### The One-Half Fraction of the 2 k Design

```Design and Analysis of
Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
1/33
Two-Level Fractional
Factorial Designs
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
2/33
Outline
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Introduction
The One-Half Fraction of the 2k factorial Design
The One-Quarter Fraction of the 2k factorial
Design
The General 2k-p Fractional Factorial Design
Alias Structures in Fractional Factorials and Other
Designs
Resolution III Designs
Resolution IV and V Designs
Supersaturated Designs
Introduction(1/5)
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As the number of factors in 2k factorial design
increases, the number of runs required for a
complete replicate of the design outgrows the
resources of most experimenters.
In 26 factorial design, 64 runs for one replicate.
Among them, 6 df for main effects, 15 df for
two-factor interaction.
That is, only 21 of them are majorly interested
in.
Introduction(2/5)
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The remaining 42 df are for three of higher
interactions.
If the experimenter can reasonably assume that
certain high-order interactions are negligible,
information on the main effects and low-order
interactions may be obtained by running only a
fraction of the complete factorial design.
Introduction(3/5)
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The Fractional Factorial Designs are among
the most widely used types of designs for
product and process design and process
improvement.
A major use of fraction factorials is in
screening experiments.
Introduction(4/5)
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Three key ideas that fractional factorial can be
used effectively:
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The sparsity of effects principle – When there are
several variables, the system or process is likely to
be driven primarily by some of the main effects an
lower-order interactions.
The projection property -- Fractional factorials can
be projected into stronger designs in the subset of
significant factors.
Introduction

Sequential experimentation – It is possible to
combine the runs of two or more fractional
factorials to assemble sequentially a larger design
to estimate the factor effects and interactions
interested.
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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Consider a 23 factorial design but an
experimenter cannot afford to run all (8) the
treatment combinations but only 4 runs.
This suggests a one-half fraction of a 23
design.
Because the design contains 23-1=4
treatment combinations, a one-half fraction
of the 23 design is often called a 23-1 design.
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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Consider a 23 factorial
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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We can have tow options:
One is the “+” sign in column ABC and the
other is the “-” sign in column ABC.
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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For the “+” in column ABC, effects a, b, c,
and abc are selected.
For the “-” in column ABC, effects ab, ac,
bc, and (1) are selected.
Since we use ABC to determine which half
to be used, ABC is called generator.
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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We look further to see if the “+” sign half is
used, the sign in column I is identical to the
one we used.
We call
I=ABC
is the defining relation in our design.
Note: C=AB is factor relation.
C=AB
I=ABC
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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In general, the defining relation for a
fractional factorials will always be the set of
all columns that are equal to the identity
column I.
If one examines the main effects:
[A]=1/2(a-b-c+abc)
[B]= 1/2(-a+b-c+abc)
[C]= 1/2(-a-b+c+abc)
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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The two-factor interactions effects:
[BC]=1/2(a-b-c+abc)
[AC]= 1/2(-a+b-c+abc)
[AB]= 1/2(-a-b+c+abc)
Thus,
A = BC, B = AC, C = AB
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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So
[A] A+BC [B] B+AC
[C] C+AB
The alias structure can be found by using the
defining relation I=ABC.
AI = A(ABC) = A2BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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The contrast for estimating the main effect
A is exactly the same as the contrast used
for estimating the BC interaction.
In fact, when estimating A, we are
estimating A+BC.
This phenomena is called aliasing and it
occurs in all fractional designs.
Aliases can be found directly from the
columns in the table of + and – signs.
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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This one-half fraction, with I=ABC, is usually
called the principal fraction.
That is, we could choose the other half of the
factorial design from Table.
This alternate, or complementary, one-half
fraction (consisting the runs (1), ab, ac, and
bc) must be chosen on purpose.
The defining relation of this design is
I=-ABC
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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So
[A]’ A-BC [B]’ B-AC
[C]’ C-AB
The alias structure can be found by using the
defining relation I=-ABC.
AI = A(-ABC) = A2BC = -BC
BI =B(-ABC) = -AC
CI = C(-ABC) = -AB
The One-Half Fraction of the 2k
Design – Definitions and Basic Principles
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In practice, it does not matter which fraction is
actually used.
Both fractions belong to the same family.
