Fractional Factorial Designs

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Fractional Factorial Designs

• 27 Factorial Design in 8 Experimental Runs to
  Measure Shrinkage in Wool Fabrics
• J.M. Cardamone, J. Yao, and A. Nunez (2004).
  Controlling Shrinkage in Wool Fabrics Effective
  Hydrogen Peroxide Systems, Textile Research
  Journal, Vol. 74 pp. 887-898

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Fractional Factorial Designs

• For large numbers of treatments (k), the total
  number of runs for a full factorial can get very
  large (2k)
• Many degrees of freedom are spent on high-order
  interactions (which are often pooled into error
  with marginal gain in added degrees of freedom)
• Fractional factorial designs are helpful when
• High-order interactions are small/ignorable
• We wish to screen many factors to find a small
  set of important factors, to be studied more
  thoroughly later
• Resources are limited
• Mechanism Confound full factorial in blocks of
  target size, then run only one block

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Fractioning the 2k - Factorial

• 2k can be run in 2q block of size 2k-q for
  q,1,k-1
• 2k-q factorial is design with k factors in 2k-q
  runs
• 1 Block of a confounded 2k factorial
• Principal Block is called the principal fraction,
  other blocks are called alternate fractions
• Procedure
• Augment table of 2-series with column of ,
  labeled I
• Defining contrasts are effects to be confounded
  together
• Generators are used to create the blocks by /-
  structure
• Generalized Interactions of Generators also have
  constant sign in blocks
• Defining Relations I A, I -B ? I -AB

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Example Wool Shrinkage

• 7 Factors ? 27 128 runs in full factorial
• A NaOH in grams/litre (1 , 3)
• B Liquor Dilution Ratio (120,130)
• C Time in minutes (20 , 40)
• D GA in grams/litre (0 , 1)
• E DD in grams/litre (0 , 3)
• F H2O2 (0 , 20 ml/L)
• G Enzyme in percent (0 , 2)
• Response Y Weight Loss
• Experiment Conducted in 2k-q 8 runs (1/16
  fraction)
• Need 24-1 Defining Contrasts/Generalized
  Interactions
• 4 Distinct Effects, 6 multiples of pairs, 4
  triples, 1 quadruple

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Defining Relations

• I ADEG BDFG ACDF -BCF
• Generalized Interactions
• (ADEG)(BDFG)ABEF,(ADEG)(ACDF)CEFG,(ADEG)(-BCF)-
  ABCDEFG
• (BDFG)(ACDF)ABCG,(BDFG)(-BCF)-CDG,(ACDF)(-BCF)-
  ABD
• (ADEG)(BDFG)(ACDF)BCDE, (ADEG)(BDFG)(-BCF)-ACE
• (ADEG)(ACDF)(-BCF)-BEG, (BDFG)(ACDF)(-BCF)-AFG
• (ADEG)(BDFG) (ACDF)(-BCF)-DEF
• Goal Choose block where ADEG,BDFG,ACDF are
  even and BCF is odd. All other generalized
  interactions will follow directly

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Aliased Effects and Design

• To Obtain Aliased Effects, multiply main effects
  by Defining Relation to obtain all effects
  aliased together
• For Factor A
• ADEGABDFGCDF-ABCFBEFACEFG-BCDEFGBCG-ACDG
  -BDABCDE-CE-ABEG-FG-ADEF