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Design and Analysis of Multi-Factored Experiments

Design and Analysis of Multi-Factored Experiments. Fractional Factorial Designs. Design of Engineering Experiments – The 2 k-p Fractional Factorial Design.

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Design and Analysis of Multi-Factored Experiments

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  1. Design and Analysis ofMulti-Factored Experiments Fractional Factorial Designs DOE Course

  2. Design of Engineering Experiments – The 2k-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 factorscreening; 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, but often with center points DOE Course

  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 DOE Course

  4. The One-Half Fraction of the 2k • Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1 • Consider a really simple case, the 23-1 • Note that I =ABC DOE Course

  5. The One-Half Fraction of the 23 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 DOE Course

  6. The Alternate Fraction of the 23-1 • 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 DOE Course

  7. Example: Run 4 of the 8 t.c.’s in 23: a, b, c, abc It is clear that from the(se) 4 t.c.’s, we cannot estimate the 7 effects (A, B, AB, C, AC, BC, ABC) present in any 23 design, since each estimate uses (all) 8 t.c’s. What can be estimated from these 4 t.c.’s? DOE Course

  8. 4A = -1 + a - b + ab - c + ac - bc + abc 4BC = 1 + a - b - ab -c - ac + bc + abc Consider (4A + 4BC)= 2(a - b - c + abc) or 2(A + BC)= a - b - c + abc Overall: 2(A + BC)= a - b - c + abc 2(B + AC)= -a + b - c + abc 2(C + AB)= -a - b + c + abc In each case, the 4 t.c.’s NOT run cancel out. DOE Course

  9. Had we run the other 4 t.c.’s: 1, ab, ac, bc, We would be able to estimate A - BC B - AC C - AB (generally no better or worse than with + signs) NOTE: If you “know” (i.e., are willing to assume) that all interactions = 0, then you can say either (1) you get 3 factors for “the price” of 2. (2) you get 3 factors at “1/2 price.” DOE Course

  10. Suppose we run those 4: 1, ab, c, abc; We would then estimate A + B C + ABC AC + BC In each case, we “Lose” 1 effect completely, and get the other 6 in 3 pairs of two effects. Members of the pair are CONFOUNDED Members of the pair are ALIASED two main effects together usually less desirable DOE Course

  11. With 4 t.c.’s, one should expect to get only 3 “estimates” (or “alias pairs”) - NOT unrelated to “degrees of freedom being one fewer than # of data points” or “with c columns, we get (c - 1) df.” In any event, clearly, there are BETTER and WORSE sets of 4 t.c.’s out of a 23. (Better & worse 23-1 designs) DOE Course

  12. Prospect in fractional factorial designs is attractive if in some or all alias pairs one of the effects is KNOWN. This usually means “thought to be zero” DOE Course

  13. Consider a 24-1 with t.c.’s 1, ab, ac, bc, ad, bd, cd, abcd Can estimate: A+BCD B+ACD C+ABD AB+CD AC+BD BC+AD D+ABC - 8 t.c.’s -Lose 1 effect -Estimate other 14 in 7 alias pairs of 2 Note: DOE Course

  14. “Clean” estimates of the remaining member of the pair can then be made. For those who believe, by conviction or via selected empirical evidence, that the world is relatively simple, 3 and higher order interactions (such as ABC, ABCD, etc.) may be announced as zero in advance of the inquiry. In this case, in the 24-1 above, all main effects are CLEAN. Without any such belief, fractional factorials are of uncertain value. After all, you could get A + BCD = 0, yet A could be large +, BCD large -; or the reverse; or both zero. DOE Course

  15. Despite these reservations fractional factorials are almost inevitable in a many factor situation. It is generally better to study 5 factors with a quarter replicate (25-2 = 8) than 3 factors completely (23 = 8). Whatever else the real world is, it’s Multi-factored. The best way to learn “how” is to work (and discuss) some examples: DOE Course

