Design of Experiments

Apr 24, 2023

(c) 2001, Ron S. Kenett, (c) 2001, Ron S. Kenett, Ph.D.Ph.D. 1

Design of Experiments

Instructor: Ron S. KenettEmail: [email protected]

Course Website: www.kpa.co.il/biostatCourse textbook: MODERN INDUSTRIAL STATISTICS,

Kenett and Zacks, Duxbury Press, 1998

mailto:[email protected]

http://www.kpa.co.il/biostat

Apr 24, 2023


Course Syllabus

•Understanding Variability•Variability in Several Dimensions•Basic Models of Probability•Sampling for Estimation of Population Quantities•Parametric Statistical Inference•Computer Intensive Techniques•Multiple Linear Regression•Statistical Process Control•Design of Experiments

Apr 24, 2023


DOE Is a Problem-Solving Tool

The color is too saturated...

When you have a serious problem...

… you need a good problem solving tool!

We get banding when...

The order wasn’t delivered

on time...

Apr 24, 2023


• Which factors have the strongest effects?• Do any of the factors interact?• Which factor settings give the best results?

DOE Helps Answer Key Questions

The media jams when we

print...

Lubricant?

Exit angle?

Spring tension?

Apr 24, 2023


Process, Responses, and Factors -1

Factors

Silver laydown, Finish time…

Time,Catalyst…

Transport speed,Capture lens...

Responses

Speed,Contrast

Yield,Purity

Image resolution,Banding

FilmBuilding

ChemicalProcess

Digital Imaging

Apr 24, 2023


Factors

Compensation plan, Sales training

Method of shipping, Order entry method

Product positioning,Price

Responses

Sales revenue, Volume of new sales

Shipping cost, Inventory level

Trial purchase,Share of market

Sales

Supply Chain

Product Develop.

Process, Responses, and Factors -2

Apr 24, 2023


Screen: Which are the biggest factors?Characterize: What is the relationship between

responses and factors?Optimize: What is the best setting of the factors?Verify: Are the results repeatable?

Ask a Series of Questions

Question Experiment(E) Q E Q E Q E (Q)

Answer A A A

Apr 24, 2023


Cycle of ExperimentationCOLLECTObserveCode

PLANQuestionsDesignScope

PRESENTAnswer QuestionsGraphicallyMathematicallyState UncertaintyRecommend

ANALYZEPlotPlotPlotModelConclude

Four Basic Steps of DOE

Apr 24, 2023


A Serious Problem...

I want my car to go

fast … like that one!

Apr 24, 2023


What Factors Affect the Speed?

Key Factor is: ___________________

Yes

Air Holes

NoSlow

Fast

Shape

Apr 24, 2023


Key Factor is: _______________________________

Yes

Air Holes

NoSlow

FastSlow

Shape

Effect of Air Holes

Apr 24, 2023


DOE Balanced Effects

Key Factor is: _______________________________

Yes

Air Holes

NoSlow

FastSlow

Shape

Slow

Apr 24, 2023


What factors affect flight time?

What factors affect the quality of flight?

The Helicopter Company

Apr 24, 2023


Length

Width

Factors

Design Wing WingLength Width

1 2.5 1.02 3.5 1.03 2.5 1.54 3.5 1.5

Response: Flight Time

Your Experiment: Plan

Apr 24, 2023


Design # Repeat Test Wing Wing Flight Quality Flight Order Length Width Time of Flight

11223344

12121212

2.52.53.53.52.52.53.53.5

1.01.01.01.01.51.51.51.5

Your Experiment: Collect

Apr 24, 2023


Calculate averages for each design

Wing Width1.0 1.5

(1)

(2)

(3)

(4)

2.5

3.5

WingLength

Avg. ___ Avg. ___

Avg. ___

Avg. ___

Your Experiment: Analyze

Apr 24, 2023


Your Experiment: Analyze

Plot the data (one line for each Wing Width)

Time

2.5Wing Length

3.5

Apr 24, 2023


Your Experiment: Present

Present Answer the original question:

Which factor most affects flight time?

Graphically Numerically

Recommendations:

Apr 24, 2023


What about Quality of Flight?

Is quality of flight related to flight time, or rotational speed, a combination, or something else entirely?

