1. 1 Chapter 8 Two-level Fractional Factorial Designs

2. 2 Fractional Factorial Design Only a fraction of the complete factorial experiment is running Motivation for fractional factorials ; as the number of factors becomes large, 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 These fractional factorial designs are among the most widely used types of designs for product and process design and for process improvement 2

3. 3 The 1/2 Fraction of the 2 k Design The design has 2 k /2 runs, referred to as a 2 k-1 Consider 2 3-1 design Suppose we select the four treatment combinations a, b, c and abc as an one-half fraction Such a 2 3-1 design is formed by selecting only those treatment combinations that have a plus in the ABC column ABC is called the generator (or word) of this fraction 3

4. 4 The defining relation for this design is I=ABC In general, the defining relation for a fractional factorial will always be the set of all columns that are equal to the identity column I Notice that A=BC, B=AC and C=AB Thus, it is impossible to differentiate between A and BC, B and AC, and C and AB They are called aliases 4 Figure 8.1 The two one-half fractions of the 2 3 design

5. 5 Notation for aliased effects: [A],[B], [C] : the linear combinations associated with the main effects [AB],[BC][AC]: the linear combinations associated with the two-factor interactions

6. 6 When the defining relation is I=ABC, alias can be found  [A]=[BC] A=AI=A(ABC)=A 2 BC=BC  [B]=[AC] B=BI=B(ABC)=AB 2 C=AC  [C]=[AB]  I=ABC is called the principal fraction  When we estimate A,B and C, we are really estimate A+BC, B+AC, and C+AB When the defining relation is I=-ABC, alias can be found  [A]=-[BC]  [B]=-[AC]  [C]=-[AB]  I=-ABC is called the alternate (or complementary) fraction  When we estimate A,B and C, we are really estimate A-BC, B-AC, and C- AB 6

7. 7 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 – a sequential experimentation  Now we have all 8 runs so as to de-aliase estimates of all effects by analyzing the eight runs as a full 2 3 design in two blocks as the following 7

8. 8 Design Resolution A term to describe the alias and confounding happening between the main and interaction effects Resolution R Design: no p-factor effect is aliased with another effect containing less than R-p factors  Resolution III Design: No main effects are aliased with any other main effect. ( Main effects are aliased with two-factor interaction and some two-factor interactions may be aliased with each other) Ex) 2 3-1 design with the defining relation I = ABC or I =- ABC (2 III 3-1 )  Resolution IV Design: No main effect is aliased with any other main effect or with any two-factor interaction. (Two factor interactions are aliased with each other) Ex) 2 4-1 design with the defining relation I = ABCD or I =- ABCD (2 IV 4-1 )  Resolution V Design: No main effect or two-factor interaction is aliased with any other main effect or with any two-factor interaction. However, two factor interactions are aliased with three-factor interactions Ex) 2 5-1 design with the defining relation I = ABCDE or I =- ABCDE (2 V 5-1 ) 8

9. 9 Resolution of a design=>The higher, the better The higher the resolution, the less restrictive the assumptions that are required regarding which interactions are negligible to obtain a unique interpretation of the data. 9

10. 10 Construction of 1/2 Fraction Write down a full 2 k-1 factorial design Add the k th factor by identifying its plus and minus levels with the signs of ABC…(K – 1) K = ABC…(K – 1) ⇒ I = ABC…K defines a column for k th factor Another way is to partition the runs into two blocks with the highest-order interaction ABC…K confounded. Each block is a 2 k-1 fractional factorial design of the highest resolution. 10

11. A chemical product is produced in a pressure vessel A factorial experiment is carried out in the pilot plant to study the factors that influence the filtration rate of this product Factor A: temperature Factor B: pressure Factor C: concentration of formaldehyde Factor D: stirring rate 11 Example

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13. With the full factorial design, we found that the main effect A,C,D and the interactions AC and AD were different from zero Now suppose that we run a half-fraction of the 2 4 design with I=ABCD 13

14. This 2 IV 4-1 design is the principal fraction, I = ABCD Using the defining relation, A = BCD, B=ACD, C=ABD, D=ABC, AB=CD, AC=BD, BC=AD 14

15. 15 A, C and D are large Since A, C and D are important factors, the significant interactions are most likely AC and AD Project this one-half design into a single replicate of the 2 3 design in factors, A, C and D

16. Figure 8.10 Projection of the 2 IV 4-1 design into a 2 3 design in the factors A, C and D 16 The regression model for predicting filtration rates is where x 1, x 3 and x 4 are the coded variables representing A, C and D, respectively.

17. Five factors in a manufacturing process for an IC were investigated for improving the process yield A =aperture setting (small, large) B =exposure time (20% below nominal, 20% above nominal) C =development time (30s, 45s) D =mask dimension (small, large) E =etch time (14.5 min, 15.5 min) Use a 2 5-1 design with I = ABCDE 17 Example

18. 18 Every main effect is aliased with four-factor interaction, and two-factor interaction is aliased with three-factor interaction.

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20. Figure 8.6 Normal probability plot of effects 20

21. Because of aliasing of AB+CDE A+BCDE C+ABDE B+ACDE If we can assume that three factor and higher interactions are negligible, we conclude that only A,B,C and AB are important A, B and C are large 21

22. Normal probability plot of the residuals 22 Plot of residuals versus predicted yield

23. Aperture-exposure time interaction 23

24. 24 Sequences of Fractional Factorials Suppose that we are investigating k=4 factors (2 4 =16 runs) It is preferable to run a 2 IV 4-1 fractional design (8 runs), analyze the results and then decide on the best set of runs to perform next If it is necessary to resolve ambiguities, we can run the alternate fraction and complete 2 4 design Both one-half fractions represent blocks of the complete design with the highest-order interaction confounded with blocks. 24

25. Figure 8.11 Possibilities for follow-up experimentation after a fractional factorial experiment 25

26. Reconsider the previous experiment. We used a 2 IV 4-1 design and tentatively identified three large main effects-A,C and D There are two large effects associated with two-factor interactions, AC+BD and AD+BC We used the fact that the main effect of B was negligible to tentatively conclude that the important interactions were AC and AD. 26 Example

27. The effect estimates obtained from the alternate fraction are [A]' =24.25 → A-BCD [B]' = 4.75 → B-ACD [C]' = 5.75 → C-ABD [D]' = 12.75 → D-ABC [AB]' = 1.25 → AB-CD [AC]' = -17.75 → AC-BD [AD]' = 14.25 → AD-BC 27 These estimates are combined with those obtained from the original one-half fraction to get the estimates of the effects