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. If you have a question that needs answering, click the Search tab and enter the key words. This finds not only answers from previous Alerts, but also other documents posted to the StatEase web site. Here's an appetizer to get this Alert off to a good start: To see experimental evidence that dogs know calculus, click http://www.maa.org/features/elvisdog.pdf. There you find data gathered by math professor Tim Pennings on how his dog Elvis showed an uncanny talent for taking the optimal path when retrieving objects thrown into the water. For simplified versions of the story, and more pictures of Elvis and his master, fetch http://www.maa.org/mathland/mathtrek%5F06%5F09%5F03.html or http://www.sciencenewsforkids.org/articles/20031008/Feature1.asp.
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: Explanation for "degrees of freedom"
2. FAQ: How StatEase does pairwise testing
3. Info alert: An opportunity to publicize your success with DOE!
4. Events alert: Talk in Toronto at the Annual Quality Congress
5. Workshop alert: See when and where to learn about DOE—A class is coming to San Jose
PS. Quote for the month—Why does physicist Freeman Dyson say that "The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often"? The answer is provide by Littlewood's Law of Miracles at the end of this Alert. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1. FAQ: Explanation for "degrees of freedom" Original Question "No matter how many books I read about statistics, I still don't understand 'degrees of freedom.' I need an explanation and many examples showing how to calculate and use them. Also, how can I be sure that all degrees of freedom in a stat problem have been accounted for?"
Answer (from StatEase Consultant Shari Kraber and Pat Whitcomb): "You are right about degrees of freedom, they can get a bit complex and it's tough to find any really good explanation. Degrees of freedom (df) are the number of independent pieces of information that can be obtained from a data set. The corrected total number of df available is N1, or one less than the total number of data points. Correcting the data for the overall mean uses 1 df. For example with 10 pieces of data, there are 9 pieces of information that are independent of the overall mean. I always check the df for the corrected total sum of squares (SS) at the bottom of the analysis of variance (ANOVA)—this should be 1 less than the number of data points you have.
Degrees of freedom account for independent pieces of information. For example, let's say that you perform an experiment in four blocks of runs. Block coefficients are corrections to the overall mean used to obtain the mean of each block. Therefore n1, where n is the number of blocks, df are used by blocks. In our example 3 df (=41) must be used to estimate the block coefficients for the predictive model since only 3 of the 4 block corrections are independent of the overall mean. Individual model terms then use up differing amounts of df, depending on the nature of your experimental factors. Main effect terms for modeling factorials correct the overall mean to obtain the mean of each factor level. Therefore n1, where n is the number of factor levels, df are used by each main effect. Thus in twolevel factorials each main effect has 21, or 1 df. If a factor has three levels the main effect has 31 or 2 df.
Factorial interaction df are calculated by multiplying the df from the main effects involved in the interactions. For example, in a twolevel design on two factors, the df for AB interaction requires 1 df because both A and B take up 1 df and 1x1=1. If factor A has 3 levels and factor B has 4 levels then the AB interaction has (31)(41)=2x3=6 df.
Now let’s put it all together—What if you studied three suppliers of four material types? In this case the df for A is 2 (=31) and B requires 3 df (=41), so the AB takes up 6 df(=2x3). In this example there are 12 different combinations of A(supplier) with B(material). Therefore our model has to support prediction of a different mean for each of those 12 combinations. Start with the overall mean (allowing estimation of 1 mean). Then the main effect for A has 2 df (allowing estimation of 2 more means. That’s 3 df total at this stage. Continuing on, the main effect for B has 3 df (allowing estimation of 3 more means. We are now at 6 total. Finally the interaction AB has 6 df, allowing estimation of 6 more means. That brings us to 12 df in total.
In ANOVA, after placing the correct df's into the block and model terms, the residual gets the 'leftovers.' If you do replicates, extra df's become available for estimation of pure error. For instance, if a center point was replicated four times, then the pure error df would be 3 (=41)."