Does everyone think about this response in the same way?

What are the implications for analysis?

Apr 24, 2023


Health Imaging Customer Preference Study• Conventional thinking:

Low noise a customer requirement

• Some data suggested thathigh resolution (MTF) was larger driver

• Experiment designed to identify key driver

Which Is the Key Driver?

MTF

Hi

Low

Low HiNoise

1

4

?

?

Apr 24, 2023


X2

X1

“Homerun”

X2

X1

“Shotgun”

X2

X1

“Factorial”

How Many Experiments ?

X2

X1

“1-at-a-time”

Apr 24, 2023


?Typical Questions

Full factorial experiments can require many runs

The problem is greater if the response is nonlinear

Talk with an expert to decide how the experiment should be structured

Full Factorial?

FractionalFactorial?

Linear Response?

Curvature?

Apr 24, 2023


A A A A A B B B B BA B

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10 11 12

Is B better than AIs B better than A? ?

Apr 24, 2023


-3

-2

-1

0

1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11 12

Split Lot DesignSplit Lot DesignA

B

A

B

A

B

A

B

A

B

A

B

Apr 24, 2023


Factors and Levels

Factors Paste Dielectric Oven

Levels-1 +1-1 +1-1 +1

Apr 24, 2023


Paste DielectOven Response-1 -1 -1 1191 -1 -1 44-1 1 -1 2321 1 -1 135-1 -1 1 111 -1 1 34-1 1 1 451 1 1 80

44

34

135

80

119

11

232

45

-1 1

Paste

Dielect

Oven-1

1

-1

1

Cube Plot - Means for Response

23 Full Factorial Design

44

34

135

80

119

11

232

45

-1 1

Paste

Dielect

Oven-1

1

-1

1


Apr 24, 2023


OvenDielectPaste

130

110

90

70

50

Resp

onse

Main Effects for Response

44

34

135

80

119

11

232

45

-1 1

Paste

Dielect

Oven-1

1

-1

1


Main Effects PlotMain Effects Plot

Apr 24, 2023


Oven

Dielect

Paste

1

-1

1

-1

Interaction Plot for Response

44

34

135

80

119

11

232

45

-1 1

Paste

Dielect

Oven-1

1

-1

1


Interaction PlotsInteraction Plots

Apr 24, 2023


C: Initial Gas Volume (m3)B: Piston Surface Area (m2)A: Piston Weight (Kg)

D: Spring Coefficient (N/m)E: Atmospheric Pressure (N/m2)F: Ambient Temperature (0K)G: Gas Temperature (0K)


Apr 24, 2023


0.64600.6500

0.65700.6540

0.65900.6570

0.66200.6580

1.24701.2350

1.27601.2480

1.30101.2880

1.29901.2840

0.19200.2150

0.20600.1880

0.22100.1990

0.19300.1920

0.33600.3850

0.33200.3220

0.37800.3550

0.36300.3530

0.68100.6780

0.68100.6820

0.68100.6820

0.68300.6800

1.46801.4590

1.46801.4790

1.49201.4720

1.48101.5040

0.46000.4690

0.44000.4250

0.43500.4120

0.41200.4460

0.90300.8280

0.96400.8360

0.86700.9380

0.94500.9730

0.46400.4610

0.46400.4600

0.46600.4670

0.46600.4650

0.91100.8950

0.89600.8970

0.93000.9220

0.90600.9260

0.13900.1310

0.13300.1480

0.14900.1690

0.14400.1490

0.23700.2340

0.23800.2400

0.24700.2260

0.22200.2720

0.48200.4810

0.48100.4810

0.48300.4820

0.48100.4830

1.03401.0360

1.04001.0270

1.04501.0570

1.05101.0370

0.33900.3200

0.32900.3170

0.27900.3180

0.33000.3200

0.69300.5790

0.65600.6780

0.63900.5880

0.65700.6500

G

F

E

D

CB

A 1-1

1-1 1-1

1

-1

1

-1

1

-1 1

-1

Cube Plot - Means for Average


Apr 24, 2023


GFEDCBA

0.8

0.7

0.6

0.5

0.4

Avera

geMain Effects Plot - Means for Average

A: Piston Weight

(Kg)

B: Piston Surface Area

(m2)