Two of them form a complete 23 design.
The two groups of runs can be combined to
form a full factorial – an example of
sequential experimentation
The One-Half Fraction of the 2k
Design – Design Resolution
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The 23-1 design is called a resolution III design.
In this design, main effects are aliased with
two-factor interactions.
In general, a design is of resolution R if no p
factor effect is aliased with another effect
containing less than R-p factors.
For a 23-1 design, no one (p) factor effect is
aliased with one (less than 3(R) – 1(p)) factor
effect.
The One-Half Fraction of the 2k
Design – Design Resolution
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Resolution III designs – These are designs in
which no main effects are aliased with any
other main effect. But main effects are aliased
with two-factor interactions and some twofactor interactions maybe aliased with each
other.
The 23-1 design is a resolution III design.
Noted as 23III1
The One-Half Fraction of the 2k
Design – Design Resolution
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Resolution IV designs – These are designs in
which no main effects are aliased with any
other main effect or with any two-factor
interaction. But two-factor interaction are
aliased with each other.
The 24-1 design with I=ABCD is a resolution
IV design.
Noted as 24IV1
The One-Half Fraction of the 2k
Design – Design Resolution
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Resolution V designs – These are designs in
which no main effects or two-factor
interaction is aliased with any other main
effect or with any two-factor interaction. But
two-factor interaction are aliased with threefactor interaction.
The 25-1 design with I=ABCDE is a resolution
V design.
Noted as 25IV1
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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Example: 23III1
C‧I=C‧ABC=AB
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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The one-half fraction of the 2k design of the
highest resolution may be constructed by
writing down a basic design consisting of the
runs for a full 2k-1 factorial and then adding the
kth factor by identifying its plus and minus
levels with the plus and minus signs of the
highest order interaction ABC..(K-1).
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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Note: Any interaction effect could be used to
generate the column for the kth factor.
However, use any effect other than ABC…(K1) will not product a design of the highest
possible resolution.
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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Any fractional factorial design of resolution R
contains complete factorial designs (possibly
replicated factorials) in any subset of R-1
factors. Important and useful !!!
Example, if an experiment has several factors
of potential interest but believes that only R-1
of them have important effects, the a fractional
factorial design of resolution R is the
appropriate choice of design.
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
Because the maximum possible
resolution of a one-half fraction of
the 2k design is R=k, every 2k-1
design will project into a full
factorial in any (k-1) of the original
k factors.
the 2k-1 design may be projected
into two replicates of a full factorial
in any subset of k-2 factors., four
replicates of a full factorial in any
subset of k-3 factors, and so on. 29
The One-Half Fraction of the 2k
Design – example (1--1/7)
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Y=filtration rate
Fours factors: A, B, C, and D.
Use 24-1 with I=ABCD
The One-Half Fraction of the 2k
Design – example (1--2/7)
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STAT > DOE > Factorial > Create Factorial Design
Number of factors  4
Design  ½ fraction  OK
Factors  Fill names for each factor
Fractional Factorial Design
Factors: 4 Base Design:
4, 8 Resolution: IV
Runs: 8 Replicates:
1 Fraction: 1/2
Blocks: 1 Center pts (total): 0
Design Generators: D = ABC
Alias Structure
I + ABCD A + BCD
AB + CD AC + BD
B + ACD
AD + BC
C + ABD
D + ABC
The One-Half Fraction of the 2k
Design – example (1--3/7)
The One-Half Fraction of the 2k
Design – example(1--4/7)
The One-Half Fraction of the 2k
Design – example (1--5/7)
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After collecting data
STAT > DOE > Factorial > Analyze Factorial Design
Response  Filtration  OK
Estimated Effects and Coefficients for Filtration (coded units)
Term
Constant
Temperature
Pressure
Conc.
Stir Rate
Temperature*Pressure
Temperature*Conc.
Temperature*Stir Rate
Effect
19.000
1.500
14.000
16.500
-1.000
-18.500
19.000
Coef
70.750
9.500
0.750
7.000
8.250
-0.500
-9.250
9.500
The One-Half Fraction of the 2k
Design – example (1--6/7)
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Obviously, no effect is significant
B is less important
Try A, C, and D  projection 23 with A, C, D
The One-Half Fraction of the 2k
Design – example (1--7/7)
Factorial Fit: Filtration versus Temperature, Conc., Stir Rate
Estimated Effects and Coefficients for Filtration (coded units)
Term
Constant
Temperature
Conc.
