Chapter 8 Two-Level Fractional
Factorial Designs
1
8.1 Introduction
• The number of factors becomes large enough to be
“interesting”, the size of the designs grows very
quickly
• After assuming some high-order interactions are
negligible, we only need to run a fraction of the
complete factorial design to obtain the information
for the main effects and low-order interactions
• Fractional factorial designs
• Screening experiments: many factors are
considered and the objective is to identify those
factors that have large effects.
2
• Three key ideas:
1. The sparsity of effects principle
– There may be lots of factors, but few are
important
– System is dominated by main effects, loworder interactions
2. The projection property
– Every fractional factorial contains full
factorials in fewer factors
3. Sequential experimentation
– Can add runs to a fractional factorial to
resolve difficulties (or ambiguities) in
interpretation
3
8.2 The One-half Fraction of the 2k
Design
• Consider three factor and each factor has two
levels.
• A one-half fraction of 23 design is called a 23-1
design
4
• In this example, ABC is called the generator of
this fraction (only + in ABC column). Sometimes
we refer a generator (e.g. ABC) as a word.
• The defining relation:
I = ABC
• Estimate the effects:
1
 A  a  b  c  abc   BC
2
1
 B   a  b  c  abc   AC
2
1
 C   a  b  c  abc   AB
2
• A = BC, B = AC, C = AB
5
• Aliases:
A  A  BC,
B
 B  AC,
C
 C  AB
• Aliases can be found from the defining relation I
= ABC by multiplication:
AI = A(ABC) = A2BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
• Principal fraction: I = ABC
6
• The Alternate Fraction of the 23-1 design:
I = - ABC
• When we estimate A, B and C using this design,
we are really estimating A – BC, B – AC, and C –
AB, i.e.  'A  A  BC,  'B  B  AC,  'C  C  AB
• Both designs belong to the same family, defined
by
I =  ABC
• 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
7
experimentation
• The de-aliased estimates of all effects by
analyzing the eight runs as a full 23 design in two
blocks. Hence




1
1
'
 A   A   A  BC  A  BC   A
2
2
1
1
'
 A   A   A  BC  A  BC   BC
2
2
• Design resolution: A design is of resolution R if no
p-factor effect is aliased with another effect
containing less than R – p factors.
• The one-half fraction of the 23 design with I =
ABC is a 2 3III1 design
8
• Resolution III Designs:
– me = 2fi
– Example: A 23-1 design with I = ABC
• Resolution IV Designs:
– 2fi = 2fi
– Example: A 24-1 design with I = ABCD
• Resolution V Designs:
– 2fi = 3fi
– Example: A 25-1 design with I = ABCDE
• In general, the resolution of a two-level fractional
factorial design is the smallest number of letters in
any word in the defining relation.
9
• The higher the resolution, the less restrictive the
assumptions that are required regarding which
interactions are negligible to obtain a unique
interpretation of the data.
• Constructing one-half fraction:
– Write down a full 2k-1 factorial design
– Add the kth factor by identifying its plus and
minus levels with the signs of ABC…(K – 1)
– K = ABC…(K – 1) => I = ABC…K
– Another way is to partition the runs into two
blocks with the highest-order interaction
ABC…K confounded.
10
11
• Any fractional factorial
design of resolution R
contains complete factorial
designs in any subset of R – 1
factors.
• A one-half fraction will
project into a full factorial in
any k – 1 of the original
factors
12
• Example 8.1:
– Example 6.2: A, C, D, AC and AD are
important.
– Use 24-1 design with I = ABCD
13
4 1
2
• This IV design is the principal fraction, I = ABCD
• Using the defining relation,
– A = BCD, B=ACD, C=ABD, D=ABC
– AB=CD, AC=BD, BC=AD
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• A, C and D are large.
• Since A, C and D are
important factors, the
significant interactions
are most likely AC and
AD.
• Project this one-half
design into a single
replicate of the 23 design
in factors, A, C and D.
(see Figure 8.4 and Page
310)
15
• Example 8.2:
– 5 factors
– Use 25-1 design with I = ABCDE (Table 8.5)
– Every main effect is aliased with four-factor
interaction, and two-factor interaction is aliased
with three-factor interaction.