C: Initial Gas

Volume (m3)

D: Spring Coefficient

(N/m)

E: Atmospheric

Pressure (N/m2)

F: Ambient Temperature

(0K)

G: Gas Temperature

(0K)

128

G

F

E

D

C

B

A

1-1

1-1

1-1

1-1

1-1

1-1

Interaction Plot for Average


Apr 24, 2023


-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

.001

.01

.05

.20

.50

.80

.95

.99

.999Pr

obab

ility

Eff_AvgAverage: 0.0198583Std Dev: 0.203497N of data: 15

Anderson-Darling Normality TestA-Squared: 0.480p-value: 0.199

Normal Probability Plot

C_IniVolA_Weight

D_Spring

B_Surfac

CD


Apr 24, 2023


6 647 1288 2569 512

10 102411 204812 409613 819214 1638415 32768

Size of Full Factorial Designs

Apr 24, 2023


-1 -1 -1 1 1 1 -11 -1 -1 -1 -1 1 1

-1 1 -1 -1 1 -1 11 1 -1 1 -1 -1 -1

-1 -1 1 1 -1 -1 11 -1 1 -1 1 -1 -1

-1 1 1 -1 -1 1 -11 1 1 1 1 1 1

A B C AB AC BC ABCA B C AB AC BC ABC

-

-

+

+

Fractional Factorial Design

Apr 24, 2023


135

34

45

119Oven

Dielect

Paste 1-1

1

-1

1

-1


Paste DielectOven Response-1 -1 -1 1191191 -1 -1 44-1 1 -1 2321 1 -1 135135-1 -1 1 111 -1 1 3434-1 1 1 45451 1 1 80

23-1 Fractional Factorial Design

Apr 24, 2023


A B C D E F G

0.5

0.6

0.7

0.8

0.9

Average

Main Effects Plot - Means for Average2 (̂7-3) = 16

1616

A B C D E F G

0.5

0.6

0.7

0.8

0.9

Average

Main Effects Plot - Means for Average

2 (̂7-2) = 3232

A B C D E F G

0.40

0.52

0.64

0.76

0.88

Average

Main Effects Plot - Means for Average

2 (̂7-1) = 6464

Apr 24, 2023


B_SurfB_Surf C_IniVolC_IniVol D_SpringD_Spring G_GasTemG_GasTem AverageAverage Std_DevStd_Dev-1 -1 -1 -1 0.789 0.21041 -1 -1 -1 0.536 0.2691-1 1 -1 -1 0.925 0.01741 1 -1 -1 0.504 0.2455-1 -1 1 -1 0.486 0.11301 -1 1 -1 0.407 0.1667-1 1 1 -1 0.528 0.11211 1 1 -1 0.446 0.1580-1 -1 -1 1 0.420 0.11941 -1 -1 1 0.649 0.2503-1 1 -1 1 0.520 0.23251 1 -1 1 0.335 0.0577-1 -1 1 1 0.492 0.13661 -1 1 1 0.458 0.1186-1 1 1 1 0.456 0.14641 1 1 1 0.424 0.12792 0 0 0 0.659 0.1476-2 0 0 0 0.546 0.19450 2 0 0 0.602 0.15380 -2 0 0 0.517 0.16740 0 2 0 0.293 0.08740 0 -2 0 0.611 0.43340 0 0 2 0.422 0.22290 0 0 -2 0.623 0.24010 0 0 0 0.412 0.12850 0 0 0 0.456 0.13770 0 0 0 0.376 0.11050 0 0 0 0.483 0.2034

CentralComposite

Designs

Apr 24, 2023


0.3-2

0.40.5

-2

0.6

-1

0.70.8

-1

0.9

0

1.0

01 12 2

0.4-2

0.5

0.6

-2-1

0.7

-1

0.8

0

0.9

01 12 2

Response Surfaces of Piston Performance

0.3-2

0.4

-2

0.5

-1

0.6

-10

0.7

01 12 2

0.35-2

0.450.550.65

-2-1

0.750.85

-1

0.95

0

1.05

01 12 2

0.3-2

0.40.5

-2-1

0.6

0.7

-1

0.8

0

0.9

01 12 2

0.4-2

0.5

0.6

-2-1

0.7

0.8

-10

0.9

01 12 2

Apr 24, 2023


Case Study A: Manufacturing

Health Imaging Development Program New mammography film

(Kodak Min R-2000 film) Emulsion making (cubes)

Factors: Ripener level, nucleation flow rate Aims: Grain size 0.80 - 0.86

Cubicity >60% Goal: What is the operating window for getting

the desired grain size and cubicity?

