PdPiiSiFilPower and Precision to Size Factorial, RSM and ...

How many runs do I need? How to use P d P i i Si F i lPower and Precision to Size Factorial, RSM and Mixture Designs

*Presentation is posted at www.statease.com/webinar.html

Please make use of the question box and my colleague will answer yourPlease make use of the question box and my colleague will answer your questions (if possible) as I continue to present. To avoid disrupting the Voice over Internet Protocol (VoIP) system, I will mute all. If we do not get to you, please accept my apology in advance. Then, I’d appreciate you sending me an email to [email protected] after the talk so we can answer your

By Brooks Henderson

an email to [email protected] after the talk so we can answer your question(s) ‘off-line.’ -- Brooks

MS Materials Science, CQEStat-Ease, Inc., Minneapolis, MN

[email protected] Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS 11

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Getting to Know YouAudience Expectations and DOE knowledgep g

First, a few quick polls to see who we have in the audience.

Open up for polling

Intermediate webinar:

Expectation of familiarity with DOE and preferably Design-Expert ® Software

Familiarity with types of designs: Factorial RSM Mixture Familiarity with types of designs: Factorial, RSM, Mixture

Otherwise, just focus on sizing tools and backfill with our other webinars and resources for further details on designs.g

How Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS 22©20

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How Many Runs Do I need?Introduction

Mission: How many runs do I really need? How to size all types of designs from an engineer’s perspective How to size all types of designs from an engineer s perspective. First time presenting and recording the full gamut of Design Types with

particular focus on sizing.

P t bi t i f i i d i i d i t **Past webinars posted at www.statease.com/webinar.html

Past webinars on topic of sizing designs or various design types* “Quality by Design (QbD) Space for Pharmaceuticals and Beyond” by Wayne

Adams in June/July, 2013 (covered FDS for TI, QbD)“H G S d i h D i E ® S f ” b Sh i i F b 2013 “How to Get Started with Design-Expert® Software” by Shari in Feb, 2013 (covered Factorial Design basics)

“Basics of Response Surface Methodology (RSM) for Process Optimization, Part 1” by Shari Kraber in 2011 (covered RSM basics)Part 1 by Shari Kraber in 2011. (covered RSM basics).

“How to Plan and Analyze a Verification DOE” by Shari Kraber in 2008 (covered power and verification studies)...Slides only

“Sizing Mixture (RSM) Designs for Adequate Precision via Fraction of Design Sizing Mixture (RSM) Designs for Adequate Precision via Fraction of Design Space (FDS)” by Pat Whitcomb in 2007 (Adv. details of FDS)...Slides only

How Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS33

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Agenda: How to Size DOEs with FDS

Using power to size factorial designs Precision in place of power

Introduce FDS Sizing designs for precision - confidence intervals (CIs) Sizing designs for precision confidence intervals (CIs)

Two component mixture Three component constrained mixture Illustrative Example Optimal Flare Illustrative Example – Optimal Flare

• Adding Intervals to graphical optimization• Sizing for Tolerance intervals

Summary


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Agenda: How to Size DOEs with FDS


Introduce FDS Sizing designs for precision (CIs) Sizing designs for precision (CIs)



Summary


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Sizing Factorial Designs

KnownFactors

UnknownFactors

Screening KnownFactors

UnknownFactors

ScreeningDuring screening and characterization the emphasis is

Screening Trivialmany

Vital fewCharacterization



characterization, the emphasis is on identifying factor effects.

What are the important design Factor effects

and interactions

Curvature?no

CharacterizationFactor effects

and interactions

Curvature?no

Characterizationfactors?

For this purpose power is an ideal metric to evaluate design

yes

ResponseSurfacemethods

Curvature?

Optimizationyes


Curvature?

