0
Upcoming SlideShare
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Standard text messaging rates apply

# Two level fractional factorial (chap 8)

1,019

Published on

Published in: Technology, Design
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
1,019
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
19
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Transcript

• 1. 1 Chapter 8 Two-Level Fractional Factorial Designs
• 2. 2 8.1 Introduction &#x2022; The number of factors becomes large enough to be &#x201C;interesting&#x201D;, the size of the designs grows very quickly &#x2022; After assuming some high-order interactions are negligible, we only need to run a fraction of the complete factorial design to obtain the information for the main effects and low-order interactions &#x2022; Fractional factorial designs &#x2022; Screening experiments: many factors are considered and the objective is to identify those factors that have large effects.
• 3. 3 &#x2022; Three key ideas: 1. The sparsity of effects principle &#x2013; There may be lots of factors, but few are important &#x2013; System is dominated by main effects, low- order interactions 1. The projection property &#x2013; Every fractional factorial contains full factorials in fewer factors 1. Sequential experimentation &#x2013; Can add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation
• 4. 4 8.2 The One-half Fraction of the 2k Design &#x2022; Consider three factor and each factor has two levels. &#x2022; A one-half fraction of 23 design is called a 23-1 design
• 5. 5 &#x2022; In this example, ABC is called the generator of this fraction (only + in ABC column). Sometimes we refer a generator (e.g. ABC) as a word. &#x2022; The defining relation: I = ABC &#x2022; Estimate the effects: &#x2022; A = BC, B = AC, C = AB ( ) ( ) ( ) ABC ACB BCA abccba abccba abccba &#xF06C;&#xF06C; &#xF06C;&#xF06C; &#xF06C;&#xF06C; =++&#x2212;&#x2212;= =+&#x2212;+&#x2212;= =+&#x2212;&#x2212;= 2 1 2 1 2 1
• 6. 6 &#x2022; Aliases: &#x2022; Aliases can be found from the defining relation I = ABC by multiplication: AI = A(ABC) = A2 BC = BC BI =B(ABC) = AC CI = C(ABC) = AB &#x2022; Principal fraction: I = ABC , ,A B CA BC B AC C AB&#x2192; + &#x2192; + &#x2192; +&#xF06C; &#xF06C; &#xF06C;
• 7. 7 &#x2022; The Alternate Fraction of the 23-1 design: I = - ABC &#x2022; When we estimate A, B and C using this design, we are really estimating A &#x2013; BC, B &#x2013; AC, and C &#x2013; AB, i.e. &#x2022; Both designs belong to the same family, defined by I = &#xF0B1; ABC &#x2022; Suppose that after running the principal fraction, the alternate fraction was also run &#x2022; The two groups of runs can be combined to form a full factorial &#x2013; an example of sequential experimentation ABCACBBCA CBA &#x2212;&#x2192;&#x2212;&#x2192;&#x2212;&#x2192; ''' ,, &#xF06C;&#xF06C;&#xF06C;
• 8. 8 &#x2022; The de-aliased estimates of all effects by analyzing the eight runs as a full 23 design in two blocks. Hence &#x2022; Design resolution: A design is of resolution R if no p-factor effect is aliased with another effect containing less than R &#x2013; p factors. &#x2022; The one-half fraction of the 23 design with I = ABC is a design ( ) ( ) ( ) ( ) BCBCABCA ABCABCA AA AA &#x2192;+&#x2212;+=&#x2212; &#x2192;&#x2212;++=+ 2 1 2 1 2 1 2 1 ' ' &#xF06C;&#xF06C; &#xF06C;&#xF06C; 13 2 &#x2212; III
• 9. 9 &#x2022; Resolution III Designs: &#x2013; me = 2fi &#x2013; Example: A 23-1 design with I = ABC &#x2022; Resolution IV Designs: &#x2013; 2fi = 2fi &#x2013; Example: A 24-1 design with I = ABCD &#x2022; Resolution V Designs: &#x2013; 2fi = 3fi &#x2013; Example: A 25-1 design with I = ABCDE &#x2022; In general, the resolution of a two-level fractional factorial design is the smallest number of letters in any word in the defining relation.
