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The 2k-p Fractional Factorial Design

• Text reference, Chapter 8• Motivation for fractional factorials is obvious; as the

number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly

• There may be many variables (often because we don’t know much about the system)

• Emphasis is on factor screening; efficiently identify the factors with large effects

• Almost always run as unreplicated factorials, but often with center points

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Why do Fractional Factorial Designs Work?

• The sparsity of effects principle– There may be lots of factors, but few are important– System is dominated by main effects, low-order

interactions

• The projection property– Every fractional factorial contains full factorials in

fewer factors

• Sequential experimentation– Can add runs to a fractional factorial to resolve

difficulties (or ambiguities) in interpretation

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The One-Half Fraction of the 2k

• Section 8-2, page 283

• Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1

• Consider a really simple case, the 23-1

• Choose {a, b, c, abc} as an one-half fraction

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• The 23-1 design contains only those treatment combinations with a “+” in the ABC column (ABC is a “generator” of the fraction)

• Note that I =ABC (defining relation)

• Main effects of A, B, and C

lA = ½(a-b-c+abc) = lBC

lB = ½(-a+b-c+abc) = lAC

lC = ½(-a-b+c+abc) = lAB

• It is impossible to differentiate between A and BC, B and AC, and C and AB – This phenomena is called aliasing and it occurs in all fractional designs (confounding)

• Aliases can be found directly from the columns in the table of + and - signs

• Notation for aliased effects:

• A = BC, B = AC, C = AB

, , A B CA BC B AC C AB

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• Aliases can be found from the defining relation I = ABC by multiplication:

AI = A(ABC) = A2BC = BC

BI =B(ABC) = AC

CI = C(ABC) = AB

• Using this fraction, instead of estimating A, we are estimating A+BC, etc.

• The two blocks/fractions can be determined by

I =+ABC

(principal fraction)

or

I =-ABC

(alternate/complementary fraction)

The One-Half Fraction of the 23

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The Alternate Fraction of the 23-1

• I = -ABC is the defining relation• Implies slightly different aliases: A = -BC, B= -

AC, and C = -AB• Both designs belong to the same family, defined by

• Suppose that after running the principal fraction, the alternate fraction was also run

• The two groups of runs can be combined to form a full factorial – an example of sequential experimentation

I ABC

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Running Both of the One-Half Fractions of the 23-1

• All the effects can be estimated by analyzing the full factorial 23 design, ordirectly from the two individual fractions. E.g.,

½(lA + l’A) = ½(A + BC + A – BC) -> A

½(lA - l’A) = ½(A + BC - A + BC) -> BC• Thus, all main effects and two factor interactions

will be estimated, but not three-factor interaction ABC. Why?

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Design Resolution• Resolution III Designs:

– Main effects are aliased with two-factor interactions– example

• Resolution IV Designs:– Two-factor interactions are aliased with each other– example

• Resolution V Designs:– Two-factor interactions are aliased with three-factor interactions– Example

• The resolution of a two-level fractional factorial design = the smallest number of letters in any word in the defining relation

3 12III

4 12IV

5 12V