Optimization

gsuitability.

methods

Confirm? Backup

Celebrate!

no

yes

Verification

methods

Confirm? Backup

Celebrate!

no

yes

Verification

yesyes


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Factorial Design – Power23 Full Factorial Δ=2 and σ=1

1212

One FactorOne Factor12

1111

1212

8

9

10

11

12

R

1

1010R1

R1

-1.00

-0.50

0.00

0.50

1.00

-1.00

-0.50

0.00

0.50

1.00

8

A: A B: B Δ

99

--1.001.00 --0.500.50 0.000.00 0.500.50 1.001.00

88

1.001.00 0.500.50 0.000.00 0.500.50 1.001.00

A: AA: AHow Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS 77©20

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Factorial Design – Power23 Full Factorial Δ=2 and σ=1

Power is reported at a 5.0% alpha level to detect the specified signal/noise ratio.Recommended power is at least 80%.

R1Signal (delta) = 2.00 Noise (sigma) = 1.00 Signal/Noise (delta/sigma) = 2.00

A B C57 2 % 57 2 % 57 2 %57.2 % 57.2 % 57.2 %


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Factorial Design – Power – Adding Runs2 Replicates of 23 Full Factorial Δ=2 and σ=1p

Power is reported at a 5.0% alpha level to detect the specified signal/noise ratio.Recommended power is at least 80%.

R1Signal (delta) = 2.00 Noise (sigma) = 1.00 Signal/Noise (delta/sigma) = 2.00

A B C95 6 % 95 6 % 95 6 %95.6 % 95.6 % 95.6 %


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Factorial Design-PowerDetermining Δy and sg y

1. Determine ΔyH bi f ff t d d t ?• How big of an effect do you need to see?

• Smaller effects are harder to find• We will size design to capture this effect or bigger, given noise

2. Determine s• How much noise will the response have? Estimate... usually from

historical datahistorical data. • More noise means lower power and effect harder to find


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Conclusions

Factorial DOE

During screening and characterization (factorials) emphasis is on identifying p y gfactor effects.

What are the important design factors?factors?

For this purpose power is an ideal metric to evaluate d i it bilitdesign suitability.

Can power be used for Mixture or RSM designs?Can power be used for Mixture or RSM designs?

How Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS

Can power be used for Mixture or RSM designs?Can power be used for Mixture or RSM designs?

1111©20

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How to Size DOEs with FDS





Summary


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Sizing RSM or Mixture Designs

KnownFactors

UnknownFactors

Screening KnownFactors

UnknownFactors

Screening




Vital fewCharacterizationWhen the goal is optimi ationWhen the goal is optimi ationFactor effects

and interactions

Curvature?no

CharacterizationFactor effects

and interactions

Curvature?no

CharacterizationWhen the goal is optimization When the goal is optimization emphasis is emphasis is on the on the fitted surface.fitted surface.

How well does the surface How well does the surface

yes


Curvature?

Optimizationyes


Curvature?

Optimization

represent true behavior?represent true behavior?

For this purpose precision (FDS) For this purpose precision (FDS) isis aa good metricgood metric for evaluatingfor evaluating methods

Confirm? Backup

Celebrate!

no

yes

Verification

methods

Confirm? Backup

Celebrate!

no

yes

Verification

is is a a good metric good metric for evaluating for evaluating design design suitability.suitability.(Assuming model adequacy; (Assuming model adequacy; i.e. insignificant lack of fit.)i.e. insignificant lack of fit.)

yesyes


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Design Properties

1. Estimate the designed for polynomial well.

2 Gi e s fficient information to allo a test for lack of fit2. Give sufficient information to allow a test for lack of fit.

Have more unique design points than coefficients in the model.

Provide an estimate of “pure” errorProvide an estimate of pure error.

