The General 2k-p Fractional
Factorial Design
• 2k-1 = one-half fraction, 2k-2 = one-quarter fraction,
2k-3 = one-eighth fraction, …, 2k-p = 1/ 2p fraction
• Design matrix for a 2k-p: Add p columns to the
basic design; select p independent generators
• Defining relation: generating relations +
generalized interactions => aliases
• Important to select generators so as to maximize
resolution, see Table 8-14 page 305
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The General 2k-p Design: Resolution may not
be sufficient to distinguish between designs
• Minimum aberration designs: to achieve fewest
number of words of min. length
• Word length patterns:
{4, 4, 4}
{4, 4, 6}
{4, 5, 5}
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The General 2k-p Fractional
Factorial Design
• Usually try to achieve highest resolution and
minimum aberration designs
• Table 8-14: Selection of 2k-p for k  15 factors
• Appendix Table X: alias structures
• Example 8-5
• Seven factors
• Estimate seven main effects and some 2-factor
interactions
• 3-factor and higher order interactions are
negligible
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Example 8-5
• Why not choose R = 7?
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The General 2k-p Fractional Factorial Design
• Techniques for resolving the ambiguities in aliased
effects:
o Use prior knowledge
One effect is not likely to be important
Interactions between factors in different
subprocesses may not be important
o Conduct follow-up experiments at different
factor settings than the original experiment
(Pages 167-175, Wu and Hamada)
Method of adding orthogonal runs
Optimal design approach
Fold-over technique
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The General 2k-p Fractional Factorial Design
• Projection
• Any subset of rk-p factors which doesn’t
appear as a word in the complete defining
relation forms a full factorial design
• Any subset of rk-p factors which appears as a
word in the complete defining relation forms a
replicated fractional factorial design
• A design of resolution R contains full factorials
in any R – 1 of the factors
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The General 2k-p Fractional Factorial Design
• Reconsider Example 8-5 (27IV3 )
• It projects into a full factorial in any four (4) of
the original seven (7) factors that is not a word
in the defining relation (28 such subsets or
designs)
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The General 2k-p Fractional Factorial Design
• Blocking
62
2
• Consider the IV fractional design
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The General 2k-p Fractional Factorial Design
• Run the design in two blocks of eight treatment
combinations each
• Select an interaction to confound
• Choose L = x1 + x2 + x4 = 0
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Example 8-6
• Machining an impeller – determine effects on
profile deviation (standard deviation of the
difference is the response)
• Eight factors were selected (many of them may
have little effects – but not sure)
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Example 8-6
• The machine has four spindles – four blocks
• Three-factor and higher interactions are not
important, but two-factor interactions may be
important
• Table 8-14 => two designs appear appropriate:
28IV 4 with 16 runs, or 28IV3 with 32 runs
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12
13
83
• Choose 2 IV
• Use a log
transformation
• Large effects: A,
B, and AD + BG
• Difficult to
separate AD and
BG
 Engineering
knowledge, or
 Adding more runs
• Assume AD is
important
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15
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Summary
• The main practical motivation for choosing a fractional
factorial design is run size economy. Aliasing of effects is
the price paid for economy.
• A 2k-p design is determined by its defining contrast
subgroup, which consists of 2p-1 defining words and the
identity I. The length of its shortest word is the resolution
of the design.
• A main effect or two-factor interaction is said to be clear if
none of its aliases are main effects or two-factor
interactions and strongly clear if none of its aliases are
main effects or two-factor or three-factor interactions.
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• Useful rules for resolution IV or V designs:
 In any resolution IV design, the main effects are clear
 In any resolution V design, the main effects are strongly
clear and the two-factor interactions are clear.
 Among the resolution IV designs with given k and p,
those with the largest number of clear two-factor
interactions are the best.
• Projective property of resolution: the projection of a
resolution R design onto any R-1 factors is a full factorial
in the R-1 factors
• Unless there is specific information on the relative
importance among the factorial effects, the minimum
aberration criterion should be used for selecting good 2k-p
designs. Minimum aberration automatically implies
maximum resolution.
• A blocked 2k-p design with more clear effects is considered
to be better.
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• The analysis of fractional factorial experiments is the same
as in the full factorial experiments except that the observed
significance of an effect should be attributed to the
combination of the effect and all its aliased effects.
• Optimal factor settings can be determined by the signs of
significant effects and the corresponding interaction plots.
• Two approaches to resolve ambiguities in aliased effects
are:
 Use a priori knowledge to dismiss some of the aliased
effects. If a lower order effect is aliased with a higher
order effect, the latter may be dismissed.
 Run a follow-up experiment
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