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

Run A B C D E F G Y1 -1 -1 -1 1 1 1 -1 1.182 1 -1 -1 -1 -1 1 1 22.303 -1 1 -1 -1 1 -1 1 23.104 1 1 -1 1 -1 -1 -1 1.735 -1 -1 1 1 -1 -1 1 27.006 1 -1 1 -1 1 -1 -1 1.727 -1 1 1 -1 -1 1 -1 0.568 1 1 1 1 1 1 1 39.00

Contrast 12.91 12.19 19.97 21.23 13.41 9.49 106.21 14.57