Stir Rate
Temperature*Conc.
Temperature*Stir Rate
Conc.*Stir Rate
Effect
19.000
14.000
16.500
-18.500
19.000
-1.000
Coef
70.750
9.500
7.000
8.250
-9.250
9.500
-0.500
SE Coef
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
T
94.33
12.67
9.33
11.00
-12.33
12.67
-0.67
P
0.007
0.050
0.068
0.058
0.052
0.050
0.626
S = 2.12132 PRESS = 288
R-Sq = 99.85% R-Sq(pred) = 90.62% R-Sq(adj) = 98.97%
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Prediction equation:

y  70.75  9.5x1  7 x3  8.25x4  9.25x1 x3  9.5x1 x4
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Coded variable : xi {1,1}
The One-Half Fraction of the 2k
Design – example (2—1/8)
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5 Factors
25-1 design
Response: Yield
37
The One-Half Fraction of the 2k
Design – example (2--2/8)
38
The One-Half Fraction of the 2k
Design – example (2--3/8)
39
The One-Half Fraction of the 2k
Design – example (2--4/8)
Factorial Fit: Yield versus Aperture, Exposure, Develop, Mask, Etch
Estimated Effects and Coefficients for Yield (coded units)
Term
Effect
Coef
Constant
30.3125
Aperture
11.1250
5.5625
Exposure
33.8750
16.9375
Develop
10.8750
5.4375
Mask
-0.8750
-0.4375
Etch
0.6250
0.3125
Aperture*Exposure
6.8750
3.4375
Aperture*Develop
0.3750
0.1875
Aperture*Mask
1.1250
0.5625
Aperture*Etch
1.1250
0.5625
Exposure*Develop
0.6250
0.3125
Exposure*Mask
-0.1250
-0.0625
Exposure*Etch
-0.1250
-0.0625
Develop*Mask
0.8750
0.4375
Develop*Etch
0.3750
0.1875
Mask*Etch
-1.3750
-0.6875
S = * PRESS = *
Analysis of Variance for Yield (coded units)
Source
DF
Main Effects
5
2-Way Interactions
10
Residual Error
0
Total
15
Seq SS
5562.8
212.6
*
5775.4
Adj SS
5562.8
212.6
*
Adj MS
1112.56
21.26
*
F
*
*
P
*
*
40
The One-Half Fraction of the 2k
Design – example (2--5/8)
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Reduced to A, B, C, AB
41
The One-Half Fraction of the 2k
Design – example (2--6/8)
Factorial Fit: Yield versus Aperture, Exposure, Develop
Estimated Effects and Coefficients for Yield (coded units)
Term
Effect
Coef
SE Coef
Constant
30.313
0.4002
Aperture
11.125
5.562
0.4002
Exposure
33.875
16.937
0.4002
Develop
10.875
5.437
0.4002
Aperture*Exposure
6.875
3.438
0.4002
T
75.74
13.90
42.32
13.59
8.59
P
0.000
0.000
0.000
0.000
0.000
Adj MS
1852.73
189.06
2.56
3.23
2.31
F
723.02
73.78
P
0.000
0.000
1.40
0.313
S = 1.60078 PRESS = 59.6364
R-Sq = 99.51% R-Sq(pred) = 98.97% R-Sq(adj) = 99.33%
Analysis of Variance for Yield (coded units)
Source
DF
Seq SS
Main Effects
3
5558.19
2-Way Interactions
1
189.06
Residual Error
11
28.19
Lack of Fit
3
9.69
Pure Error
8
18.50
Total
15
5775.44
Adj SS
5558.19
189.06
28.19
9.69
18.50
42
The One-Half Fraction of the 2k
Design – example (2--7/8)
43
The One-Half Fraction of the 2k
Design – example (2--8/8)
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Collapse into two replicate of a 23 design
44
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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Using fractional factorial designs often leads to
greater economy and efficiency in
experimentation. Particularly if the runs can be
made sequentially.
For example, suppose that we are investigating
k=4 factors (24=16 runs). It is almost always
preferable to run 24-1IV fractional design (four
runs), analyze the results, and then decide on the
best set of runs to perform next.
