Fraction of Design Space (FDS) When the goal is optimization (usuallythe case for RSM), the emphasis is onproducing a fitted surface as preciselyas possible. The response surface isdrawn by predicting the mean outcomeas a function of inputs over the regionof experimentation. How precisely thesurface can be drawn (or the mean val-ues estimated) is a function of the stan-dard error (SE) of the predicted meanresponse—the smaller the SE the bet-ter. The standard error of the predict-ed mean response at any point in thedesign space is a function of threethings:1. The experimental error (expressedas standard deviation).2. The experiment design—the num-ber of runs and their location.3. Where the point is located in thedesign space (its coordinates).

Figure 1a (see page 2) shows a 3D plotof standard error for a two-factor face-centered (alpha = 1) central compositedesign (FCD). The red dots representthe coordinates of the design points,

Stat-Teaser • News from Stat-Ease, Inc.

Workshop ScheduleA B O U T S TAT - E A S E ® S O F T W A R E , T R A I N I N G , A N D C O N S U LT I N G F O R D O EPhone 612.378.9449 Fax 612.746.2069 E-mail info@statease.com Web Site www.statease.com

September 2008 • 1

which range from minus 1 to plus 1 incoded factor units. (For two factors anFCD becomes equivalent to a full three-level factorial.) The upward (Z) axisdisplays the standard error of the pre-dicted mean expressed as a relativevalue to the actual experimental error,which remains to be determined (thinkof this as a thought experiment!). Inother words, when the standard devia-tion is known it simply becomes a mul-tiplier on the Z-axis.

The shape of the surface depends on the

FDS—A Power Tool for DesignersFDS—A Power Tool for Designersof Optimization Experimentsof Optimization Experiments

—Continued on page 2

Experiment Design Made EasySeptember 30–Oct. 2, 2008: Philadelphia, PANovember 4–6, 2008: Minneapolis, MNDecember 9–11, 2008: Dallas, TXStudy the practical aspects of design ofexperiments (DOE). Learn about simple,but powerful, two-level factorial designs.$1495 ($1195 each, 3 or more)

Response Surface Methods for Process OptimizationSeptember 23–25, 2008: Minneapolis, MNMaximize profitability by discovering optimal process settings via RSM. $1495($1195 each, 3 or more)

Mixture Design for Optimal FormulationsOctober 21–23, 2008: Minneapolis, MNFind the ideal recipes for your mixtureswith high-powered statistical tools.$1495 ($1195 each, 3 or more)

DOE for DFSS: Variation by DesignNovember 11–12, 2008: Minneapolis, MNUse DOE to create products and processesrobust to varying conditions, and tolerance analysis to assure your specifica-tions are met. A must for Design for SixSigma (DFSS). $1195 ($995 each, 3 or more)

Designed Experiments for Life SciencesNovember 18–20, 2008: Minneapolis, MNLearn how to apply DOE to Life Scienceproblems. $2050 ($1650 each, 3 or more)

PreDOE: Basic Statistics forExperimenters (Web-Based)PreDOE is an entry-level course for thosewho need to go back to the basics. Seehttp://www.statease.com/clas_pre.htmlfor more information. $95

Attendance is limited to 16. Contact Elicia* at612.746.2038 or workshops@statease.com.

*See page 4 for a profile on Elicia andher white lab, Kaylee.

©2008 Stat-Ease, Inc. All rights reserved.

We are devoting the majority of this issue to a white paper by Pat Whitcomb that detailsa major new statistical tool for evaluating alternative experiment designs. His article pro-vides the pinnacle to a two-part series on power written by Stat-Ease Consultant ShariKraber, which we published in the September and December 2007 issues of the Stat-Teaser—“No More Under-Sized Factorials” and “When Power is Too Low in FactorialDesigns;” respectively. In a sidebar asking “Can I Use Power for Mixture and RSM?,”Shari advised that “the power calculation is inappropriate for response surface (RSM)and mixture (MIX) design objectives, often reporting very low values.” Shari promised“a future article on Fraction of Design Space (FDS) graphs which are a better tool toassess the capability of RSM and MIX designs.” Here it is!