– Table 8.6 (Page 312)
– Figure 8.6: the normal probability plot of the
effect estimates
– A, B, C and AB are important
– Table 8.7: ANOVA table
– Residual Analysis
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– Collapse into two replicates of a 23 design
• Sequences of
fractional factorial:
Both one-half
fractions represent
blocks of the
complete design
with the highestorder interaction
confounded with
blocks.
17
• Example 8.3:
– Reconsider Example 8.1
– Run the alternate fraction with I = – ABCD
– Estimates of effects
– Confirmation experiment
18
8.3 The One-Quarter Fraction of the
2k Design
• A one-quarter fraction of the 2k design is called a
2k-2 fractional factorial design
• Construction:
– Write down a full factorial in k – 2 factors
– Add two columns with appropriately chosen
interactions involving the first k – 2 factors
– Two generators, P and Q
– I = P and I = Q are called the generating
relations for the design
19
– All four fractions are the family.
20
•
•
•
•
The complete defining relation: I = P = Q = PQ
P, Q and PQ are called words.
Each effect has three aliases
A one-quarter fraction of the 26-2 with I = ABCE
and I = BCDF. The complete defining relation is
I = ABCE = BCDF = ADEF
21
• Another way to construct such design is to derive
the four blocks of the 26 design with ABCE and
BCDF confounded , and then choose the block
with treatment combination that are + on ABCE
and BCDF
• The 26-2 design with I = ABCE and I = BCDF is
the principal fraction.
• Three alternate fractions:
– I = ABCE and I = - BCDF
– I = -ABCE and I = BCDF
– I = - ABCE and I = -BCDF
22
• This 26IV2fractional factorial will project into
– A single replicate of a 24 design in any subset of
four factors that is not a word in the defining
relation.
– A replicate one-half fraction of a 24 in any
subset of four factors that is a word in the
defining relation.
• 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.
23
• Example 8.4:
– Injection molding process with six factors
– Design table (see Table 8.10)
– The effect estimates, sum of squares, and
regression coefficients are in Table 8.11
– Normal probability plot of the effects
– A, B, and AB are important effects.
– Residual Analysis (Page 322 – 325)
24
8.4 The General 2k-p Fractional
Factorial Design
• A 1/ 2p fraction of the 2k design
• Need p independent generators, and there are 2p –
p – 1 generalized interactions
• Each effect has 2p – 1 aliases.
• A reasonable criterion: the highest possible
resolution, and less aliasing
• Minimum aberration design: minimize the number
of words in the defining relation that are of
minimum length.
25
• Minimizing aberration of resolution R ensures that
a design has the minimum # of main effects
aliased with interactions of order R – 1, the
minimum # of two-factor interactions aliased with
interactions of order R – 2, ….
• Table 8.14
26
• Example 8.5
– Estimate all main effects and get some insight
regarding the two-factor interactions.
– Three-factor and higher interactions are
negligible.
– 27IV2 and 27IV3 designs in Appendix Table XII
(Page 666)
73
2
– IV 16-run design: main effects are aliased with
three-factor interactions and two-factor
interactions are aliased with two-factor
interactions
– 27IV2 32-run design: all main effects and 15 of 21
two-factor interactions
27
• Analysis of 2k-p Fractional Factorials:
– For the ith effect:
2(Contrasti ) Contrasti
i 

, N  2Np
N
N /2
• Projection of the 2k-p Fractional Factorials
– Project into any subset of r  k – p of the
original factors: a full factorial or a fractional
factorial (if the subsets of factors are appearing
as words in the complete defining relation.)
– Very useful in screening experiments
73
2
– For example IV 16-run design: Choose any
four of seven factors. Then 7 of 35 subsets are
appearing in complete defining relations.
28
• Blocking Fractional Factorial:
– Appendix Table XII
– Consider the 26IV2 fractional factorial design with
I = ABCE = BCDF = ADEF. Select ABD (and its
aliases) to be confounded with blocks. (see
Figure 8.18)
• Example 8.6
– There are 8 factors
84
83
– 2 IV or 2 IV
– Four blocks
– Effect estimates and sum of squares (Table 8.17)
– Normal probability plot of the effect estimates
29
(see Figure 8.19)
• A, B and AD + BG are important effects
• ANOVA table for the model with A, B, D and AD
(see Table 8.18)
• Residual Analysis (Figure 8.20)
• The best combination of operating conditions: A –,
B + and D –
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8.5 Resolution III Designs
• Designs with main effects aliased with two-factor
interactions
• A saturated design has k = N – 1 factors, where N is
the number of runs.