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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If it is necessary to resolve ambiguities, we can
always run the alternate fraction and complete
the 24 design.
When this method is used to complete the
design, both one-half fractions represent blocks
of the complete design with the highest order
interaction (ABCD) confounded with blocks.
Sequential experimentation has the result of
losing only the highest order interaction.
Possible
Strategies for
Follow-Up
Experimentation
Following a
Fractional
Factorial Design
47
The One-Half Fraction of the 2k
Design – example (3—1/4)
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From Example 1, 24-1IV design
Use I=-ABCD
STAT>DOE>Factorial>Create Factorial Design
Create base design first
2-level factorial(specify generators)
Number of factors  3
Design  Full factorial
Generators  D=-ABC  OK
48
The One-Half Fraction of the 2k
Design – example (3—2/4)
49
The One-Half Fraction of the 2k
Design – example (3—3/4)
50
The One-Half Fraction of the 2k
Design – example (3—4/4)
Estimated Effects and Coefficients for Filtration (coded units)
Term
Effect
Coef
Constant
69.375
Temperature
24.250
12.125
Pressure
4.750
2.375
Conc.
5.750
2.875
Stir Rate
12.750
6.375
Temperature*Pressure 1.250
0.625
Temperature*Conc.
-17.750
-8.875
Temperature*Stir Rate 14.250
7.125
S = * PRESS = *
Analysis of Variance for Filtration (coded units)
Source
DF
Seq SS
Main Effects
4
1612
2-Way Interactions
3
1039
Residual Error
0
*
Total
7
2652
Adj SS
1612
1039
*
Adj MS
403.1
346.5
*
F
*
*
P
*
*
51
The One-Half Fraction of the 2k
Design – Construction and analysis of the one-half fraction
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Adding the alternate fraction to the principal
fraction may be thought of as a type of
confirmation experiment that will allow us to
strengthen our initial conclusions about the twofactor interaction effects.
A simple confirmation experiment is to compare
the results from regression and actual runs.
The One-Quarter Fraction of
the 2k Design
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For a moderately large number of factors,
smaller fractions of the 2k design are frequently
useful.
One-quarter fraction of the 2k design
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2k-2 runs
called 2k-2 fractional factorial
The One-Quarter Fraction of
the 2k Design
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Constructed by writing down a basic design
consisting of runs associated with a full factorial
in k-2 factors and then associating the two
additional columns with appropriately chosen
interactions involving the first k-2 factors.
Thus, two generators are needed.
I=P and I=Q are called generating relations for
the design.
The One-Quarter Fraction of
the 2k Design
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The signs of P and Q determine which one of the
one-quarter fractions is produced.
All four fractions associated with the choice of
generators ±P or ±Q are members of the same
family.
+P and +Q are principal fraction.
I=P=Q=PQ
P, Q, and PQ are defining relation words
The One-Quarter Fraction of
the 2k Design
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6 2
IV
Example: 2
P=ABCE, Q=BCDF, PQ=ADEF
Thus A=BCE=ABCDF=DEF
When estimating A, one is really estimating
A+BCE+DEF+ABCDF
The One-Quarter Fraction of
the 2k Design
Complete defining relation: I = ABCE = BCDF = ADEF
57
The One-Quarter Fraction of
the 2k Design
58
The One-Quarter Fraction of
the 2k Design
6 2
IV

2
Factor relations:
E=ABC, F=BCD

I=ABCE=BCDF=ADEF
The One-Quarter Fraction of
the 2k Design


STAT>DOE>Factorial>Create factorial
Design
Design  Full factorialOKOK
60
The One-Quarter Fraction of
the 2k Design
61
The One-Quarter Fraction of
the 2k Design



Alternate fractions of 26-2 design
P=ABCE, -Q=-BCDF
-P=-ABCE, Q=BCDF
-P=-ABCE, -Q=-BCDF
[A]  A+BCE-DEF-ABCDF
The One-Quarter Fraction of
the 2k Design


A 26-2 design will project into a single
replicate of a 24 design in any subset of fours
factors that is not a word in the defining
relation.
It also collapses to a replicated one-half
fraction of a 24 in any subset of four factors
that is a word in the defining relation.