• 10. 10 &#x2022; 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. &#x2022; Constructing one-half fraction: &#x2013; Write down a full 2k-1 factorial design &#x2013; Add the kth factor by identifying its plus and minus levels with the signs of ABC&#x2026;(K &#x2013; 1) &#x2013; K = ABC&#x2026;(K &#x2013; 1) =&gt; I = ABC&#x2026;K &#x2013; Another way is to partition the runs into two blocks with the highest-order interaction ABC&#x2026;K confounded.
• 11. 11
• 12. 12 &#x2022; Any fractional factorial design of resolution R contains complete factorial designs in any subset of R &#x2013; 1 factors. &#x2022; A one-half fraction will project into a full factorial in any k &#x2013; 1 of the original factors
• 13. 13 &#x2022; Example 8.1: &#x2013; Example 6.2: A, C, D, AC and AD are important. &#x2013; Use 24-1 design with I = ABCD
• 14. 14 &#x2022; This design is the principal fraction, I = ABCD &#x2022; Using the defining relation, &#x2013; A = BCD, B=ACD, C=ABD, D=ABC &#x2013; AB=CD, AC=BD, BC=AD 4 1 2IV &#x2212;
• 15. 15 &#x2022; A, C and D are large. &#x2022; Since A, C and D are important factors, the significant interactions are most likely AC and AD. &#x2022; Project this one-half design into a single replicate of the 23 design in factors, A, C and D. (see Figure 8.4 and Page 310)
• 16. 16 &#x2022; Example 8.2: &#x2013; 5 factors &#x2013; Use 25-1 design with I = ABCDE (Table 8.5) &#x2013; Every main effect is aliased with four-factor interaction, and two-factor interaction is aliased with three-factor interaction. &#x2013; Table 8.6 (Page 312) &#x2013; Figure 8.6: the normal probability plot of the effect estimates &#x2013; A, B, C and AB are important &#x2013; Table 8.7: ANOVA table &#x2013; Residual Analysis &#x2013; Collapse into two replicates of a 23 design
• 17. 17 &#x2022; Sequences of fractional factorial: Both one-half fractions represent blocks of the complete design with the highest- order interaction confounded with blocks.
• 18. 18 &#x2022; Example 8.3: &#x2013; Reconsider Example 8.1 &#x2013; Run the alternate fraction with I = &#x2013; ABCD &#x2013; Estimates of effects &#x2013; Confirmation experiment
• 19. 19 8.3 The One-Quarter Fraction of the 2k Design &#x2022; A one-quarter fraction of the 2k design is called a 2k-2 fractional factorial design &#x2022; Construction: &#x2013; Write down a full factorial in k &#x2013; 2 factors &#x2013; Add two columns with appropriately chosen interactions involving the first k &#x2013; 2 factors &#x2013; Two generators, P and Q &#x2013; I = P and I = Q are called the generating relations for the design &#x2013; All four fractions are the family.
• 20. 20
• 21. 21 &#x2022; The complete defining relation: I = P = Q = PQ &#x2022; P, Q and PQ are called words. &#x2022; Each effect has three aliases &#x2022; A one-quarter fraction of the 26-2 with I = ABCE and I = BCDF. The complete defining relation is I = ABCE = BCDF = ADEF
• 22. 22 &#x2022; Another way to construct such design is to derive the four blocks of the 26 design with ABCE and BCDF confounded , and then choose the block with treatment combination that are + on ABCE and BCDF &#x2022; The 26-2 design with I = ABCE and I = BCDF is the principal fraction. &#x2022; Three alternate fractions: &#x2013; I = ABCE and I = - BCDF &#x2013; I = -ABCE and I = BCDF &#x2013; I = - ABCE and I = -BCDF
• 23. 23 &#x2022; This fractional factorial will project into &#x2013; A single replicate of a 24 design in any subset of four factors that is not a word in the defining relation. &#x2013; A replicate one-half fraction of a 24 in any subset of four factors that is a word in the defining relation. &#x2022; In general, any 2k-2 fractional factorial design can be collapsed into either a full factorial or a fractional factorial in some subset of r &#xF0A3; k &#x2013;2 of the original factors. 26 2 &#x2212; IV
• 24. 24 &#x2022; Example 8.4: &#x2013; Injection molding process with six factors &#x2013; Design table (see Table 8.10) &#x2013; The effect estimates, sum of squares, and regression coefficients are in Table 8.11 &#x2013; Normal probability plot of the effects &#x2013; A, B, and AB are important effects. &#x2013; Residual Analysis (Page 322 &#x2013; 325)
• 25. 25 8.4 The General 2k-p Fractional Factorial Design &#x2022; A 1/ 2p fraction of the 2k design &#x2022; Need p independent generators, and there are 2p &#x2013;p &#x2013; 1 generalized interactions &#x2022; Each effect has 2p &#x2013; 1 aliases. &#x2022; A reasonable criterion: the highest possible resolution, and less aliasing &#x2022; Minimum aberration design: minimize the number of words in the defining relation that are of minimum length.