3. Remain insensitive to outliers, influential values and bias from model misspecification.

4. Be robust to errors in control of the component levels.

5. Provide a check on model assumptions, e.g., normality of errors.6 Generate useful information throughout the region of interest6. Generate useful information throughout the region of interest,

i.e., provide a good distribution of standard error:

7. Do not contain an excessively large number of trials.( ) 2ˆVar Y σ


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Introducing FDSFraction of Design Space – what is precision?g p p

Produce Useful Predictions [confidence intervals (CI), etc.]:

( )= ± = ±CI Y t s Y ds( )α± ±Y,df2

d is the half width of the interval

CI Y t s Y dYs


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Introducing FDSFraction of Design Space – what is precision?g p p

The size of the CI depends on the Std. Error ( )of the DOEYs

0.600

0.800

1.000

sign

0.000

0.200

0.400

td E

rror

of D

es

0.50

1.00

St

-1.00

-0.50

0.00

-1.00 -0.50

0.00 0.50

1.00

A: A

B: B


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FDSFraction of Design Spaceg p

Fraction of Design Space: Calculates the volume of the design space having a

Standard Error, ,less than or equal to a specified value

Ysvalue. The ratio of this volume to the total volume of the design

volume is the fraction of design space.g Produces a single plot showing the cumulative fraction

of the design space on the x-axis (from zero to one) th th isversus the on the y-axis.Ys


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FDS PlotFraction of Design Space

ProgramProgramdetailsdetailsg p

1. Pick random points in the design space.

2. Calculate the standard error of the expected value

( )01y T T

SEX X

−

3. Plot the standard error as a fraction of the design space.

( )0y T T0 0x X X x

s=

FDS G hFDS G hFDS GraphFDS Graph

Mea

nM

ean

0.600.600.700.700.800.800.900.901.001.00

Std

Err

MS

tdE

rr M

0.000.000.100.100.200.200.300.300.400.400.500.50

Fraction of Design SpaceFraction of Design Space0.000.00 0.250.25 0.500.50 0.750.75 1.001.00


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FDS – StdErrTwo-Factor RSM Designg

1.001.00StdErr of DesignStdErr of Design FDS GraphFDS Graph

0.500.50

1.001.00

0 700 70

0.800.80

0.900.90

1.001.00

0.000.00

B: B

B: B 55

Std

Err

Mea

nS

tdE

rr M

ean

0.400.40

0.500.50

0.600.60

0.700.70

0.430.430.500.50

0.610.61

--0.500.50 0.430.43

0.500.50

0 610 61

SS0.100.10

0.200.20

0.300.30

--1.001.00 --0.500.50 0.000.00 0.500.50 1.001.00

--1.001.00

A: AA: A

0.610.61

Fraction of Design SpaceFraction of Design Space

0.000.00 0.250.25 0.500.50 0.750.75 1.001.00

0.000.00


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Summary


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Two Component MixtureLinear Model

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Two Component MixTwo Component Mix

66

77

22

55R1

R1

44 22

33

00 0.250.25 0.50.5 0.750.75 11Actual AActual A

11 0.750.75 0.50.5 0.250.25 00Actual BActual B


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Two Component MixTwo Component MixThe solid center line is the fitted model;The solid center line is the fitted model;is the expected value or mean is the expected value or mean y

66

ppprediction.prediction.

The curved dotted lines are the The curved dotted lines are the computer generated confidence computer generated confidence

y

55R1

R1 ±y d

gglimits, or the actual precision.limits, or the actual precision.

d d Is the halfIs the half--width of the desired width of the desired confidence interval, or the desired confidence interval, or the desired

Note: The actual precisionNote: The actual precision

44

precision. It is used to create the precision. It is used to create the outer straight lines.outer straight lines.

Note: The actual precision Note: The actual precision of the fitted value depends of the fitted value depends on where we are predicting.on where we are predicting.

33

00 0.250.25 0.50.5 0.750.75 11Actual AActual A

11 0.750.75 0.50.5 0.250.25 00Actual BActual B


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Build the two component mixture:

Choose “Simplex Lattice”

22

22

Change the “Order” to “Linear” and

22

the “Replicates” to “2”


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Click on “Click on “Evaluate” and then “” and then “Graphs”:”:


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Want a linear surface to represent the true response value within ± 0 64 with 95% confidencevalue within ± 0.64 with 95% confidence.

