Design and Analysis of Engineering Experiments:

4. Introduction to Factorial Design

Dicky Dermawanwww.dickydermawan.net78.net

dickydermawan@gmail.com

ITK-226 Statistika & Rancangan Percobaan

DOX 6E Montgomery 2

Design of Engineering ExperimentsPart 1 – Introduction

Chapter 1, Text

• Why is this trip necessary? Goals of the course

• An abbreviated history of DOX• Some basic principles and terminology• The strategy of experimentation• Guidelines for planning, conducting and

analyzing experiments

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Strategy of Experimentation

• “Best-guess” experiments– Used a lot– More successful than you might suspect, but there are

disadvantages…• One-factor-at-a-time (OFAT) experiments

– Sometimes associated with the “scientific” or “engineering” method

– Devastated by interaction, also very inefficient• Statistically designed experiments

– Based on Fisher’s factorial concept

4

One-factor-at-a-time (OFAT) experiments

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Planning, Conducting & Analyzing an Experiment

1. Recognition of & statement of problem

2. Choice of factors, levels, and ranges

3. Selection of the response variable(s)

4. Choice of design

5. Conducting the experiment

6. Statistical analysis

7. Drawing conclusions, recommendations

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Planning, Conducting & Analyzing an Experiment

• Get statistical thinking involved early• Your non-statistical knowledge is crucial to success• Pre-experimental planning (steps 1-3) vital• Think and experiment sequentially• See Coleman & Montgomery (1993) Technometrics

paper + supplemental text material

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Design of Engineering ExperimentsPart 4 – Introduction to Factorials

• Text reference, Chapter 5-8• General principles of factorial experiments• The two-factor factorial with fixed effects• The ANOVA for factorials• Extensions to more than two factors• Quantitative and qualitative factors –

response curves and surfaces

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Some Basic Definitions

Definition of a factor effect: The change in the mean response when the factor is changed from low to high

40 52 20 3021

2 230 52 20 40

112 2

52 20 30 401

2 2

A A

B B

A y y

B y y

AB

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The Case of Interaction:

50 12 20 401

2 240 12 20 50

92 2

12 20 40 5029

2 2

A A

B B

A y y

B y y

AB

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Regression Model & The Associated Response Surface

0 1 1 2 2 12 1 2

1 2 1 2 1 2

The least squares fit is

ˆ 35.5 10.5 5.5 0.5 35.5 10.5 5.5

y x x x x

y x x x x x x

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The Effect of Interaction on the Response Surface

Suppose that we add an interaction term to the model:

Interaction is actually a form of curvature

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Example 5-1 The Battery Life ExperimentText reference pg. 165

A = Material type; B = Temperature (A quantitative variable)

1. What effects do material type & temperature have on life?

2. Is there a choice of material that would give long life regardless of temperature (a robust product)?

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The General Two-Factor Factorial Experiment

a levels of factor A; b levels of factor B; n replicates

This is a completely randomized design

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Statistical (effects) model:

1,2,...,

( ) 1, 2,...,

1, 2,...,ijk i j ij ijk

i a

y j b

k n

Other models (means model, regression models) can be useful

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Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 177

2 2 2... .. ... . . ...

1 1 1 1 1

2 2. .. . . ... .

1 1 1 1 1

( ) ( ) ( )

( ) ( )

a b n a b

ijk i ji j k i j

a b a b n

ij i j ijk iji j i j k

y y bn y y an y y

n y y y y y y

breakdown:

1 1 1 ( 1)( 1) ( 1)

T A B AB ESS SS SS SS SS

df

abn a b a b ab n

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ANOVA Table – Fixed Effects Case

Design-Expert will perform the computations

Text gives details of manual computing (ugh!) – see pp. 169 & 170

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Design-Expert Output – Example 5-1

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Residual Analysis – Example 5-1

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Residual Analysis – Example 5-1

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Interaction Plot DESIGN-EXPERT Plot

Life

X = B: TemperatureY = A: Material

A1 A1A2 A2A3 A3

A: MaterialInteraction Graph

Life

B: Temperature

15 70 125

20

62

104

146

188

2

2

22

2

2

Exercise

The 2k Factorial Design

Factorial designs are widely used in experiments involving several factors where it is necessary to study the joint effect of the factors on a response.

The most important cases is that of k factors, each at only 2 levels. A complete replicate of such a design requires 2 x 2 x 2 x … x 2 = 2k observations.

This extremely important class of design is particularly useful in the early stages of experimental works when there are likely to be many factors to be investigated. It provides the smallest number of runs with which k factors can be studied in a complete factorial design. Consequently, 2k factorial designs are widely used in factor screening experiments.

