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Fractional Factorial Designs. 2 7 – 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

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Fractional factorial designs l.jpg

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 (1:20,1:30)

    • 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:

  • A=DEG=ABDFG=CDF=-ABCF=BEF=ACEFG=-BCDEFG=BCG=-ACDG=-BD=ABCDE=-CE=-ABEG=-FG=-ADEF


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