The overall standard deviation for this response is 0.55.

Enter:Enter:

d d = 0.64= 0.64

s = 0.55s = 0.55

a (a (αα) = 1 ) = 1 –– 0.95 = 0.050.95 = 0.05


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Only 53% of the design space Only 53% of the design space is precise enough to predict is precise enough to predict the mean within the mean within ±± 0.64.0.64.


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Two Component MixTwo Component Mix

53% of the design space 53% of the design space has the desired precision, has the desired precision, i.e. is inside the solid i.e. is inside the solid straight (red) linesstraight (red) lines

66

77

53%53%

straight (red) lines.straight (red) lines.

55R1

R1 ±y d

44

33

00 0.250.25 0.50.5 0.750.75 11Actual AActual A

11 0.750.75 0.50.5 0.250.25 00Actual BActual B


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Sizing for PrecisionFDS ≥ ?

How good is good enough? Rules of thumb:F l ti t FDS ≥ 80%• For exploration want FDS ≥ 80%

• For verification want FDS ≥ 90% or higher!

Wh t b d t i i i ?What can be done to improve precision?• Manage expectations; i.e. increase d• Decrease noise; i.e. decrease sec ease o se; e dec ease s• Increase risk of Type I error; i.e. increase alpha• Increase the number of runs in the design


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Summary


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Three component constrained MixtureDesign for Quadratic Modelg

0.0 ≤ A ≤ 1.0A: AA: A1.0001.000

0.0 ≤ B ≤ 1.00.0 ≤ C ≤ 0.1

Total = 1 0Total 1.0

0.0000.000 0.0000.000

B: BB: B1.0001.000

C: CC: C1.0001.0000.0000.000

StdErr of DesignStdErr of Design

How Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS

StdErr of DesignStdErr of Design

3030©20

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Define PrecisionHalf Width of the Confidence Interval

Goal: Generate useful information throughout the region of interest?

Question: Will predictions, using the quadratic model from this design, be precise enough for our purposes?

T k th t th i i fi it b f t To know the truth requires an infinite number of runs; most likely this will exceed our budget.

So the question is how precisely do we need to estimate theSo the question is how precisely do we need to estimate the response?

The trade off is more precision requires more runs.


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Constrained MixtureQuadratic Model (1 replicate)( p )

Determine d:

• Want quadratic surface to represent the true response value within ±10 with 95% confidence.

Determine s:

• The standard deviation for this response is 7.8.p


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Constrained MixtureQuadratic Model (1 replicate)( p )

Only 58% of the design space hasOnly 58% of the design space has StdErrStdErr ≤≤ 0.540.54

d= d= 1010ss = 7.8= 7.8

0 050 05αα = 0.05= 0.05


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Constrained MixtureDX Tip – Rebuild the designp g

FDS is low, try replicating the whole design.

In Design-Expert, you can do this by rebuilding the design

Go to File->New Design

Click “No” to “Save changes?”, but click Yes to “Use previous design info?”

You will be taken back to the Design build screen where you entered your factors and everything is pre-populated.y y g p p p


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Constrained MixtureQuadratic Model (Try 2 replicates)( y p )

100% of the design space has100% of the design space has StdErrStdErr ≤≤ 0.610.61

Now, you can rebuild Now, you can rebuild the same design with the same design with more runs.more runs.

Let’s replicate it and Let’s replicate it and check the FDScheck the FDS


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Precision Depends On

The size of the confidence interval half-width (d):

A larger half-width (d) increases the FDS.

The size of the experimental error s:

A smaller s increases the FDS.

The a risk chosen:

A larger a increases the FDS.

Choose design appropriate to the problem:

Size the design for the precision required.