The 22 Factorial DesignEffect of Reactant Concentration and Catalyst Amount

on The Conversion in A Chemical Process

A = Concentration

“-” = 15% “+” = 25%

B = Catalyst

“-” = 1 lb “+” = 2 lb

The 22 Factorial Design: An Example

Effect of Factors:

-Main effect : effect of factor A; effect of factor B

- Interaction effect : AB

Anova & Effect of Factor ijkijjiijk )(y

The 22 Factorial Design: An Example

Magnitude & direction of effect of factors:

-Main effect:

A = {½ [(36-28)+(31-18)] + ½ [(32-25)+(30-19)] + ½ [(32-27)+(29-23)]}/3 = 8,33

B = {½ [(18-28)+(31-36)] + ½ [(19-25)+(30-32)] + ½ [(23-27)+(29-32)]}/3 = -5,00

- Interaction :

AB= {½ [(31-18) - (36-28)]+ ½ [(30-19) - (32-25)] + ½ [29-23)-(32-27)] }/3 = 1.67

Effect of a main factor is the change in response produce by a change in the level of that factor, averaged over the levels of the other factors.

Interaction effect AB is the average difference between the effect of A at the high level B and the effect of A at low level of B

A = Concentration

“-” = 15% “+” = 25%

B = Catalyst

“-” = 1 lb “+” = 2 lb

Regression Model, Surface Response & Contour Plot

Suppose we conclude that interaction AB is not significant. The regression model is:

x1 is coded variable representing natural variable A, i.e. reactant concentration

x2 is coded variable representing natural variable B, i.e. catalyst amount

22110 xxy

2)low.(conc)high.(conc

2)low.(conc)high.(conc

1

.concx

2)low.(cat)high.(cat

2)low.(cat)high.(cat

2

.catx

The fitted regression model is:

21 x2

5x

2

33,85,27y

Residual & Model Adequacy

Model adequacy checking:

Normal Probability Plot of Residual

Residual vs Predicted Conversion

Error calculation: yy

Exercise

Analysis Procedure for 2k Factorial Design

1. Estimate factor effect

2. Form initial model

3. Perform statistical testing

4. Refine model

5. Analyze residual

6. Interpret results

The 23 Factorial Design

Effect of Percentage of Carbonation, Operating Pressure and Line Speed on Uniformity of Filling Height of A Soft Drink Bottler

  Coded Factor Response Factor LevelRun A B C Rep I Rep II   Low (-1) High (+)1 -1 -1 -1 -3 -1 A (%) 10 122 1 -1 -1 0 1 B (psi) 25 303 -1 1 -1 -1 0 C (b/min) 200 2504 1 1 -1 2 35 -1 -1 1 -1 06 1 -1 1 2 17 -1 1 1 1 18 1 1 1 6 5      

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment

  Coded Factor  Run Label

Filtration Rate [gph]Run A B C D

1 -1 -1 -1 -1 (1) 452 1 -1 -1 -1 a 713 -1 1 -1 -1 b 484 1 1 -1 -1 ab 655 -1 -1 1 -1 c 686 1 -1 1 -1 ac 607 -1 1 1 -1 bc 808 1 1 1 -1 abc 659 -1 -1 -1 1 d 4310 1 -1 -1 1 ad 10011 -1 1 -1 1 bd 4512 1 1 -1 1 abd 10413 -1 -1 1 1 cd 7514 1 -1 1 1 acd 8615 -1 1 1 1 bcd 7016 1 1 1 1 abcd 96

FactorsA (Temperature)B (pressure)C (HCHO concentration)D (stirring rate)

1. Estimate factor effect

2. Form initial model

3. Perform statistical testing

4. Refine model

5. Analyze residual

6. Interpret results

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment: Contrast Constant  A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD(1) -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1a 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1b -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1ab 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1c -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1ac 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1bc -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1abc 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1d -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1ad 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1bd -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1abd 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1cd -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1acd 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1bcd -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1abcd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment: Effect estimateModel Term Effect Estimate Sum of Square % Contribution

A 21.625 1870.5625 33%B 3.125 39.0625 1%C 9.875 390.0625 7%D 14.625 855.5625 15%AB 0.125 0.0625 0%AC -18.125 1314.0625 23%AD 16.625 1105.5625 19%BC 2.375 22.5625 0%BD -0.375 0.5625 0%CD -1.125 5.0625 0%ABC 1.875 14.0625 0%ABD 4.125 68.0625 1%ACD -1.625 10.5625 0%BCD -2.625 27.5625 0%ABCD 1.375 7.5625 0%

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment:

Normal Probability Plot of Effect

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment: Significant Effect

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment: Error Checking

The Unreplicated 24 Factorial Design

Pilot Plan Filtration Rate Experiment:

Interpretation – Surface Response

The Unreplicated 24 Factorial DesignExercise

The Unreplicated 24 Factorial DesignExercise (cont’)

Fractional Factorial Design: A 25-1 Design

• Motivation for fractional factorials is obvious; as the number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly

