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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). – PowerPoint PPT presentation

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Title: Fractional Factorial Designs


1
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

2
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

3
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

4
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

5
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

6
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
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