6. Fractional Factorial Designs
(Ch.8. Two-Level Fractional Factorial Designs)
Hae-Jin Choi
School of Mechanical Engineering,
Chung-Ang University
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Introduction to The 2k-p Fractional Factorial 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)
 Almost always run as unreplicated factorials
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Why do Fractional Factorial Designs Work?
 The sparsity of effects principle
 There may be lots of factors, but few are important
 System is dominated by main effects, low-order interactions
 The projection property
 Every fractional factorial contains full factorials in fewer factors
 Sequential experimentation
 Can add runs to a fractional factorial to resolve difficulties (or
ambiguities) in interpretation
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The One-Half Fraction of the 2k
 Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1
 Consider a really simple case, the 23-1
 Note that I =ABC
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The One-Half Fraction of the 23
For the principal fraction, notice that the contrast for estimating the main effect A is
exactly the same as the contrast used for estimating the BC interaction.
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
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Projection of Fractional Factorials
Every fractional factorial
contains full factorials in
fewer factors
The “flashlight” analogy
A one-half fraction will
project into a full
factorial in any k – 1 of
the original factors
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Aliasing in the One-Half Fraction of the 23
 A = BC, B = AC, C = AB (or me = 2fi)
 Aliases can be found from the defining relation I = ABC
by multiplication
 ABC is called the generator.
 AI = A(ABC) = A2BC = BC
 BI =B(ABC) = AC
 CI = C(ABC) = AB
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Aliasing in the One-Half Fraction of the 23
 Main effect
 Two factor interaction
effect
1
A   a  b  c  abc 
2
1
B  a  b  c  abc 
2
1
C  a  b  c  abc 
2
1
BC   a  b  c  abc 
2
1
AC  a  b  c  abc 
2
1
AB  a  b  c  abc 
2
 Alias structure of effects
[ A] → A + BC , [B ] → B + AC , [C ] → C + AB
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The Alternate Fraction of the 23-1
 I = -ABC is the defining relation
 Implies slightly different aliases: A = -BC,
B= -AC, and C =
-AB
 Both designs belong to the same family, defined by
I = ± ABC
[ A]' → A − BC , [B ]' → B − AC , [C ]' → C − AB
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Design Resolution
 Resolution III Designs:
 me = 2fi (i.e., main effect = 2 factor interaction)
3−1
 example 2 III
 Resolution IV Designs:
 2fi = 2fi
 example 24IV−1
 Resolution V Designs:
 2fi = 3fi
 example 25−1
V
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Construction of a One-half Fraction
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Resin Plant Experiment – the 24-1 Design
 A chemical product is produced in a pressure vessel. A factorial
experiment is carried out in the pilot plant to study the factors
thought to influence the filtration rate of this product .
 The factors are A = temperature, B = pressure, C = mole ratio, D=
stirring rate
 A 24-1 fractional factorial was used to investigate the effects of four
factors on the filtration rate of a resin
 Generator I = ABCD
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Resin Plant Experiment – the 24-1 Design
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Aliasing the 2IV4-1 Factorial Design
 Resolution IV design with the generator
 I=ABCD
 Main effect is aliased with three factor interaction
 A=A2BCD=BCD;
 B=AB2CD=ACD;
 C=ABC2D=ABD;
 D=ABCD2=ABC;
 Two factor interaction is aliased with other two factor interaction
 AB=CD;
 AC=BD;
 AD=BC;
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Resin Plant Experiment – the 24-1 Design
Interpretation of results often relies on making some assumptions
Ockham’s razor
Confirmation experiments can be important
Adding the alternate fraction – see page 301
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Resin Plant Experiment – MINITAB Results
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Resin Plant Experiment – MINITAB Results
Zero degree of
freedom for residuals
y  0  1 x1  2 x2  3 x3  4 x4  5 x1 x2  6 x1 x3  7 x1 x4
y  0  1 x1  3 x3  4 x4  6 x1 x3  7 x1 x4  
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2 degree of freedom for residuals
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Resin Plant Experiment – MINITAB Results
ŷ  ˆ0  ˆ1 x1  ˆ3 x3  ˆ4 x4  ˆ6 x1 x3  ˆ7 x1 x4
19.00 
14.00 
16.50 
 18.50 
19.00 




yˆ  70.75  
x
x
x
x
x
xx




 2  1  2  3  2  4  2  1 3  2  1 4
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Resin Plant Experiment – MINITAB Results
19.00 
14.00 
16.50 
 18.50 
19.00 








yˆ  70.75  
x
x
x
x
x
xx
 2  1  2  3  2  4  2  1 3  2  1 4
For example the residual at x1  1, x2  1, x3  1, x4  1
  y  yˆ


19.00 
14.00 
16.50 
 18.50 
19.00 





 100   70.75  






(1)
(
1)
(1)
(1)(
1)
(1)(1)








 2 


 2 
 2 
 2 
 2 
 100 100.25  0.25
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Resin Plant Experiment – MINITAB Results
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Manufacturing Process for a Circuit
 Five factors in a manufacturing process for an integrated circuit
were investigated in a 25-1 design with the objective of improving
the process yield.
Select ABCDE as the generator (Resolution V design)
I=ABCDE ; E=ABCD ;
Every main effect is aliased with a four-factor interaction. E.g., [A] -> A+BCDE
Every two factor interaction is aliased with a three-factor interaction. E.g., [AB]-> AB+CDE
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Manufacturing Process – MINITAB Results
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Manufacturing Process – MINITAB Results
A, B, C, and AB are significant
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Manufacturing Process – MINITAB Results
 Selecting only A, B, C, and AB
 This implies 23 Design with 2 replicates at each experimental point
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Manufacturing Process – MINITAB Results
 ANOVA
 Residual analysis
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Manufacturing Process – MINITAB Results
 Interaction Plot of AB
 Cube Plot
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The Sequential Experimentation
 Suppose that after running the principal fraction, the alternate
fraction was also run
 The two groups of runs can be combined to form a full
factorial – an example of sequential experimentation
 De-aliased estimates of the effects can be obtained by adding
and subtracting
1
1
([ A]  [ A]')  ( A  BC  A  BC )  A
2
2
1
1
([ A]  [ A]')  ( A  BC  A  BC )  BC
2
2
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The Sequential Experimentation
 If it is necessary to resolve ambiguities, we can run the alternate
fraction and complete 2k design.
Run 1
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Run 2
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Resin Plant Experiment – Alternate Fraction
 Recall the resin plant experiment
 Generator I=-ABCD
[ A]  19  A  BCD (from main fraction)
1
[ A]'  (43  71 48  104  68  86  70  65)
4
 24.25  A  BCD (from alternative fraction)
Main Effect of original design
1
A  [ A]  [ A]'  21.63
2
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The One-Quarter Fraction of the 2k
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The One-Quarter Fraction of the 26-2
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The General 2k-p Fractional Factorial 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
Important to select generators so as to maximize
resolution, see the table in the next slide
Projection – a design of resolution R contains full factorials
in any R – 1 of the factors
Effects of factors are
Effecti 
Contrasti
( N / 2)
where
N = number of observations
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The General 2k-p Design
 Resolution may not be sufficient
 Minimum abberation designs
Our choice
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