Pat Whitcomb, President

2 • September 2008

design—the location of the run pointsand their number. For example, Figure1b illustrates the impact on standarderror from replication of the four ver-tices (corners) and four axial (edge) runs—the additional eight runs cause the SEsurface to become flatter and lower.However, predictions around theperimeter of the design space still exhib-it higher standard errors than near thecenter. This is not necessarily bad,assuming the experimenter centers thedesign at the most likely point for thepotential optimum.

A contour plot of the unreplicatedFCD’s standard error is shown in Figure2a. This 2D representation will be easi-er to work with for numerical purposes.The three contours (0.43, 0.50 and 0.61)enclose 25%, 50% and 75% of the designspace. In other words 0.25 fraction ofthe design space has a relative standarderror less than 0.43, and so forth.

The fraction of design space plot isshown in Figure 2b. It displays the areaor volume of the design space having amean standard error less than or equalto a specified value. (I’ve called out theSEs for FDSs of 0.25, 0.5 and 0.75 so youcan see the connection to the contourplot.) The ratio of this volume to thetotal volume is the fraction of designspace. Thus the FDS plot is a singleplot showing the cumulative fraction ofthe design space on the x-axis (fromzero to one) versus a standard error onthe y-axis. It is a great tool to comparedesigns— look for lower (less error) andflatter (more uniform) profiles.

Sizing for PrecisionFor a given number of runs (dictated bythe experimental budget), a design willideally provide a fitted response surfacethat is precise throughout the region ofinterest. Statistically, precision isdefined as the half-width (difference“d”) of the confidence interval on the

Stat-Teaser • News from Stat-Ease, Inc.

—Continued from page 1

predicted mean value. Mathematicallythis is expressed as . As with anystatistical interval, it depends on thedegree of specified risk alpha (generallydisplayed by statisticians as the Greeksymbol α, but we express it in English as“a”), which is typically set at 5% (0.05)to provide a confidence level of 95%.

To keep things really simple, let’s workwith a one-factor linear response sur-face experiment to illustrate how to sizea design for precision. The solid centerline in Figure 3 represents the predictedmean values for the fitted model.

The curved dotted lines are the com-

-1.00

-0.50

0.00

0.50

1.00

-1.00

-0.50

0.00

0.50

1.00

0.00

0.20

0.40

0.60

0.80

1.00

A: A B : B

-1.00

-0.50

0.00

0.50

1.00

-1.00

-0.50

0.00 0.50

1.00

0.00

0.20

0.40

0.60

0.80

1.00

A: A

B: B

Figures 1a and 1b: 1a (left) represents an unreplicated face-centered design(FCD) and 1b (right) is the same design with 8 outer points replicated

-1.00 -0.50 0.00 0.50 1.00-1.00

-0.50

0.00

0.50

1.00 StdErr of Design

A: A

5

0.43

0.50

0.61

B: B

FDS Graph

Fraction of Design Space

0.00 0.25 0.50 0.75 1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.43 0.50

0.61

Std

. Err

or

Figures 2a and 2b: 2a (left) shows a contour plot of an unreplicated FCD.In 2b (right) see the fraction of design space (FDS) plot for the FCD

3

4

5

6

7

O ne Factor Experim ent

0

0.25 0.5 0.75 1 Factor A

±y dR1

Figure 3: Linear fit with confidencelimits displayed

September 2008 • 3 Stat-Teaser • News from Stat-Ease, Inc.

FDS—A Powerful Tool Continued...puter-generated confidence limits, orthe actual precision. Notice how theyflare out at the extremes of the low andhigh levels of the factor (scaled from 0to 1 in this case). The desired precisionis shown by ±d, the half-width of theconfidence interval, used to create theouter straight lines. As shown in theFCD example (Figures 1a, 1b, 2a and2b), the precision obtained depends onwhere in the design space the predictionis made. In Figure 3 you see that lessprecision is obtained near the extremes.Precision in this case is best at the center(A = 0.5) but it remains adequate in therange from 0.25 to 0.75. However, thedesired precision is not obtained for 0 to0.25 or from 0.75 to 1. In other words,only about 50% of the design space meetsthe goal for precision. The FDS graphfor this example is shown in Figure 4.

The legend for “reference Y” indicatesthat only 51% of the design space is pre-cise enough to predict the mean within±0.90 (entered for “d” in the floatingFDS graph tool).

