  A version of this study appeared in a recent issue of BioProcess International magazine.

Also see for an article that deals with thorny issues confronting every experimenter -- how to handle individual results that do not appear to fit with the rest of the data.  This write-up formed the basis for a talk I gave at the Annual Quality Congress.


5. Events alert: Link to a schedule of appearances by Stat-Ease, now including 2004 conferences and shows

Click for a list of appearances by Stat-Ease professionals.  We hope to see you sometime in the coming new year of 2004!


6. Workshop alert: 2004 schedule is now posted to our web site

See for schedule and site information on all Stat-Ease workshops open to the public.  To enroll, click the "register online" link on our web site or call Stat-Ease at 1.612.378.9449.  If spots remain available, bring along several colleagues and take advantage of quantity discounts in tuition, or consider bringing in an expert from Stat-Ease to teach a private class at your site.  Call us to get a quote.


I hope you learned something from this issue. Address your general questions and comments to me at:



Mark J. Anderson, PE, CQE
Principal, Stat-Ease, Inc. (
Minneapolis, Minnesota USA

PS. Quote for the month -- Why pioneering pilot (and native son of Minnesota) Charles Lindbergh loved aviation:

"Science, freedom, beauty...what more could you ask of life?"
Coincidentally, the day I wrote this, this news broke about Lindbergh's other love.  Here it is: (Update 3/07: This link no longer active.)

Trademarks: Design-Ease, Design-Expert and Stat-Ease are registered trademarks of Stat-Eae, Inc.

Acknowledgements to contributors:

—Students of Stat-Ease training and users of Stat-Ease software
—Fellow Stat-Ease consultants Pat Whitcomb and Shari Kraber (see for resumes)
—Statistical advisor to Stat-Ease: Dr. Gary Oehlert (
—Stat-Ease programmers, especially Tryg Helseth (
—Heidi Hansel, Stat-Ease marketing director, and all the remaining staff


Interested in previous FAQ DOE Alert e-mail newsletters?
To view a past issue, choose it below.

#1 Mar 01
, #2 Apr 01, #3 May 01, #4 Jun 01, #5 Jul 01 , #6 Aug 01, #7 Sep 01, #8 Oct 01, #9 Nov 01, #10 Dec 01, #2-1 Jan 02, #2-2 Feb 02, #2-3 Mar 02, #2-4 Apr 02, #2-5 May 02, #2-6 Jun 02, #2-7 Jul 02, #2-8 Aug 02, #2-9 Sep 02, #2-10 Oct 02, #2-11 Nov 02, #2-12 Dec 02, #3-1 Jan 03, #3-2 Feb 03, #3-3 Mar 03, #3-4 Apr 03, #3-5 May 03, #3-6 Jun 03
, #3-7 Jul 03, #3-8 Aug 03, #3-9 Sep 03 #3-10 Oct 03, #3-11 Nov 03, #3-12 Dec 03 (see above)

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