• 26. 26 &#x2022; Minimizing aberration of resolution R ensures that a design has the minimum # of main effects aliased with interactions of order R &#x2013; 1, the minimum # of two-factor interactions aliased with interactions of order R &#x2013; 2, &#x2026;. &#x2022; Table 8.14
• 27. 27 &#x2022; Example 8.5 &#x2013; Estimate all main effects and get some insight regarding the two-factor interactions. &#x2013; Three-factor and higher interactions are negligible. &#x2013; designs in Appendix Table XII (Page 666) &#x2013; 16-run design: main effects are aliased with three-factor interactions and two-factor interactions are aliased with two-factor interactions &#x2013; 32-run design: all main effects and 15 of 21 two-factor interactions 3727 2and2 &#x2212;&#x2212; IVIV 27 2 &#x2212; IV 37 2 &#x2212; IV
• 28. 28 &#x2022; Analysis of 2k-p Fractional Factorials: &#x2013; For the ith effect: &#x2022; Projection of the 2k-p Fractional Factorials &#x2013; Project into any subset of r &#xF0A3; k &#x2013; 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.) &#x2013; Very useful in screening experiments &#x2013; For example 16-run design: Choose any four of seven factors. Then 7 of 35 subsets are appearing in complete defining relations. pNii i N N Contrast N Contrast &#x2212; === 2, 2/ )(2 &#xF06C; 37 2 &#x2212; IV
• 29. 29 &#x2022; Blocking Fractional Factorial: &#x2013; Appendix Table XII &#x2013; Consider the fractional factorial design with I = ABCE = BCDF = ADEF. Select ABD (and its aliases) to be confounded with blocks. (see Figure 8.18) &#x2022; Example 8.6 &#x2013; There are 8 factors &#x2013; &#x2013; Four blocks &#x2013; Effect estimates and sum of squares (Table 8.17) &#x2013; Normal probability plot of the effect estimates (see Figure 8.19) 26 2 &#x2212; IV 3848 2or2 &#x2212;&#x2212; IVIV
• 30. 30 &#x2022; A, B and AD + BG are important effects &#x2022; ANOVA table for the model with A, B, D and AD (see Table 8.18) &#x2022; Residual Analysis (Figure 8.20) &#x2022; The best combination of operating conditions: A &#x2013;, B + and D &#x2013;
• 31. 31 8.5 Resolution III Designs &#x2022; Designs with main effects aliased with two-factor interactions &#x2022; A saturated design has k = N &#x2013; 1 factors, where N is the number of runs. &#x2022; For example: 4 runs for up to 3 factors, 8 runs for up to 7 factors, 16 runs for up to 15 factors &#x2022; In Section 8.2, there is an example, design. &#x2022; Another example is shown in Table 8.19: design I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG 13 2 &#x2212; III 47 2 &#x2212; III
• 32. 32 &#x2022; This design is a one-sixteenth fraction, and a principal fraction. I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG= AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG &#x2022; Each effect has 15 aliases.