For those of you who would like toreproduce this one-factor example FDSgraph in your Design-Expert® pro-gram (version 7.13 or later is needed),here are its parameters:

• Go to the “Response Surface” tab andchoose the “One Factor” design.• Set the factor range from 0 to 1.• Change the “Model” from its defaultof “Quadratic” to “Linear” and buildthe design.• The design should now be comprisedof two runs at each extreme (0 and 1)plus one center point (0.5) for a total of5 runs.• Go to “Evaluation, Graphs” for FDS andenter a desired precision of ±0.90 (d = 0.90).• Enter the overall standard deviation forthe response, estimated to be 0.55 (s = 0.55).

FDS GuidelinesTypically, experimenters are trying todo as few runs as possible, and yet theydesire high precision in predicting theresponse. The trick is to balance thesemissions when confronted with experi-mental error coming from variation inthe process, in the sampling and in theresponse measurement (the testing).There are no hard and fast rules forwhat’s required for the fraction of designspace based on desired precision; but thefollowing guidelines should help.

If you are mapping out an experimentalregion to explore for a significantly bet-ter response (optimization), be satisfiedwith a design that produces a FDS over80 percent. However, if the experimentis the culmination of a series of develop-ments and thus it must provide verificationof manufacturing setup, then we adviseyou achieve an FDS above 95 percent.

Oops—the one-factor example shownabove in Figure 4 falls short. What canbe done to improve precision in a caselike this? You could try to manageexpectations by negotiating an increasein the difference “d,” in other words behappy with less precise predictions.This might not very popular with yourmanagement and clients. Another tackmight be to reduce the noise (“s”).

Unfortunately this is likely to be sys-tematic—a state of nature—and thusnot amenable to improvement.

If you are willing to take on more risk,you could increase the alpha (“a”)—thatwould help for FDS.

However, assuming you have somebudget in reserve, perhaps the firstthing to do is try adding more runs andsee how much this improves your FDS.For example, in this case, by adding onlytwo more runs at the two extremes, theFDS improves to 100 percent as shownin Figure 5. The augmented designobtains the desired precision throughoutthe entire design space. Life is good!

ConclusionAt the early stages of experimentation,screening and characterization, wherefactorials are the design of choice, theemphasis is on identifying main effectsand interactions. For this purpose,power is an ideal metric to evaluatedesign suitability. However, when thegoal is optimization (usually the case forRSM), the emphasis shifts to creating afitted surface within a desired precision.Then fraction of design space (FDS)becomes a powerful tool for gradingdesign suitability.—Pat Whitcomb, pat@statease.com

Figure 4: FDS graphs for one-factorresponse surface experiment

Figure 5: The FDS is greatly improvedafter adding only two more runs

PresortedStandard

U.S. POSTAGE PAIDMinneapolis, MNPermit No. 28684

Address Service Requested

Stat-Ease, Inc., Hennepin SquareSuite 480, 2021 E. Hennepin Ave.Minneapolis, MN 55413-2726

09/0

8

Employee Profile — Elicia BechardStat-Ease is pleased to introduce EliciaBechard, Workshop Coordinator, whohas been with us for a little more than ayear. Her cheerful, sunny presence, aswell as great organizational and inter-personal skills, have made her a wel-come addition at Stat-Ease.

Education and ExperienceElicia is a graduate of the University ofMinnesota, Morris with a BA degree inPsychology and Human Services, and aminor in French. Prior to coming toStat-Ease her jobs included customer ser-vice, and 8 years at a local Methodistchurch as a Youth Director for 6th gradethrough college-age students.

HobbiesSome of Elicia’s favorite things includegardening, making jewelry and othercrafts, reading books, cooking, andbeing an Auntie.

Elicia Bechard, Workshop Coordinator

Interesting Facts Northern Minnesota has a large wilder-ness area called the Boundary WatersCanoe Area (BWCA). Elicia loves theoutdoors and has been there nine times.She once jumped off a 30+ foot cliff—which cured her of the desire of everdoing it again!

She and her husband, Kevin, recentlyadopted a white lab puppy, Kaylee, incelebration of their first weddinganniversary. Kaylee has been very busy“training” them!

Workshop TipsIf you are interested in signing up for aStat-Ease workshop, Elicia advises youto register early, communicate anydietary restrictions ahead of time, con-tact her for quantity discounts at workshops@statease.com, and don’t forget to e-mail stathelp@statease.comafter class if you have any questions!

Kaylee at 2months old