The One-Quarter Fraction of
the 2k Design


Projection of the design into subsets of the
original six variables
Any subset of the original six variables that is not
a word in the complete defining relation will result
in a full factorial design



Consider ABCD (full factorial)
Consider ABCE (replicated half fraction)
Consider ABCF (full factorial)
64
The One-Quarter Fraction of
the 2k Design


In general, any 2k-2 fractional factorial design
can be collapsed into either a full factorial or
a fractional factorial in some subset of r≦k-2
of the original factors.
Those subset of variables that form full
factorials are not words in the complete
defining relation.
The One-Quarter Fraction of
the 2k Design— example(4—1/10)





Injection molding process
Response: Shrinkage
Factors: Mold temp, screw speed, holding
time, cycle time, gate size, holding pressure.
Each at two levels
To run a 26-2 design, 16 runs
The One-Quarter Fraction of
the 2k Design— example(4—2/10)
The One-Quarter Fraction of
the 2k Design— example(4—3/10)

Full model
The One-Quarter Fraction of
the 2k Design— example(4—4/10)
Factorial Fit: Shrinkage versus Temperature, Screw, ...
Estimated Effects and Coefficients for Shrinkage (coded units)
Term
Constant
Temperature
Screw
Hold Time
Cycle Time
Gate
Pressure
Temperature*Screw
Temperature*Hold Time
Temperature*Cycle Time
Temperature*Gate
Temperature*Pressure
Screw*Cycle Time
Screw*Pressure
Temperature*Screw*Cycle Time
Temperature*Hold Time*Cycle Time
Effect
13.875
35.625
-0.875
1.375
0.375
0.375
11.875
-1.625
-5.375
-1.875
0.625
-0.125
-0.125
0.125
-4.875
Coef
27.313
6.937
17.812
-0.437
0.688
0.187
0.187
5.938
-0.813
-2.688
-0.937
0.313
-0.062
-0.063
0.062
-2.437
The One-Quarter Fraction of
the 2k Design— example(4—5/10)

Reduced model
The One-Quarter Fraction of
the 2k Design— example(4—6/10)

Reduced model
Factorial Fit: Shrinkage versus Temperature, Screw
Estimated Effects and Coefficients for Shrinkage (coded units)
Term
Effect
Coef
SE Coef
Constant
27.313
1.138
Temperature
13.875
6.937
1.138
Screw
35.625
17.812
1.138
Temperature*Screw
11.875
5.938
1.138
T
24.00
6.09
15.65
5.22
P
0.000
0.000
0.000
0.000
Adj MS
2923.31
564.06
20.73
20.73
F
141.02
27.21
S = 4.55293 PRESS = 442.222
R-Sq = 96.26% R-Sq(pred) = 93.36% R-Sq(adj) = 95.33%
Analysis of Variance for Shrinkage (coded units)
Source
DF
Seq SS
Main Effects
2
5846.6
2-Way Interactions
1
564.1
Residual Error
12
248.7
Pure Error
12
248.8
Total
15
6659.4
Adj SS
5846.6
564.1
248.7
248.8
P
0.000
0.000
The One-Quarter Fraction of
the 2k Design— example(4—7/10)


Reduced model
Normal plot
The One-Quarter Fraction of
the 2k Design— example(4—8/10)



Reduced model
Residuals vs
Hold time
Less scatter in
low hold time
than it is high
The One-Quarter Fraction of
the 2k Design— example(4—9/10)
S 2 (i  )
Fi  ln 2 
S (i )
*

F*C is large
The One-Quarter Fraction of
the 2k Design— example(4—10/10)
The General 2k-p Fractional
Factorial Design – choose a design



2k-1 = one-half fraction, 2k-2 = one-quarter
fraction, 2k-3 = one-eighth fraction, …, 2k-p =
1/ 2p fraction
Add p columns to the basic design; select p
independent generators
The defining relation for the design consists
of the p generators initially chosen and their
2p-p-1 generalized interactions.
The General 2k-p Fractional
Factorial Design – choose a design



Important to select generators so as to
maximize resolution
For example, the 26-2IV design, generators:
E=ABC, F=BCD, producing IV design. 
maximum resolution
If E=ABC, F=ABCD is chosen,
I=ABCE=ABCDF=DEF, resolution III.