• 33. 33 &#x2022; Assume that three-factor and higher interactions are negligible. &#x2022; The saturated design in Table 8.19 can be used to obtain resolution III designs for studying fewer than 7 factors in 8 runs. For example, for 6 factors in 8 runs, drop any one column in Table 8.19 (see Table 8.20) 47 2 &#x2212; III
• 34. 34 &#x2022; When d factors are dropped , the new defining relation is obtained as those words in the original defining relation that do not contain any dropped letters. &#x2022; If we drop B, D, F and G, then the treatment combinations of columns A, C, and E correspond to two replicates of a 23 design.
• 35. 35 &#x2022; Sequential assembly of fractions to separate aliased effects: &#x2013; Fold over of the original design &#x2013; Switching the signs in one column provides estimates of that factor and all of its two-factor interactions &#x2013; Switching the signs in all columns dealiases all main effects from their two-factor interaction alias chains &#x2013; called a full fold-over
• 36. 36 &#x2022; Example 8.7 &#x2013; Seven factors to study eye focus time &#x2013; Run design (see Table 8.21) &#x2013; Three large effects &#x2013; Projection? &#x2013; The second fraction is run with all the signs reversed &#x2013; B, D and BD are important effects 47 2 &#x2212; III
• 37. 37 &#x2022; The defining relation for a fold-over design &#x2013; Each separate fraction has L + U words used as generators. &#x2013; L: like sign &#x2013; U: unlike sign &#x2013; The defining relation of the combining designs is the L words of like sign and the U &#x2013; 1 words consisting of independent even products of the words of unlike sign. &#x2013; Be careful &#x2013; these rules only work for Resolution III designs
• 38. 38 &#x2022; Plackett-Burman Designs &#x2013; These are a different class of resolution III design &#x2013; Two-level fractional factorial designs for studying k = N &#x2013; 1 factors in N runs, where N = 4 n &#x2013; N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, &#x2026; &#x2013; The designs where N = 12, 20, 24, etc. are called nongeometric PB designs &#x2013; Construction: &#x2022; N = 12, 20, 24 and 36 (Table 8.24) &#x2022; N = 28 (Table 8.23)
• 39. 39 &#x2022; The alias structure is complex in the PB designs &#x2022; For example, with N = 12 and k = 11, every main effect is aliased with every 2FI not involving itself &#x2022; Every 2FI alias chain has 45 terms &#x2022; Partial aliasing can greatly complicate interpretation &#x2022; Interactions can be particularly disruptive &#x2022; Use very, very carefully (maybe never)
• 40. 40 &#x2022; Projection: Consider the 12-run PB design &#x2013; 3 replicates of a full 22 design &#x2013; A full 23 design + a design &#x2013; Projection into 4 factors is not a balanced design &#x2013; Projectivity 3: collapse into a full fractional in any subset of three factors. 13 2 &#x2212; III
• 41. 41 &#x2022; Example 8.8: &#x2013; Use a set of simulated data and the 11 factors, 12- run design &#x2013; Assume A, B, D, AB, and AD are important factors &#x2013; Table 8.25 is a 12-run PB design &#x2013; Effect estimates are shown in Table 8.26 &#x2013; From this table, A, B, C, D, E, J, and K are important factors. &#x2013; Interaction? (due to the complex alias structure) &#x2013; Folding over the design &#x2013; Resolve main effects but still leave the uncertain about interaction effects.
• 42. 42 8.6 Resolution IV and V Designs &#x2022; Resolution IV: if three-factor and higher interactions are negligible, the main effects may be estimated directly &#x2022; Minimal design: Resolution IV design with 2k runs &#x2022; Construction: The process of fold over a design (see Table 8.27) 13 2 &#x2212; III
• 43. 43 &#x2022; Fold over resolution IV designs: (Montgomery and Runger, 1996) &#x2013; Break as many two-factor interactions alias chains as possible &#x2013; Break the two-factor interactions on a specific alias chain &#x2013; Break the two-factor interactions involving a specific factor &#x2013; For the second fraction, the sign is reversed on every design generators that has an even number of letters
• 44. 44 &#x2022; Resolution V designs: main effects and the two- factor interactions do not alias with the other main effects and two-factor interactions.