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Design of experiment
ERT 427
2k-p fractional factorial
Miss Hanna Ilyani Zulhaimi
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OUTLINE
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Limitation of full factorial design
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The concept of fractional factorial, 2k-p
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One-half fraction factorial design, 2k-1
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One-quarter fraction factorial design, 2k-2
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General 2k-p fractional factorial design
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Fundamental Principles Regarding
Factorial Effects
Suppose there are k factors (A,B,...,J,K) in an experiment. All possible
factorial effects include
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Hierarchical Ordering principle
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effects of order 1: A, B, ..., K (main effects)
effects of order 2: AB, AC, ....,JK (2-factor interactions)
Lower order effects are more likely to be important than higher order
effects.
Effects of the same order are equally likely to be important
Effect Heredity Principle
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In order for an interaction to be significant, at least one of its parent factors
should be significant.
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Fractional Factorial Designs
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Purpose: full factorial design can be very expensive
n  Large number of factors ⇒ too many experiments
n  May not have sources (time, money etc) for full factorial design
n  Costly (Degrees of freedom wasted on estimating higher order terms)
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Often only lower order effects are important
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Number of runs required for full factorial grows quickly – Consider 2k
design
n  If k=7→128 runs required
n  Can estimate 127 effects
n  Only 7 df for main effects, 21 for 2-factor interactions – the remaining
99 df are for interactions of order ≥ 3
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A fraction of the full factorial design ( i.e. a subset of all possible level
combinations) is sufficient.
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Example 1
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Suppose mechanical engineer wants to design a new car and
consider the following nine factors each with 2 levels
1. Engine Size
6. Shape
2. Number of cylinders
7. Tires
3. Drag
8. Suspension
4. Weight
9. Gas Tank Size
5. Automatic vs Manual
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Only have resources for conducting is 64 runs
n  If you drop three factors for a 26 full factorial design, those
factor and their interactions with other factors cannot be
investigated.
n  Want investigate all nine factors in the experiment
n  A fraction of 29 factorial design will be used.
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Confounding (aliasing) will happen because using a subset
How to construct the fraction?
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Example 2
Filtration rate experiment:
Recall that there are four factors in the experiment (A,
B, C and D), each of 2 levels. Suppose the available resource is
enough for conducting 8 runs. 24 full factorial design consists of
all the 16 level combinations of the four factors. We need to
choose half of them.
The chosen half is called 24−1 fractional factorial design
Which half we should select (construct)?
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Full Factorial of 2 Level for k=4
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Continue…
For 24−1 Fractional Factorial Design
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number of factors: k = 4
n  the
fraction index: p = 1
n  Thus, the
number of runs (level combinations):
n  Construct
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24−1 designs via “confounding”
Generate D with a high order interaction of A, B and C, where:
D = ABC
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Fractional Factorial for 24-1
§  The chosen fraction includes the following 8 level combinations:
(−,−,−,−), (+,−,−,+), (−,+,−,+), (+,+,−,−), (−,−,+,+), (+,−,+,−),
(−,+,+,−), (+,+,+,+)
Note: 1 corresponds to + and −1 corresponds to −
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Aliasing in 23−1 Design (One-half
Fraction)
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Consider a situation with 3 factors, each at 2 level but the
experiment cannot afford to run at 8 treatment combination.
This will suggest one-half fraction of 23 design, which
contains 23-1=4 treatment combinations.
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Note that I=ABC
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Aliasing in 23−1 Design (One-half
Fraction)
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 or confounding
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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One-Half Fraction of 23
I = ±ABC
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It is called the defining relation, or ABCD is called a defining
word.
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In previous case, we select 4 treatment combination as our onehalf fraction. Each half fraction with have
n  Principal fraction (positive)
n  Alternate fraction (negative)
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The experiment will be run either principal or alternate fraction.
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The two groups of runs can be combined to form a full factorial.
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Exersice:
Construct a half fraction of 2 level factorial design
with 4 factors.
Hint: Generator,
D= ABC
Defining relation, I = +ABCD
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Design Resolution
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Design of resolution III, IV and IV are particularly important.
The definition is as follows:
Resolution
Definition
Resolution III No main effects are aliased with other main effects,
but main effect are aliased with two-factor
interaction.
Resolution IV No main effects are aliased with other main effects
or 2-factor interaction, but two-factor interaction
are aliased each other.
Resolution V
No main effect or two-factor interaction is aliased
with other main effect or two-factor interaction, but
two factor interactions are aliased with three-factor
interactions.
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The 2III3-1 Design
+ The
2IV4-1
Design
16
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The 2v5-1 Design
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(continued)
+ Guide to choice of fractional factorial
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designs
Factors
2
3
4
5
6
7
8
4 runs
Full
1/2 (III)
-
-
-
-
-
8
2 rep
Full
1/2 (IV)
1/4 (III)
1/8 (III)
1/16 (III)
-
16
4 rep
2 rep
Full
1/2 (V)
1/4 (IV)
1/8 (IV)
1/16 (IV)
32
8 rep
4 rep
2 rep
Full
1/2 (VI)
1/4 (IV)
1/8 (IV)
64
16 rep
8 rep
4 rep
2 rep
Full
1/2 (VII)
1/4 (V)
128
32 rep
16 rep
8 rep
4 rep
2 rep
Full
1/2 (VIII)
+ (continued)
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Factors
9
10
11
12
13
14
15
4 runs
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
16
1/32 (III)
1/64 (III)
1/128 (III)
1/256 (III)
32
1/16 (IV)
1/32 (IV)
1/64 (IV)
1/128 (IV) 1/256 (IV) 1/512 (IV) 1/1024 (IV)
64
1/8 (IV)
1/16 (IV)
1/32 (IV)
1/64 (IV)
1/128 (IV) 1/256 (IV) 1/512 (IV)
128
1/4 (VI)
1/8 (V)
1/16 (V)
1/128 (IV)
1/64 (IV)
1/512 (III) 1/1024 (III) 1/2048 (III)
1/128 (IV) 1/128 (IV)
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Guide (continued)
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Resolution V and higher à safe to use (main and two-factor
interactions OK)
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Resolution IV à think carefully before proceeding (main OK,
two factor interactions are aliased with other two factor
interactions)
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Resolution III à Stop and reconsider (main effects aliased with
two-factor interactions).
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See design generators for selected designs in the attached
table.
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Guide (continued)
What is the maximum resolution criterion?
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Fractional factorial design with maximum resolution is optimal!
Why?
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The higher the resolution, the less restrictive the assumptions that
are required
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Interactions are negligible to obtain a unique interpretation of
result
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One Quarter Fraction: 2k−2 Design
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Parts manufactured in an injection molding process are showing
excessive shrinkage. A quality improvement team has decided to use a
designed experiment to study the injection molding process so that
shrinkage can be reduced. The team decides to investigate six factors
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A: mold temperature B: screw speed
C: holding time
D: cycle time
E: gate size
F : holding pressure
each at two levels, with the objective of learning about main effects and
interactions. They decide to use 16-run fractional factorial design.
• a full factorial has 26=64 runs.
• 16-run is one quarter of the full factorial
How to construct the fraction?
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One Quarter Fraction: 2k−2 Design
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General 2k−p Fractional Factorial
Designs
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k factors, 2k level combinations, but want to run a 2−p fraction only.
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Select the first k − p factors to form a full factorial design (basic
design).
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Alias the remaining p factors with some high order interactions of the
basic design.
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Defining contrasts subgroup: G = { defining words}
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Define alias structure that meet with the concern.
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Use maximum resolution to choose the optimal design.
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Choose important effect to form models, pool unimportant effects into
error component
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Thank you J