• For example: 4 runs for up to 3 factors, 8 runs for up
to 7 factors, 16 runs for up to 15 factors
31
• In Section 8.2, there is an example, 2 III design.
• Another example is shown in Table 8.19: 27III4design
I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG
= AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG
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• This design is a one-sixteenth fraction, and a
principal fraction.
I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG =
ABEF = BEG= AFG = DEF = ADEG = CEFG = BDFG =
ABCDEFG
• Each effect has 15 aliases.
32
• Assume that three-factor and higher interactions
are negligible.
7 4
2
• The saturated III design in Table 8.19 can be used
to obtain resolution III designs for studying fewer
than 7 factors in 8 runs. For example, for 6 factors
in 8 runs, drop any one column in Table 8.19 (see
Table 8.20)
33
• When d factors are dropped , the new defining
relation is obtained as those words in the original
defining relation that do not contain any dropped
letters.
• If we drop B, D, F and G, then the treatment
combinations of columns A, C, and E correspond
to two replicates of a 23 design.
34
• Sequential assembly of fractions to separate
aliased effects:
– Fold over of the original design
– Switching the signs in one column provides
estimates of that factor and all of its two-factor
interactions
– Switching the signs in all columns dealiases all
main effects from their two-factor interaction
alias chains – called a full fold-over
35
• Example 8.7
– Seven factors to study eye focus time
7 4
2
– Run III design (see Table 8.21)
– Three large effects
– Projection?
– The second fraction is run with all the signs
reversed
– B, D and BD are important effects
36
• The defining relation for a fold-over design
– Each separate fraction has L + U words used as
generators.
– L: like sign
– U: unlike sign
– The defining relation of the combining designs
is the L words of like sign and the U – 1 words
consisting of independent even products of the
words of unlike sign.
– Be careful – these rules only work for
Resolution III designs
37
• Plackett-Burman Designs
– These are a different class of resolution III
design
– Two-level fractional factorial designs for
studying k = N – 1 factors in N runs, where N
=4n
– N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
– The designs where N = 12, 20, 24, etc. are
called nongeometric PB designs
– Construction:
• N = 12, 20, 24 and 36 (Table 8.24)
• N = 28 (Table 8.23)
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• The alias structure is complex in the PB designs
• For example, with N = 12 and k = 11, every main
effect is aliased with every 2FI not involving itself
• Every 2FI alias chain has 45 terms
• Partial aliasing can greatly complicate
interpretation
• Interactions can be particularly disruptive
• Use very, very carefully (maybe never)
39
• Projection: Consider the
12-run PB design
– 3 replicates of a full 22
design
31
3
2
– A full 2 design + a III
design
– Projection into 4 factors is
not a balanced design
– Projectivity 3: collapse
into a full fractional in any
subset of three factors.
40
• Example 8.8:
– Use a set of simulated data and the 11 factors, 12run design
– Assume A, B, D, AB, and AD are important
factors
– Table 8.25 is a 12-run PB design
– Effect estimates are shown in Table 8.26
– From this table, A, B, C, D, E, J, and K are
important factors.
– Interaction? (due to the complex alias structure)
– Folding over the design
– Resolve main effects but still leave the uncertain
about interaction effects.
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8.6 Resolution IV and V Designs
• Resolution IV: if three-factor and higher
interactions are negligible, the main effects may
be estimated directly
• Minimal design: Resolution IV design with 2k
runs
• Construction: The process of fold over a 2 3III1
design (see Table 8.27)
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• Fold over resolution IV designs: (Montgomery
and Runger, 1996)
– Break as many two-factor interactions alias
chains as possible
– Break the two-factor interactions on a specific
alias chain
– Break the two-factor interactions involving a
specific factor
– For the second fraction, the sign is reversed on
every design generators that has an even
number of letters
43
• Resolution V designs: main effects and the twofactor interactions do not alias with the other main
effects and two-factor interactions.
44