  16. Design and Analysis ofMulti-Factored Experiments Aliasing Structure DOE Course

  17. Construction of a One-half Fraction The basic design; the design generator DOE Course

  18. Example: 25-1 : A, B, C, D, E Step 1: In a 2k-p, we “lose” 2p-1. Here we lose 1. Choose the effect to lose. Write it as a “Defining relation” or “Defining contrast.” I = ABDE Step 2: Find the resulting alias pairs: *A=BDE AB=DE ABC=CDE B=ADE AC=4 BCD=ACE C=ABCDE AD=BE BCE=ACD D=ABE AE=BD E=ABD BC=4 CD=4 CE=4 - lose 1 - other 30 in 15 alias pairs of 2 - run 16 t.c.’s 15 estimates *AxABDE=BDE DOE Course

  19. See if they are (collectively) acceptable. Another option (among many others): I = ABCDE A=4 AB=3 B=4 AC=3 C=4 AD=3 D=4 AE=3 E=4 BC=3 BD=3 BE=3 CD=3 CE=3 DE=3 DOE Course

  20. Next step: Find the 2 blocks (only one of which will be run) • Assume we choose I=ABDE I II 1 c a ac ab abc b bc de cde ade acde abde abcde bde bcde ad acd d cd bd bcd abd abcd ae ace e ce be bce abe abce Same process as a Confounding Scheme DOE Course

  21. 1 a b ab c ac bc abc e ae be abe ce ace bce abce Next: Pick which block to run. (say, block II) Next: Go out and collect the data. Next: Analyze it. a.) find a proper Yates’ order. i.) pick a letter and for a moment call it “DEAD.” (assume we pick “d”) ii.) use the remaining (“live”) letters to form the STANDARD Yates’ order: (see right column) Now append the dead letter as needed to form the block chosen to be run: DOE Course

  22. There’s only one way to do it, i.e. either adding “d” or not adding “d”; one way will work, one won’t. DOE Course

  23. Example 2: 25-2 A, B, C, D, E Must “lose” 3; other 28 in 7 alias groups of 4 In a 25 , there are 31 effects; with 8 t.c., there are 7 df & 7 estimates available DOE Course

  24. Choose the 3: Like in confounding schemes, 3rd must be product of first 2: I = ABC = BCDE = ADE A = BC = 5 = DE B = AC = 3 = 4 C = AB = 3 = 4 D = 4 = 3 = AE E = 4 = 3 = AD BD = 3 = CE = 3 BE = 3 = CD = 3 Assume we use this design. Find alias groups: DOE Course

  25. Let’s find the 4 blocks: I =ABC = BCDE = ADE Assume we run the Principal block (block 1) DOE Course

  26. Run the 8 t.c.’s and analyze: Since it’s a 25-2, we designate 2 letters as “DEAD” (say b, d), write a standard Yates’ order in the other (3) (live) letters, and append the dead letters to form the t.c.’s being run: I = ABC = BCDE = ADE t.c. yield (1) (2) (3) Estimate 1 . - . a ( bd) -2+5-2 A . c (b) -2+3-4 C . ac (d) AC - +4-3 B e (d) . -4+3-2 E D . ae (b) AE -3+4- . ce ( bd) CE -3+2-3 . ace ACE -2+3-2 DOE Course

  27. Good to know rule: t.c. with even # letters in common with even-lettered effect is a + for that effect; t. c. with odd # letters in common with odd-lettered effect is a + for that effect; otherwise a - (minus) abd in ABCDE: 3 in common with 5 ODD with ODD + abce in ABCDFG: 3 in common with 6 ODD with EVEN - # letters in effect Even Odd # letters t.c. has in Common with Effect Even Odd DOE Course

  28. Examples DOE Course

  29. Example Interpretation of results often relies on making some assumptions Ockham’srazor Confirmation experiments can be important See the projection of this design into 3 factors DOE Course

  30. 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 DOE Course

  31. Possible Strategies for Follow-Up Experimentation Following a Fractional Factorial Design DOE Course

  32. The One-Quarter Fraction of the 2k DOE Course

  33. The One-Quarter Fraction of the 26-2 Complete defining relation: I = ABCE = BCDF = ADEF DOE Course

  34. Analysis of Fractional Factorials • Easily done by computer • Same method as full factorial except that effects are aliased • All other steps same as full factorial e.g. ANOVA, normal plots, etc. • Important not to use highly fractionated designs - waste of resources because “clean” estimates cannot be made. DOE Course

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