The General 2k-p Fractional
Factorial Design – choose a design



Sometimes resolution alone is insufficient to
distinguish between designs.
For 27-2IV design, all of the design are
resolution IV but with different alias
structures.
Design A has more extensive two-factor
aliasing and design C the least.  Choose
design C
The General 2k-p Fractional
Factorial Design – choose a design
The General 2k-p Fractional
Factorial Design – choose a design

The next table
shows the
suggested
generators for
better designs.
The General 2k-p Fractional
Factorial Design – choose a design
The General 2k-p Fractional
Factorial Design— example(5—1/4)




7 factors are interested.
Two-factor interactions are to be explored.
Resolution IV is assumed be appropriate.
Two choices: 27-2IV (32 runs) and 27-3IV (16
runs)
The General 2k-p Fractional
Factorial Design— example(5—2/4)

27-3IV (16 runs)
Fractional Factorial Design
Factors: 7 Base Design:
7, 32 Resolution: IV
Runs: 32 Replicates:
1 Fraction: 1/4
Blocks: 1 Center pts (total):
0
Design Generators: F = ABCD, G = ABDE
Alias Structure
I + CEFG + ABCDF + ABDEG
A + BCDF + BDEG + ACEFG
D + ABCF + ABEG + CDEFG
G + CEF + ABDE + ABCDFG
AD + BCF + BEG + ACDEFG
AG + BDE + ACEF + BCDFG
BE + ADG + BCFG + ACDEF
CD + ABF + DEFG + ABCEG
CG + EF + ABCDE + ABDFG
DG + ABE + CDEF + ABCFG
BCE + BFG + ACDG + ADEF
CDG + DEF + ABCE + ABFG
B + ACDF + ADEG + BCEFG
E + CFG + ABDG + ABCDEF
AB + CDF + DEG + ABCEFG
AE + BDG + ACFG + BCDEF
BC + ADF + BEFG + ACDEG
BF + ACD + BCEG + ADEFG
CE + FG + ABCDG + ABDEF
DE + ABG + CDFG + ABCEF
ACE + AFG + BCDG + BDEF
BCG + BEF + ACDE + ADFG
C + EFG + ABDF + ABCDEG
F + CEG + ABCD + ABDEFG
AC + BDF + AEFG + BCDEG
AF + BCD + ACEG + BDEFG
BD + ACF + AEG + BCDEFG
BG + ADE + BCEF + ACDFG
CF + EG + ABD + ABCDEFG
DF + ABC + CDEG + ABEFG
ACG + AEF + BCDE + BDFG
CDE + DFG + ABCG + ABEF
The General 2k-p Fractional
Factorial Design— example(5—3/4)

27-3IV (16 runs)
Fractional Factorial Design
Factors: 7 Base Design:
7, 16 Resolution: IV
Runs: 16 Replicates:
1 Fraction: 1/8
Blocks: 1 Center pts (total):
0
Design Generators: E = ABC, F = BCD, G = ACD
Alias Structure
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG
A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG
C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG
E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF
G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG
AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG
AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF
AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG
ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG
B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG
D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG
F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG
AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG
AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG
AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG
BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG
The General 2k-p Fractional
Factorial Design— example(5—4/4)



Choose better design among costs,
information and resolution.
Appendix X provides a good reference for
choosing “better” design
Do not choose a design according to one
single criterion unless is “order” by your boss.
The General 2k-p Fractional
Factorial Design – Analysis



Use computer soft
wares.
Projection – a
design of resolution
R contains full
factorials in any R –
1 of the factors
27-3IV design
The General 2k-p Fractional
Factorial Design – Analysis



It will project into a full factorial in any four
of the original seven factors that is not a word
in the defining relation
C(7, 5)=35 subsets of four factors. 7 of them
(ABCE, BCDF, ACDG, ADEF, BDEG,
ABFG, and CEFG) appeared in defining
relations.
The rest of 28 four-factor subset would form
24 designs.
The General 2k-p Fractional
Factorial Design – Analysis



Obviously, A, B, C, D are one of them.
