1. 1 Chapter 8 Two-level Fractional Factorial Designs

2. 2 Construction of 1/4 Fraction One quarter fraction of the 2 k design contains 2 k-2 runs and is called a 2 k-2 fractional factorial Write down a full factorial in k-2 factors Add two columns with appropriately chosen interactions involving the first k-2 factors two generators P and Q I = P and I = Q ; “generating relations” for the design The signs of P and Q determine which one of the one-quarter fractions is produced  I=P, I=Q : principal fraction  I=P, I=-Q  I=-P, I=Q alternate fractions  I=-P, I=-Q 2 4 fractions: members of the same family

3. 3 The complete defining relation consists of all the columns that are equal to the identity column I i.e. the defining relation is I=P=Q=PQ (PQ: generalized interaction) P, Q and PQ are called words Each effect has three aliases Potentially important effects should not be aliased with each other => Be careful in choosing the generators 3

4. 4 For example, a one-quarter fraction of the 2 6-2 with I = ABCE and I = BCDF. The complete defining relation is I = ABCE = BCDF = ADEF Every main effect is aliased by three- and five factor interactions ex) A= BCE=ABCDF=DEF => When we estimate A, we are really estimate A+BCE+DEF+ABCDF Two-factor interactions are aliased with each other higher order interactions. ex) AB=CE=ACDF=BDEF  Resolution IV design.

5. 5 Design construction : method 1  Write down the basic design (a full 2 4 design in A,B,C, and D)  Two factors E and F are added by associating their plus and minus levels with the plus and minus signs of the interactions ABC and BCD, respectively. (Aliased as E=BCD, F=BCD) Construct such design: method 2  Derive the four blocks of the 2 6 design with ABCE and BCDF confounded  Choose the block with treatment combination that are + on ABCE and BCDF The 2 6-2 design with I = ABCE and I = BCDF is the principal fraction Three alternate fractions: I = ABCE and I = - BCDF I = -ABCE and I = BCDF I = - ABCE and I = -BCDF

6. 6 This 2 IV 6-2 fractional factorial will project into  A single replicate of a 2 4 design in any subset of four factors that is not a word in the defining relation  Two replicates of one-half fraction of a 2 4 in any subset of four factors that is a word in the defining relation In general, any 2 k-2 fractional factorial design can be collapsed into either a full factorial or a fractional factorial in some subset of r ≤ k–2 of the original factors 6

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8. A quality improvement team uses a designed experiment to study the injection molding process so that shrinkage can be reduced Six factors are considered  mold temperature (A)  screw speed (B)  holding time (C)  cycle time (D)  gate size (E)  holding pressure (F) 8 Example

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11. Normal probability plot of effects 11 Plot of AB (mold temperature-screw speed) interaction

12. Normal probability plot of residuals 12 Residuals versus holding time

13. 13 General 2 k-p fractional factorial A 1/ 2 p fraction of the 2 k design Need p independent generators, and there are 2 p –p – 1 generalized interactions Each effect has 2 p – 1 aliases A reasonable criterion: the highest possible resolution, and less aliasing Minimum aberration design: minimize the number of words in the defining relation that are of minimum length. Minimizing aberration of resolution R ensures that a design has the minimum # of main effects aliased with interactions of order R – 1, the minimum # of two-factor interactions aliased with interactions of order R – 2, …. 13

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15. We have seven factors and we want to estimate all main effects and get some insight regarding the two-factor interactions. Three-factor and higher interactions are negligible. Resolution IV design would be proper 2 IV 7-3 and 2 IV 7-2 designs in Appendix Table XII (Page 666) 2 IV 7-3 design: main effects are aliased with three-factor interactions and two- factor interactions are aliased with two-factor interactions 2 IV 7-2 design: all main effects and 15 of 21 two-factor interactions The i th effect is estimated by Project into any subset of r ≤ k – p of the original factors: a full factorial or a fractional factorial (if the subsets of factors are appearing as words in the complete defining relation) Very useful in screening experiments For example 2 IV 7-3 design: choose any four of seven factors. Then 7 of 35 subsets are appearing in complete defining relations. 15 Example

16. 16 Resolution III Designs Designs with main effects aliased with two-factor interactions Used for screening (for example, 5-7 variables in 8 runs, 9-15 variables in 16 runs) When k=N-1, the fractional factorial design is a saturated design Example: 2 III 3-1 2 III 7-4 16

17. Sequential assembly of fractions to separate aliased effects Switching the signs in one column provides estimates of that factor and all of its two-factor interactions – called a single factor fold over Switching the signs in all columns dealiases all main effects from their two- factor interaction alias chains – called a full fold over Consider 2 III 7-4 with Table 8-19 Suppose that a second fractional design with the signs reversed in the column for factor D is also run We have isolated the main effect of D and all of its two-factor interactions 17

18. A human performance analyst is conducting an experiment to study eye focus time and has built an apparatus in which several factors can be controlled during the test The factors he initially regards as important A: sharpness of vision B: distance from target to eye C: target shape D: illumination level E: target size F: target density G: subject 18 Example

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20. 20 The interpretation of the result is not unique This 2 III 7-4 design does not project into a full 2 3 factorial in ABD It projects into two replicates of a 2 3-1 design which is a resolution III design A is aliased with BD, B is aliased with AD and D is aliased with AB To separate the main effects and the two-factor interactions, the full fold over design is used

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22. 22 Plackett-Burman Designs These are a different class of resolution III design The number of runs, N, need only be a multiple of four N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,… N =12,20,24,28 and 36 are of interest Table 8-23 presents are used to construct the Plackett-Burman Designs 22

23. 23 The alias structure is complex in the PB designs For example, with N = 12 and k = 11, every main effect is aliased with every two-factor interaction not involving itself Partial aliasing can greatly complicate interpretation Interactions can be particularly disruptive Use very, very carefully

24. 24 Example

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