ACase Study

Apr 24, 2023


Experimental Design

3 x 3 factorial design

30

18

6

0.38869%

0.63465%

0.754 60%

0.98761%

0.82147%

0.65580%

1.25948%

Flow(cc/min)

13.2 39.6 66Ripener (cc) ACase Study

0.54376%

0.96746%

Apr 24, 2023


0.500.60

0.70

0.800.86

0.90

1.00

1.10

1.20

Size

Mapping the Effects: Grain Size

FLO

W (c

c/m

in)

RIPENER (cc)

ACase Study

Apr 24, 2023


Cubicity

77%

72%

67%

62%

57%

52%

47%

Mapping the Effects: Cubicity

FLO

W (c

c/m

in)

ACase StudyRIPENER (cc)

Apr 24, 2023


Where Is the Overlap?

FLO

W (c

c’s/

min

)

RIPENER (cc’s)

77%

72%

67%62%

57%

52%

47%

0.50 0.60 0.70

0.80

0.86 0.90

1.00

1.10

1.20

ACase Study

Apr 24, 2023


Case Study B: Business Units

Consumer Imaging New opportunity for branded

photofinishing in the United States Differentiation, pricing, segmentation all

important considerations

Question: How should a Kodak brandedphotofinishing option be priced?What share of choice can we expect?

BCase Study

Apr 24, 2023


Methodology

Mall intercept interviews with 100 consumers per cell who:

Have used at least three rolls of 35 mm film in the past year, and

Are responsible for deciding on photofinishing

Respondents complete “choice” exercises that simulate purchase decisions

BCase Study

Apr 24, 2023


CELL 1:

Price Levels for Mass Merchant

Store Processing:

Overnight $3.49 $3.99 $4.99

Same Day $6.24 $6.99 $7.49

KODAK Premium Processing Bundle:

Photofile $8.29 $9.29 $10.29

Price Levels

BCase Study

CELL 2:

Price Levels for Drug/Food

Store Processing:

Overnight $4.49 $4.99 $5.99

Same Day $6.99 $7.99 $8.49

KODAK Premium Processing Bundle:

Photofile $8.29 $9.29 $10.29

Apr 24, 2023


Choice Example

9 cards per personIn-Store

Overnight Processing

$3.99

In-Store Same Day Processing

$6.99

KODAK Premium

Processing

$9.29

Choice cards in study simulated Point of Sale (POS)

BCase Study

Apr 24, 2023


Results: Mass Merchant

57%63%63%

34%23%26%

10% 10% 14%0%

20%

40%

60%

80%

100%

$8.29 $9.29 $10.29Price of KPP

Next Day KPP Same Day

BCase Study

Cell 1 - Mass Merchant(Same Day =$6.99 [median] and Next Day = $3.99 [median])

Apr 24, 2023


Cell 2 - Drug/Food(Same Day =$7.99 [median] and Next Day = $4.99 [median])

Results: Drug/Food

58%67%

60%

35%22%26%

7%14% 11%0%

20%

40%

60%

80%

100%

$8.29 $9.29 $10.29

Price of KODAK Premium Processing Next Day KODAK Premium Processing Same Day

BCase Study

Apr 24, 2023


Cell 1 - Mass Merchant(Same Day =$6.24 [low] and Next Day = $3.49 [low])

Results: Mass Merchant

58%63%64%

29%19%22%

13% 14% 18%0%

20%

40%

60%

80%

100%

$8.29 $9.29 $10.29


BCase Study

Apr 24, 2023


Cell 2 - Drug/Food(Same Day =$6.99 [low] and Next Day = $4.49 [low])

Results: Drug/Food

61%69%

62%

31%19%22%

8%16% 13%0%

20%

40%

60%

80%

100%

$8.29 $9.29 $10.29


BCase Study

Apr 24, 2023


Model: Share of Choice Calculator

INPUTS:Market: Mass MerchantPrice for Same Day Processing: $6.99Price for Next Day Processing: $3.99Price for KODAK Premium Processing : $9.29

MODEL ESTIMATES:Share of Choice for KODAK Premium Processing :

26%Share of Choice for Same Day: 10%Share of Choice for Next Day: 63% BCase Study

Apr 24, 2023


a 25-1 Fractional Factorial Experiment

1. 1. Purity by SE-HPLCPurity by SE-HPLC2. 2. O.D.O.D.

