6. Fractional Factorial Designs (Ch.8. Two-Level Fractional Factorial Designs)
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1 6. Fractional Factorial Designs (Ch.8. Two-Level Fractional Factorial Designs) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1
2 Introduction to The 2 k-p Fractional Factorial Design 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 Emphasis is on factor screening; efficiently identify the factors with large effects There may be many variables (often because we don t know much about the system) Almost always run as unreplicated factorials 2
3 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
4 The One-Half Fraction of the 2 k Notation: because the design has 2 k /2 runs, it s referred to as a 2 k-1 Consider a really simple case, the Note that I =ABC 4
5 The One-Half Fraction of the 2 3 For the principal fraction, notice that the contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction. This phenomena is called aliasing and it occurs in all fractional designs Aliases can be found directly from the columns in the table of + and - signs 5
6 Projection of Fractional Factorials Every fractional factorial contains full factorials in fewer factors The flashlight analogy A one-half fraction will project into a full factorial in any k 1 of the original factors 6
7 Aliasing in the One-Half Fraction of the 2 3 A = BC, B = AC, C = AB (or me = 2fi) Aliases can be found from the defining relation I = ABC by multiplication ABC is called the generator. AI = A(ABC) = A 2 BC = BC BI =B(ABC) = AC CI = C(ABC) = AB 7
8 Aliasing in the One-Half Fraction of the 2 3 Main effect 1 A a b c abc 2 1 B a b c abc 2 1 C a b c abc 2 Two factor interaction effect 1 BC a b c abc 2 1 AC a b c abc 2 1 AB a b c abc 2 Alias structure of effects [ A] A + BC, [ B] B + AC, [ C] C + AB 8
9 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 I = ± ABC [ A]' A BC, [ B]' B AC, [ C]' C AB 9
10 Design Resolution Resolution III Designs: me = 2fi (i.e., main effect = 2 factor interaction) 3 1 example Resolution IV Designs: 2fi = 2fi example Resolution V Designs: 2fi = 3fi example 2 III IV V 10
11 Construction of a One-half Fraction 11
12 Resin Plant Experiment the Design A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product. The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate A fractional factorial was used to investigate the effects of four factors on the filtration rate of a resin Generator I = ABCD 12
13 Resin Plant Experiment the Design 13
14 Aliasing the 2 IV 4-1 Factorial Design Resolution IV design with the generator I=ABCD Main effect is aliased with three factor interaction A=A 2 BCD=BCD; B=AB 2 CD=ACD; C=ABC 2 D=ABD; D=ABCD 2 =ABC; Two factor interaction is aliased with other two factor interaction AB=CD; AC=BD; AD=BC; 14
15 Resin Plant Experiment the Design Interpretation of results often relies on making some assumptions Ockham s razor Confirmation experiments can be important Adding the alternate fraction see page
16 Resin Plant Experiment MINITAB Results 16
17 Resin Plant Experiment MINITAB Results Zero degree of freedom for residuals y x x x x xx xx xx y x x x xx xx degree of freedom for residuals 17
18 Resin Plant Experiment MINITAB Results ŷ ˆ ˆ x ˆ x ˆ x ˆ xx ˆ xx yˆ x x x xx xx
19 Resin Plant Experiment MINITAB Results yˆ x x x xx xx For example the residual at x 1, x 1, x 1, x y yˆ (1) ( 1) (1) (1)( 1) (1)(1)
20 Resin Plant Experiment MINITAB Results 20
21 Manufacturing Process for a Circuit Five factors in a manufacturing process for an integrated circuit were investigated in a design with the objective of improving the process yield. Select ABCDE as the generator (Resolution V design) I=ABCDE ; E=ABCD ; Every main effect is aliased with a four-factor interaction. E.g., [A] -> A+BCDE Every two factor interaction is aliased with a three-factor interaction. E.g., [AB]-> AB+CDE 21
22 Manufacturing Process MINITAB Results 22
23 Manufacturing Process MINITAB Results A, B, C, and AB are significant 23
24 Manufacturing Process MINITAB Results Selecting only A, B, C, and AB This implies 2 3 Design with 2 replicates at each experimental point 24
25 Manufacturing Process MINITAB Results ANOVA Residual analysis 25
26 Manufacturing Process MINITAB Results Interaction Plot of AB Cube Plot 26
27 The Sequential Experimentation 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 De-aliased estimates of the effects can be obtained by adding and subtracting 1 1 ([ A] [ A]') ( A BC A BC) A ([ A] [ A]') ( A BC A BC) BC
28 The Sequential Experimentation If it is necessary to resolve ambiguities, we can run the alternate fraction and complete 2 k design. Run 1 Run 2 28
29 Resin Plant Experiment Alternate Fraction Recall the resin plant experiment Generator I=-ABCD [ A] 19 A BCD (from main fraction) 1 [ A]' ( ) A BCD (from alternative fraction) Main Effect of original design 1 A [ A ] [ A ]'
30 The One-Quarter Fraction of the 2 k 30
31 The One-Quarter Fraction of the
32 The General 2 k-p Fractional Factorial Design 2 k-1 = one-half fraction, 2 k-2 = one-quarter fraction, 2 k-3 = one-eighth fraction,, 2 k-p = 1/ 2 p fraction Add p columns to the basic design; select p independent generators Important to select generators so as to maximize resolution, see the table in the next slide Projection a design of resolution R contains full factorials in any R 1 of the factors Effects of factors are Effect where N i Contrasti ( N / 2) = number of observations 32
33 The General 2 k-p Design Resolution may not be sufficient Minimum abberation designs Our choice 33
34 34
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