Chapter%208%20Two-Level%20Fractional%20Factorial%20Designs

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Chapter 8 Two-Level Fractional Factorial Designs
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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.

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• Three key ideas
• 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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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

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• 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

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• Aliases
• 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

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• 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

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• 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

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• 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.

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• 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) gt I ABCK
• Another way is to partition the runs into two
  blocks with the highest-order interaction ABCK
  confounded.

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• 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

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• Example 8.1
• Example 6.2 A, C, D, AC and AD are important.
• Use 24-1 design with I ABCD

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• This design is the principal fraction, I
  ABCD
• Using the defining relation,
• A BCD, BACD, CABD, DABC
• ABCD, ACBD, BCAD

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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)

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• 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

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• Sequences of fractional factorial Both one-half
  fractions represent blocks of the complete design
  with the highest-order interaction confounded
  with blocks.

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• Example 8.3
• Reconsider Example 8.1
• Run the alternate fraction with I ABCD
• Estimates of effects
• Confirmation experiment

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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.

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• 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

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• 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

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• 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.

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• 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)

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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.

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• 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

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• 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

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• 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.

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• 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)

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• 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
• 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

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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.

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• 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)

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• 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.

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• 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

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• 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

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• 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

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• 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)

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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)

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• 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.

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• 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.

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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
  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

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• Resolution V designs main effects and the
  two-factor interactions do not alias with the
  other main effects and two-factor interactions.