2. 2
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.
3. 3
• 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, low-
order interactions
1. The projection property
– Every fractional factorial contains full
factorials in fewer factors
1. Sequential experimentation
– Can add runs to a fractional factorial to
resolve difficulties (or ambiguities) in
interpretation
4. 4
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
5. 5
• 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:
• A = BC, B = AC, C = AB
( )
( )
( ) ABC
ACB
BCA
abccba
abccba
abccba



=++−−=
=+−+−=
=+−−=
2
1
2
1
2
1
6. 6
• Aliases:
• Aliases can be found from the defining relation I
= ABC by multiplication:
AI = A(ABC) = A2
BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
• Principal fraction: I = ABC
, ,A B CA BC B AC C AB→ + → + → +  
7. 7
• 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.
• 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
experimentation
ABCACBBCA CBA −→−→−→ '''
,, 
8. 8
• The de-aliased estimates of all effects by
analyzing the eight runs as a full 23
design in two
blocks. Hence
• 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 design
( ) ( )
( ) ( ) BCBCABCA
ABCABCA
AA
AA
→+−+=−
→−++=+
2
1
2
1
2
1
2
1
'
'


13
2 −
III
9. 9
• 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.
10. 10
• 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.
11. 11
12. 12
• 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
13. 13
• Example 8.1:
– Example 6.2: A, C, D, AC and AD are
important.
– Use 24-1
design with I = ABCD
14. 14
• This 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
4 1
2IV
−
15. 15
• 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)
16. 16
• 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
– Collapse into two replicates of a 23
design
17. 17
• Sequences of
fractional factorial:
Both one-half
fractions represent
blocks of the
complete design
with the highest-
order interaction
confounded with
blocks.
18. 18
• Example 8.3:
– Reconsider Example 8.1
– Run the alternate fraction with I = – ABCD
– Estimates of effects
– Confirmation experiment
19. 19
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
– All four fractions are the family.
20. 20
21. 21
• 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
22. 22
• 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
23. 23
• This fractional 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.
26
2 −
IV
24. 24
• 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)
25. 25
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.
26. 26
• 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
27. 27
• Example 8.5
– Estimate all main effects and get some insight
regarding the two-factor interactions.
– Three-factor and higher interactions are
negligible.
– designs in Appendix Table XII
(Page 666)
– 16-run design: main effects are aliased with
three-factor interactions and two-factor
interactions are aliased with two-factor
interactions
– 32-run design: all main effects and 15 of 21
two-factor interactions
3727
2and2 −−
IVIV
27
2 −
IV
37
2 −
IV
28. 28
• Analysis of 2k-p
Fractional Factorials:
– For the ith effect:
• 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
– For example 16-run design: Choose any
four of seven factors. Then 7 of 35 subsets are
appearing in complete defining relations.
pNii
i N
N
Contrast
N
Contrast −
=== 2,
2/
)(2

37
2 −
IV
29. 29
• Blocking Fractional Factorial:
– Appendix Table XII
– Consider the 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
–
– Four blocks
– Effect estimates and sum of squares (Table 8.17)
– Normal probability plot of the effect estimates
(see Figure 8.19)
26
2 −
IV
3848
2or2 −−
IVIV
30. 30
• 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 –
31. 31
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
• In Section 8.2, there is an example, design.
• Another example is shown in Table 8.19: design
I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG
= AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG
13
2 −
III
47
2 −
III
32. 32
• 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.
33. 33
• Assume that three-factor and higher interactions
are negligible.
• The saturated 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)
47
2 −
III
34. 34
• 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.
35. 35
• 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
36. 36
• Example 8.7
– Seven factors to study eye focus time
– Run 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
47
2 −
III
37. 37
• 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
38. 38
• 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
= 4 n
– 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)
39. 39
• 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)
40. 40
• Projection: Consider the
12-run PB design
– 3 replicates of a full 22
design
– A full 23
design + a
design
– Projection into 4 factors is
not a balanced design
– Projectivity 3: collapse
into a full fractional in any
subset of three factors.
13
2 −
III
41. 41
• Example 8.8:
– Use a set of simulated data and the 11 factors, 12-
run 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.
42. 42
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
design (see Table 8.27)
13
2 −
III
43. 43
• 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
44. 44
• Resolution V designs: main effects and the two-
factor interactions do not alias with the other main
effects and two-factor interactions.
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