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• Adding Intervals to graphical optimization

• Sizing for Tolerance intervals Summary


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Optimal Flare4 Component IV Optimal Mix DOEp p

In manufacturing a particular type of flare the chemical constituents are: A magnesium B sodium nitrate C strontium nitrate and D binderA - magnesium, B - sodium nitrate, C - strontium nitrate, and D - binder. Experience dictates the following constraints:

0.40 ≤ A ≤ 0.60

0.10 ≤ B ≤ 0.50

0.10 ≤ C ≤ 0.50

0.03 ≤ D ≤ 0.08

Total = 1.00

The problem is to find the blend (A, B, C, D) which gives the maximum illumination (light), measured in candles.

Experience suggests a special cubic model for illumination Experience suggests a special cubic model for illumination.

Mix section 4Mix section 4 3838©20

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Optimal Flare

1. Chose an Optimal mixture and enter the four components and th i t i t d t t l f 1their constraints and a total of 1:


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Optimal Flare

2. Use “Best” to build an “IV-optimal” design and change the d f lt “Q d ti ” t “S i l C bi ” d ldefault “Quadratic” to a “Special Cubic” model:

3. One response “illumination” with units of “candles”.

Hint: Be sure to change Quadratic to Special Cubic model.


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Optimal FlareSizing for precision with FDS

Does the design just built have adequate precision?

g p

Determine d:

• Want FDS ≥ 80% with precision of ± 6

Determine s:

• Standard deviation for illumination (estimated from SPC d t ) i 4 5data) is 4.5


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Only Only 54% 54% of the design space hasof the design space has StdErrStdErr ≤≤ 0.540.54

g p

Design-Expert® Software

Min Std Error Mean: 0.398Avg Std Error Mean: 0.638Max Std Error Mean: 1.503ConstrainedPoints = 50000t(0.05/2,10) = 2.22814

FDS Graph2.000

t(0.05/2,10) 2.22814d = 6, s = 4.5FDS = 0.54Std Error Mean = 0.598

rror M

ean

1.000

1.500S

td E

r

0.500d= 6d= 6ss = = 4.54.5

0.00 0.20 0.40 0.60 0.80 1.00

0.000

Mix section 4Mix section 4 4242

Fraction of Design Space

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To improve precision what type of points should be added; model, l k f fit li t ?

g p

lack of fit or replicates?

Answer: Increasing model points is the most effective way to g p yimprove precision. Assuming there are already 5 each for lack of fit and replicates, then increase the number of model points.model points.


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Rebuild your design and increase the number of model points until hi FDS ≥ 80% f d 6 d 4 5

g p

you achieve a FDS ≥ 80% for d = 6 and s = 4.5.

When rebuilding your design say:• “No” to “Save changes ?”• No to Save changes…?• “Yes” to “Use previous design info?”

The default was 14 model points; so lets try 4 extra runs for e de au as ode po s; so e s y e a u s oa total of 18 model points.

Design-Expert® Software

Min Std Error Mean: 0.374Avg Std Error Mean: 0.549Max Std Error Mean: 0.954ConstrainedPoints = 50000

FDS Graph1.000

t(0.05/2,14) = 2.14479d = 6, s = 4.5FDS = 0.81Std Error Mean = 0.622

Std

Erro

r Mea

n

0.200

0.400

0.600

0.800

Mix section 4Mix section 4 4444Fraction of Design Space

0.00 0.20 0.40 0.60 0.80 1.00

0.000

0.200

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Optimal FlareAnalyze and Optimizey p

ANOVA:ANOVA:


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Optimal FlareNumerical Optimizationp

We designed for We designed for d = 6d = 6, so got a , so got a

We found the optimum!We found the optimum!

The CI is: The CI is:

gg gglittle better than we desiredlittle better than we desired

267.2 267.2 ±± 5.55.5


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Optimal FlareGraphical Optimizationp p

Now, let’s see if there is a Now, let’s see if there is a region that can meet specs.region that can meet specs.

Th ifi tiTh ifi tiThe specifications are:The specifications are:LightLight ≥ 260 candles≥ 260 candles

Using Graphical optimization Using Graphical optimization with a goal of with a goal of Light ≥ 260Light ≥ 260provides the answer.provides the answer.


