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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

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

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

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

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

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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