Consider the following situation:
If the 4 of 7 factors are more important than
the rest of 3 factors, we would assign the
more important four factors to A, B, C, D and
the less important 3 factors to E, F, and G.
The General 2k-p Fractional
Factorial Design – Blocking



Sometimes the runs needed in fraction
factorial can not be made under homogeneous
conditions.
We confound the fractional factorial with
blocks.
Appendix X contain recommended blocking
arrangements for fractional factorial designs.
The General 2k-p Fractional
Factorial Design – Blocking



For example, 26-2IV
According to the suggestion in
Appendix X(f), ABD and its
aliases to be confounded with
block.
STAT>DOE>Create Factorial
Design
The General 2k-p Fractional
Factorial Design – Blocking

Design Full factorial
Generators:

OK

The General 2k-p Fractional
Factorial Design – Blocking
The General 2k-p Fractional
Factorial Design – Blocking
bc
The General 2k-p Fractional
Factorial Design— example(6—1/4)






5 axes CNC machine
Response=profile deviation
8 factors are interested.
Four spindles are treated as blocks
Assumed tree factor and higher interactions
are negligible
From Appendix X, 28-4IV (16 runs)and 28-3IV
(32 runs) are feasible.
The General 2k-p Fractional
Factorial Design— example(6—1/4)





However, if 28-4IV (16 runs) is used, twofactor effects will confound with blocks
If EH interaction is unlikely, 28-3IV (32 runs)
is chosen.
STAT>DOE>Create Factorial Design
Choose 2 level factorial (default generators)
Number of factors  8, number of blocks 
4  OK
The General 2k-p Fractional
Factorial Design— example(6—1/4)
Fractional Factorial Design
Factors: 8 Base Design:
8, 32 Resolution with blocks: III
Runs: 32 Replicates:
1 Fraction:
1/8
Blocks: 4 Center pts (total): 0
* NOTE * Blocks are confounded with two-way interactions.
Design Generators: F = ABC, G = ABD, H = BCDE
Block Generators: EH, ABE
The General 2k-p Fractional
Factorial Design— example(6—1/4)
The General 2k-p Fractional
Factorial Design— example(6—1/4)
The General 2k-p Fractional
Factorial Design— example(6—1/4)



Analyze Design
A*D + B*G + E*F*H
Inseparable
If prior knowledge
implies that AD is
possible, one can use
reduced model
The General 2k-p Fractional
Factorial Design— example(6—1/4)

Reduced model: A, B, D, and AD
The General 2k-p Fractional
Factorial Design— example(6—1/4)

Reduced model: A, B, D, and AD
Factorial Fit: ln(std_dev) versus Block, A, B, D
Estimated Effects and Coefficients for ln(std_dev) (coded units)
Term
Effect
Coef
SE Coef
T
Constant
1.2801
0.02310
55.41
Block 1
-0.0053
0.04001
-0.13
Block 2
-0.0280
0.04001
-0.70
Block 3
0.0406
0.04001
1.02
A
0.2903
0.1451
0.02310
6.28
B
-0.2005
-0.1003
0.02310
-4.34
D
0.1081
0.0541
0.02310
2.34
A*D
-0.3741
-0.1871
0.02310
-8.10
P
0.000
0.896
0.491
0.320
0.000
0.000
0.028
0.000
S = 0.130680 PRESS = 0.728635
R-Sq = 84.47% R-Sq(pred) = 72.39% R-Sq(adj) = 79.94%
Analysis of Variance for ln(std_dev) (coded units)
Source
Blocks
Main Effects
2-Way Interactions
Residual Error
Total
DF
3
3
1
24
31
Seq SS
0.02014
1.08929
1.11970
0.40986
2.63899
Adj SS
0.02014
1.08929
1.11970
0.40986
Adj MS
0.00671
0.36310
1.11970
0.01708
F
0.39
21.26
65.57
P
0.759
0.000
0.000
The General 2k-p Fractional
Factorial Design— example(6—1/4)


Reduced model: A, B, D, and AD
Estimated equation

y  0.1451x1  0.1003x2  0.0541x4  0.1871x1 x4
The General 2k-p Fractional
Factorial Design— example(6—1/4)
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