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Uncertainty is a BIG ProblemSpecifications as Boundsp

If the flare manufacturing process is t d th b d th 50%operated on the boundary, then 50%

of the flares produced fall outside the specification.

SpecificationSpecification


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Optimal FlareGraphical Optimization - Adding CIsp p g

Adding Adding Confidence IntervalsConfidence Intervals to to the optimization provides a bufferthe optimization provides a bufferthe optimization provides a bufferthe optimization provides a buffer


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Optimal FlareGraphical Optimization -Adding Tolerance Intervals (Tis)Adding Tolerance Intervals (Tis)

What about Tolerance Intervals?To ensure a certain proportion of the population (say 99%)

will meet specs, use tolerance intervalsCaution:Caution:

• Tolerance intervals are wider than confidence intervals• Before running the design, be sure to size for

T l i t l b h i th d d tiTolerance intervals by changing the dropdown option.


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Optimal FlareGraphical Optimization -Adding Tolerance Intervals (Tis)Adding Tolerance Intervals (Tis)

Caution: Caution: Tolerance Intervals are wider than Tolerance Intervals are wider than Confidence Intervals You may wipe out yourConfidence Intervals You may wipe out yourConfidence Intervals. You may wipe out your Confidence Intervals. You may wipe out your design space, if not properly planned for.design space, if not properly planned for.Add Tolerance IntervalsAdd Tolerance Intervals

The design space is all The design space is all grayed out after adding grayed out after adding TiTi f 99% f thf 99% f thTisTis for 99% of the for 99% of the population to the criteria.population to the criteria.


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Optimal FlareSummaryy

For Exploration, size your design with FDS using confidence i t l (d f lt)intervals (default)

Use Graphical Optimization to find a region that meets specs

Aft fi di i b t b k ff f th b d iAfter finding a region, be sure to back off from the boundaries by adding a Confidence Interval (CI) or Tolerance Interval (TI)

TIs require more runs than CIs, so be sure to choose the right s equ e o e u s a C s, so be su e o c oose e gone when sizing via the FDS graph.

Use CIs for most exploration

Use TIs for verifying specs and finding a region that meets specs.


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Design Sizing Considerations

Factorial DOE Mixture Design andR S f M h dFactorial DOE Response Surface Methods

During screening and characterization (factorials)

When the goal is optimization (usually the case for mixture ( )

emphasis is on identifying factor effects.

What are the important design

( ydesign & RSM) emphasis is on the fitted surface.

How well does the surfaceWhat are the important design factors?

For this purpose power is an id l t i t l t

How well does the surface represent true behavior?

For this purpose precision (FDS) i d t i tideal metric to evaluate

design suitability.(FDS) is a good metric to evaluate design suitability.


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Summary


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How to Size DOEs with FDSSummaryy

1. Size your DOE appropriately for its type and purpose:Factorial Designs size via Power Factorial Designs – size via Power.The power to detect each individual effect/coefficient is key

RSM Functional design – size via confidence intervals.A fid i t l d d i t i fidA confidence interval pads our design space to give us confidence that the process mean is within boundaries.

For QbD or to verify specifications – size via tolerance intervals.A t l i t l i fid th t t t d ti f thA tolerance interval give us confidence that a stated portion of the population is within specifications.

2. For Power (factorial designs), define: Δy : difference you want the DOE to detect s: standard deviation excluding factor effects (historical data)


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How to Size DOEs with FDSSummaryy

3. For FDS (RSM and Mixture), define:d: half width of the desired interval (confidence or tolerance) d: half-width of the desired interval (confidence or tolerance)

s: same as above4. Tolerance intervals require a larger design (sample size).q g g ( p )

Now Now youyou can answer the question...How can answer the question...How many runs do we really need?many runs do we really need?


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How to get helpg p

Search publications posted at www.statease.com.

In Stat-Ease software press for Screen Tips, view reports in annotated mode look for context sensitive Helpreports in annotated mode, look for context-sensitive Help (right-click) or search the main Help system.

