design and analysis of engineering experiments: 4. introduction to factorial design
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ITK-226 Statistika & Rancangan Percobaan. Design and Analysis of Engineering Experiments: 4. Introduction to Factorial Design. Dicky Dermawan www.dickydermawan.net78.net [email protected]. Design of Engineering Experiments Part 1 – Introduction Chapter 1, Text. - PowerPoint PPT PresentationTRANSCRIPT
Design and Analysis of Engineering Experiments:
4. Introduction to Factorial Design
Dicky Dermawanwww.dickydermawan.net78.net
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
DOX 6E Montgomery 3
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
DOX 6E Montgomery 5
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
DOX 6E Montgomery 6
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
DOX 6E Montgomery 7
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
DOX 6E Montgomery 8
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
DOX 6E Montgomery 9
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
DOX 6E Montgomery 10
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
DOX 6E Montgomery 11
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
DOX 6E Montgomery 12
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)?
DOX 6E Montgomery 13
The General Two-Factor Factorial Experiment
a levels of factor A; b levels of factor B; n replicates
This is a completely randomized design
DOX 6E Montgomery 14
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
DOX 6E Montgomery 15
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
DOX 6E Montgomery 16
ANOVA Table – Fixed Effects Case
Design-Expert will perform the computations
Text gives details of manual computing (ugh!) – see pp. 169 & 170
DOX 6E Montgomery 17
Design-Expert Output – Example 5-1
DOX 6E Montgomery 18
Residual Analysis – Example 5-1
DOX 6E Montgomery 19
Residual Analysis – Example 5-1
DOX 6E Montgomery 20
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