1. 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

2. Fractional Factorial Designs For large numbers of treatments (k), the total number of runs for a full factorial can get very large (2 k ) 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 2 k - Factorial 2 k can be run in 2 q block of size 2 k-q for q=,1…,k-1 2 k-q factorial is design with k factors in 2 k-q runs 1 Block of a confounded 2 k 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  2 7 = 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 = H 2 O 2 (0, 20 ml/L)  G = Enzyme in percent (0, 2)  Response: Y = % Weight Loss Experiment: Conducted in 2 k-q = 8 runs (1/16 fraction) Need 2 4 -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