• Emphasis is on factor screening; efficiently identify the factors with large effects

• There may be many variables (often because we don’t know much about the system)

Basic designDefining Relation = Design GeneratorPrincipal & The Alternate Fraction of the 25-1

Confounding & Aliasing

Construction of a One-half Fraction

Basic Design: Full 25 Design

RunCoded Factor

A B C D E1 -1 -1 -1 -1 -12 1 -1 -1 -1 -13 -1 1 -1 -1 -14 1 1 -1 -1 -15 -1 -1 1 -1 -16 1 -1 1 -1 -17 -1 1 1 -1 -18 1 1 1 -1 -19 -1 -1 -1 1 -110 1 -1 -1 1 -111 -1 1 -1 1 -112 1 1 -1 1 -113 -1 -1 1 1 -114 1 -1 1 1 -115 -1 1 1 1 -116 1 1 1 1 -117 -1 -1 -1 -1 118 1 -1 -1 -1 119 -1 1 -1 -1 120 1 1 -1 -1 121 -1 -1 1 -1 122 1 -1 1 -1 123 -1 1 1 -1 124 1 1 1 -1 125 -1 -1 -1 1 126 1 -1 -1 1 127 -1 1 -1 1 128 1 1 -1 1 129 -1 -1 1 1 130 1 -1 1 1 131 -1 1 1 1 132 1 1 1 1 1

A fractional 25-1 Design is a half fraction of Full 25 factorial design

Factor Level    Low (-1) High (+)A Aperture setting small largeB Exposure time -20% +20%C Develop time 30 s 45 sD Mask dimension small largeE Etch time 14.5 min 15.5 min

Defining Relation = Design Generator

  Coded Factor  Run A B C D E = ABCD1 -1 -1 -1 -1 12 1 -1 -1 -1 -13 -1 1 -1 -1 -14 1 1 -1 -1 15 -1 -1 1 -1 -16 1 -1 1 -1 17 -1 1 1 -1 18 1 1 1 -1 -19 -1 -1 -1 1 -110 1 -1 -1 1 111 -1 1 -1 1 112 1 1 -1 1 -113 -1 -1 1 1 114 1 -1 1 1 -115 -1 1 1 1 -116 1 1 1 1 1

  Coded Factor  Run A B C D E = -ABCD1 -1 -1 -1 -1 -12 1 -1 -1 -1 13 -1 1 -1 -1 14 1 1 -1 -1 -15 -1 -1 1 -1 16 1 -1 1 -1 -17 -1 1 1 -1 -18 1 1 1 -1 19 -1 -1 -1 1 110 1 -1 -1 1 -111 -1 1 -1 1 -112 1 1 -1 1 113 -1 -1 1 1 -114 1 -1 1 1 115 -1 1 1 1 116 1 1 1 1 -1

Principal Fraction of the 25-1 The Alternate Fraction

Confounding & Aliasing

Since E = ABCD:• Effect of E & effect of ABCD are

indistinguishable or • ABCDE = EE = E2 = I, thus I= ABCDE

A = AI = A2BCDE = BCDE, or

also:

AB = ABI = A2B2CDE = CDE, thus

ABCDEE

BCDEAA

ACDEBB ABDECC

CDEABAB

Data Analysis

RunBASIC DESIGN  

YieldA B C D

E = ABCD

1 -1 -1 -1 -1 1 82 1 -1 -1 -1 -1 93 -1 1 -1 -1 -1 344 1 1 -1 -1 1 525 -1 -1 1 -1 -1 166 1 -1 1 -1 1 227 -1 1 1 -1 1 458 1 1 1 -1 -1 609 -1 -1 -1 1 -1 610 1 -1 -1 1 1 1011 -1 1 -1 1 1 3012 1 1 -1 1 -1 5013 -1 -1 1 1 1 1514 1 -1 1 1 -1 2115 -1 1 1 1 -1 4416 1 1 1 1 1 63

1. Estimate factor effect

2. Form initial model

3. Perform statistical testing

4. Refine model

5. Analyze residual

6. Interpret results

Estimates of Factor Effect & Initial Model

Normal Probability Plot of Effects

Perform Statistical Testing: Analysis of Variance Table

21321 xx4375.3x4375.5x9375.16x5625.53125.30y

Factor Level    Low (-1) High (+)A Aperture setting small largeB Exposure time -20% +20%C Develop time 30 s 45 sD Mask dimension small largeE Etch time 14.5 min 15.5 min

Refine Model

)5.37s,time.Dev(725.0Exp%675.075.24y

Small Apperture Setting:

Large Apperture Setting:

20

.Exp%x2

5.7

5.37time.Devx3

eargl

small

1

1x1

)5.37s,time.Dev(725.0Exp%01875.1875.35y

Analysis of Residual:Normal Plot of Residual

Analysis of Residual:Plot of Residual vs Predicted Yield

Exercise

Exercise