Responses

Apr 24, 2023


Factors

A. A. Freeze TemperatureFreeze TemperatureB. B. Thaw TemperatureThaw TemperatureC. C. VolumeVolumeD. D. ConcentrationConcentrationE. E. BulkBulk

Apr 24, 2023


LevelsFACTORFACTOR LowLow HighHighA:FreezeTempA:FreezeTemp-196-196 -70-70B:ThawRempB:ThawRemp 55 2323C:VolumeC:Volume 1.81.8 200200D:ConcprotD:Concprot 0.350.35 0.50.5E:BulkE:Bulk aa bb

Apr 24, 2023


Choosing the Experimental Array - 1

Fractional Factorial DesignFractional Factorial Design

Factors: 5 Base Design: 5, 16 Resolution: VFactors: 5 Base Design: 5, 16 Resolution: VRuns: 32 Replicates: 2 Runs: 32 Replicates: 2 Fraction: 1/2 Fraction: 1/2

Blocks: none Center pts (total): 0Blocks: none Center pts (total): 0

Design Generators: E = ABCD Design Generators: E = ABCD

Apr 24, 2023


Choosing the Experimental Array - 2

Design Generators: E = ABCDDesign Generators: E = ABCD

Alias StructureAlias Structure

I + ABCDEI + ABCDE

A + BCDE, B + ACDE, C + ABDE, D + ABCE, E A + BCDE, B + ACDE, C + ABDE, D + ABCE, E + ABCD+ ABCDAB + CDE, AC + BDE, AD + BCE, AE + BCD, AB + CDE, AC + BDE, AD + BCE, AE + BCD, BC + ADEBC + ADEBD + ACE, BE + ACD, CD + ABE, CE + ABD, BD + ACE, BE + ACD, CD + ABE, CE + ABD, DE + ABCDE + ABC

Apr 24, 2023


FreezeTempFreezeTemp ThawTempThawTemp VolumeVolume ConcprotConcprot BulkBulk-196-196 55 1.81.8 0.350.35 bb-70-70 55 1.81.8 0.350.35 aa-196-196 2323 1.81.8 0.350.35 aa-70-70 2323 1.81.8 0.350.35 bb-196-196 55 200.0200.0 0.350.35 aa-70-70 55 200.0200.0 0.350.35 bb-196-196 2323 200.0200.0 0.350.35 bb-70-70 2323 200.0200.0 0.350.35 aa-196-196 55 1.81.8 0.500.50 aa-70-70 55 1.81.8 0.500.50 bb-196-196 2323 1.81.8 0.500.50 bb-70-70 2323 1.81.8 0.500.50 aa-196-196 55 200.0200.0 0.500.50 bb-70-70 55 200.0200.0 0.500.50 aa-196-196 2323 200.0200.0 0.500.50 aa-70-70 2323 200.0200.0 0.500.50 bb

Apr 24, 2023


-196 -700.35 0.50FreezeTemp

ThawTempVolume

Concprot

Bulk

5

23

b

a

1.8200.0

2^(5-1) Fractional Factorial Design

Apr 24, 2023


Protocol of ExperimentTwo Independent samples for each runTwo Independent samples for each runEach sample will be tested in duplicateEach sample will be tested in duplicate

a) Take the two bulksa) Take the two bulksb) Dilute the two bulks at 0.5 and 0.35 mg/mlb) Dilute the two bulks at 0.5 and 0.35 mg/mlc) Take all the samples at the two freezing temperature conditionsc) Take all the samples at the two freezing temperature conditionsd) Take all the samples at the two thawing temperature conditionsd) Take all the samples at the two thawing temperature conditionse) Start the analysise) Start the analysis

Documents

Design of Experiments