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## Fractional Factorial Designs

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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). – PowerPoint PPT presentation

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Title: Fractional Factorial Designs

1
Fractional Factorial Designs
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
• 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 (120,130)
• 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

5
Defining Relations
• I ADEG BDFG ACDF -BCF
• Generalized Interactions
ABCDEFG
• (BDFG)(ACDF)ABCG,(BDFG)(-BCF)-CDG,(ACDF)(-BCF)-
ABD
• Goal Choose block where ADEG,BDFG,ACDF are
even and BCF is odd. All other generalized

6
Aliased Effects and Design
• To Obtain Aliased Effects, multiply main effects
by Defining Relation to obtain all effects
aliased together
• For Factor A