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
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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:
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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
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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:
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(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
1
2
3
4
5
6
7
8
Contrast
A
-1
1
-1
1
-1
1
-1
1
12.91
B
-1
-1
1
1
-1
-1
1
1
12.19
C
-1
-1
-1
-1
1
1
1
1
19.97
D
1
-1
-1
1
1
-1
-1
1
21.23
E
1
-1
1
-1
-1
1
-1
1
13.41
F
1
1
-1
-1
-1
-1
1
1
9.49
G
-1
1
1
-1
1
-1
-1
1
106.21
Y
1.18
22.30
23.10
1.73
27.00
1.72
0.56
39.00
14.57