Shari and Pat have tackled a very tough topic and wrestled it down to the ground. Here's a link to another explanation of degrees of freedom by Gerard E. Dallal of Tufts University:
http://www.tufts.edu/~gdallal/dof.htm from his "The Little Handbook of Statistical Practice" (table of contents at http://www.StatisticalPractice.com).  Mark
(Learn more about degrees of freedom and ANOVA by attending the threeday computerintensive workshop "Experiment Design Made Easy." See http://www.statease.com/clas_edme.html For a course description. Link from this page to the course outline and schedule. Then, if you like, enroll online.)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2. ExpertFAQ: How StatEase does pairwise testing "I'm trying to help a student with multiple comparisons done with StatEase software. I found the table of pairwise ttests in the ANOVA, but I'm not sure how the calculations are done."
Answer: *(For details, see "Multiple Comparison Methods for Means" by Rafter, Abell and Braselton in Society for Industrial and Applied Mathematics (SIAM) Review, Vol. 44, No. 2, 2002, pp 259278, posted at http://sci2s.ugr.es/keel/pdf/specific/articulo/35723.pdf for an abstract.)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3. Info alert: An opportunity to publicize your success with DOE! Gain technical recognition in your company and scientific field. Share a DOE with our writer at: Rename proprietary factors if needed to maintain confidentiality. A thirtyminute interview, some emails, a review, and it's done!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 4. Events alert: Talk in Toronto at the Annual Quality Congress I will present a talk titled "Screening Process Factors in the Presence of Interactions" at session W202 of the American Society of Quality (ASQ) Annual Quality Congress (AQC) in Toronto on Wednesday, May 26. For information on this event, see http://aqc.asq.org/.
My talk, coauthored by Pat Whitcomb, introduces a new, more efficient type of fractional twolevel factorial design of experiments (DOE) tailored for screening of process factors. These designs are referred to as "Min Res IV" because they require a minimal number of factor combinations (runs) to resolve main effects from twofactor interactions (resolution IV). We set the stage for these stateoftheart minimumrun mediumresolution designs by first reviewing traditional screening approaches—standard twolevel fractional factorials (2^kp) and PlackettBurman designs. To provide an element of realism, we will show how well each screening design identifies main effects in the presence of twofactor interactions (2FI’s) affecting a hypothetical machine operated by computer numerical control (CNC).
Our ultimate objective is to equip technical professionals with costeffective statistical tools for making breakthroughs in process efficiency and product quality.
I hope you can make it to AQC in Toronto for my talk. If not, do not fret—the proceedings will be posted to our web site and I will provide a link in a future issue of the DOE FAQ Alert.
Click http://www.statease.com/events.html for a list of appearances by StatEase professionals. We hope to see you sometime in the near future!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 5. Workshop alert: See when and where to learn about DOE—A class is coming to San Jose I will teach the Experiment Design Made Easy (EDME) workshop in San Jose, California on June 810.* You will see this and other scheduled classes at http://www.statease.com/clas_pub.html. To enroll, click the "register online" link on our web site or call StatEase 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 StatEase to teach a private class at your site. Call us to get a quote.
*PS. I was just told that the San Jose EDME may be sold out. If so, please consider going to the next one scheduled July 1315 at our home training center in Minneapolis.  Mark
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I hope you learned something from this issue. Address your general questions and comments to me at: mark@statease.com. Sincerely, Mark Mark J. Anderson, PE, CQE PS. Quote for the month—Why does physicist Freeman Dyson say that "The paradoxical feature of the laws of probability is that they make unlikely events happen unexpectedly often"?
"Littlewood was a famous mathematician... at Cambridge University.... He defined a miracle as an event that has special significance when it occurs...with a probability of one in a million...Littlewood's Law of Miracles states that in the course of any normal person's life, miracles happen at a rate of roughly one per month. The proof of the law is simple. During the time that we are awake and actively engaged in living our lives, roughly for eight hours each day, we see and hear things happening at a rate of about one per second. So the total number of events that happen to us is about thirty thousand per day, or about a million per month...The chance of a miracle is about one per million events. Therefore we should expect about one miracle to happen, on the average, every month."
 From a review by Freeman Dyson of the book "Debunked! ESP, Telekinesis, Other Pseudoscience" by Georges Charpak and Henri Broch at http://www.nybooks.com/articles/16991. I became aware of this paradox from a fascinating wordgame you can subscribe to at http://www.mootgame.com/joinlist.html. Mark
Trademarks: DesignEase, DesignExpert and StatEase are registered trademarks of StatEase, Inc. Acknowledgements to contributors: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Interested
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