Explore Experiment Design Forum http://forum.statease.comp p g p

E-mail [email protected] for answers from Stat-Ease’s staff of statistical consultants.

Call 612.378.9449 and ask for “statistical help.”


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Practical Paperbacks on DOE*by Mark Anderson and Pat Whitcomby

User Review of DOE Simplified:

As an engineer (just beginning self study on the topic of DOE) I found this book very useful. The authors provide practical insight that I was unable to find in other DOE or statistics books. This is not a book for advanced statisticians, however, it is a great book for someone trying to understand and apply the principles of DOEa great book for someone trying to understand and apply the principles of DOE.

* Published by Productivity Press, New York.How Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS 5858

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Statistics Made Easy®

For all the new features in v8 of Design-Expert software, see www.statease.com/dx8descr.html

Best of luck for your experimenting!experimenting!

Thanks for listening!-- Brooks

Brooks Henderson, Brooks Henderson, MS Materials Science, CQEMS Materials Science, CQE, Q, Q

StatStat--Ease, Inc.Ease, [email protected]@statease.com

*Pdf of this Powerpoint presentation posted at www.statease.com/webinar.html.f f p p p /For future webinars, subscribe to DOE FAQ Alert at www.statease.com/doealert.html. How Many Runs Do I need? Sizing a DOE with FDSHow Many Runs Do I need? Sizing a DOE with FDS 5959

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ReferencesFDS, and TI,

1. Alyaa R. Zahran, Christine M. Anderson-Cook and Raymond H. Myers, “Fraction of Design Space to Assess Prediction”, Journal of Quality Technology, Vol. 35, No. 4, g p , Q y gy, , ,October 2003.

2. Heidi B. Goldfarb, Christine M. Anderson-Cook, Connie M. Borror and Douglas C. Montgomery, “Fraction of Design Space plots for Assessing Mixture and Mixture-P D i ” J l f Q lit T h l V l 36 N 2 O t b 2004Process Designs”, Journal of Quality Technology, Vol. 36, No. 2, October 2004.

3. Steven DE Gryze, Ivan Langhans and Martina Vandebroek, “Using the Intervals for Prediction: A Tutorial on Tolerance Intervals for Ordinary Least-Squares Regression”, Chemometrics and Intelligent Laboratory Systems 87 (2007) 147 – 154Chemometrics and Intelligent Laboratory Systems, 87 (2007) 147 154.

4. Gerald J. Hahn and William Q. Meeker, Statistical Intervals; A Guide for Practitioners, (1991) John Wiley and Sons, Inc,

5 Gary Oehlert and Patrick Whitcomb (2001) Sizing Fixed Effects for Computing Power in5. Gary Oehlert and Patrick Whitcomb (2001), Sizing Fixed Effects for Computing Power in Experimental Designs, Quality and Reliability Engineering International, July 27, 2001.


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Additional References

[1] Gary W. Oehlert (2000), A First Course in Design and Analysis of Experiments, W. H. Freeman and Company.

[2] Douglas C. Montgomery (2005), 6th edition, Design and Analysis of Experiments, John Wiley and Sons, Inc.

[3] Alyaa R. Zahran, Christine M. Anderson-Cook and Raymond H. Myers, Fraction of Design Space to Assess Prediction, Journal of Quality Technology, Vol. 35, No. 4, October 2003

[4] Raymond H. Myers and Douglas C. Montgomery (2002), 2nd edition, Response Surface Methodology, John Wiley and Sons, Inc.

[5] George E. P. Box and Norman R. Draper (1987), Empirical Model-Building and Response Surfaces, John Wiley and Sons, Inc.

[6] John A. Cornell (2002), 3rd edition, Experiments with Mixtures, John Wiley and Sons, Inc.[7] Wendell F. Smith (2005), Experimental Design for Formulation, ASA-SIAM Series on

Statistics and Applied probability.


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