1 The 2 k-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

2 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

3 The One-Half Fraction of the 2 k Section 8-2, page 283 Notation: because the design has 2 k /2 runs, it’s referred to as a 2 k-1 Consider a really simple case, the Choose {a, b, c, abc} as an one-half fraction

4 The 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 l A = ½(a-b-c+abc) = l BC l B = ½(-a+b-c+abc) = l AC l C = ½(-a-b+c+abc) = l AB 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

5 Aliases can be found from the defining relation I = ABC by multiplication: AI = A(ABC) = A 2 BC = 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 2 3

6 The Alternate Fraction of the 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

7 Running Both of the One-Half Fractions of the All the effects can be estimated by analyzing the full factorial 2 3 design, or directly from the two individual fractions. E.g., ½( l A + l’ A ) = ½(A + BC + A – BC) -> A ½( l A - 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?

8 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