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Fractional Factorial Designs

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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  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 (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

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

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