1 / 62

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.

jana
Download Presentation

Design and Analysis of Multi-Factored Experiments

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 and constructing a FFD DOE Course

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

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

  23. An easier way to construct a one-half fraction The basic design; the design generator DOE Course

  24. Examples DOE Course

  25. 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

  26. 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

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

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

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

  30. 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

  31. Design and Analysis of Multi-Factored Experiments Design Resolution and Minimal-Run Designs DOE Course

  32. Design Resolution for Fractional Factorial Designs • The concept of design resolution is a useful way to catalog fractional factorial designs according to the alias patterns they produce. • Designs of resolution III, IV, and V are particularly important. • The definitions of these terms and an example of each follow. DOE Course

  33. 1. Resolution III designs • These designs have no main effect aliased with any other main effects, but main effects are aliased with 2-factor interactions and some two-factor interactions may be aliased with each other. • The 23-1 design with I=ABC is a resolution III design or 2III3-1. • It is mainly used for screening. More on this design later. DOE Course

  34. 2. Resolution IV designs • These designs have no main effect aliased with any other main effect or two-factor interactions, but two-factor interactions are aliased with each other. • The 24-1 design with I=ABCD is a resolution IV design or 2IV4-1. • It is also used mainly for screening. DOE Course

  35. 3. Resolution V designs • These designs have no main effect or two factor interaction aliased with any other main effect or two-factor interaction, but two-factor interactions are aliased with three-factor interactions. • A 25-1 design with I=ABCDE is a resolution V design or 2V5-1. • Resolution V or higher designs are commonly used in response surface methodology to limit the number of runs. DOE Course

  36. Guide to choice of fractional factorial designs DOE Course

  37. Guide (continued) DOE Course

  38. Guide (continued) • Resolution V and higher  safe to use (main and two-factor interactions OK) • Resolution IV  think carefully before proceeding (main OK, two factor interactions are aliased with other two factor interactions) • Resolution III  Stop and reconsider (main effects aliased with two-factor interactions). • See design generators for selected designs in the attached table. DOE Course

  39. More on Minimal-Run Designs • In this section, we explore minimal designs with one few factor than the number of runs; for example, 7 factors in 8 runs. • These are called “saturated” designs. • These Resolution III designs confound main effects with two-factor interactions – a major weakness (unless there is no interaction). • However, they may be the best you can do when confronted with a lack of time or other resources (like $$$). DOE Course

  40. If nothing is significant, the effects and interactions may have cancelled itself out. • However, if the results exhibit significance, you must take a big leap of faith to assume that the reported effects are correct. • To be safe, you need to do further experimentation – known as “design augmentation” - to de-alias (break the bond) the main effects and/or two-factor interactions. • The most popular method of design augmentation is called the fold-over. DOE Course

  41. Case Study: Dancing Raisin Experiment • The dancing raisin experiment provides a vivid demo of the power of interactions. It normally involves just 2 factors: • Liquid: tap water versus carbonated • Solid: a peanut versus a raisin • Only one out of the four possible combinations produces an effect. Peanuts will generally float, and raisins usually sink in water. • Peanuts are even more likely to float in carbonated liquid. However, when you drop in a raisin, they drop to the bottom, become coated with bubbles, which lift the raisin back to the surface. The bubbles pop and the up-and-down process continues. DOE Course

  42. BIG PROBLEM – no guarantee of success • A number of factors have been suggested as causes for failure, e.g., the freshness of the raisins, brand of carbonated water, popcorn instead of raisin, etc. • These and other factors became the subject of a two-level factorial design. • See table on next page. DOE Course

  43. Factors for initial DOE on dancing objects DOE Course

  44. The full factorial for seven factors would require 128 runs. To save time, we run only 1/16 of 128 or a 27-4 fractional factorial design which requires only 8 runs. • This is a minimal design with Resolution III. At each set of conditions, the dancing performance was rated on a scale of 1 to 10. • The results from this experiment is shown in the handout. DOE Course

  45. Results from initial dancing-raisin experiment • The half-normal plot of effects is shown. DOE Course

  46. Three effects stood out: cap (E), age of object (G), and size of container (B). • The ANOVA on the resulting model revealed highly significant statistics. • Factors G+ (stale) and E+ (capped liquid) have a negative impact, which sort of make sense. However, the effect of size (B) does not make much sense. • Could this be an alias for the real culprit (effect), perhaps an interaction? • Take a look at the alias structure in the handout. DOE Course

  47. Alias Structure • Each main effect is actually aliased with 15 other effects. To simplify, we will not list 3 factor interactions and above. • [A] = A+BD+CE+FG • [B] = B+AD+CF+EG • [C] = C+AE+BF+DG • [D] = D+AB+CG+EF • [E] = E+AC+BG+DF • [F] = F+AG+BC+DE • [G] = G+AF+BE+CD • Can you pick out the likely suspect from the lineup for B? The possibilities are overwhelming, but they can be narrowed by assuming that the effects form a family. DOE Course

  48. The obvious alternative to B (size) is the interaction EG. However, this is only one of several alternative “hierarchical” models that maintain family unity. • E, G and EG (disguised as B) • B, E, and BE (disguised as G) • B, G, and BG (disguised as E) • The three interaction graphs are shown in the handout. DOE Course

  49. Notice that all three interactions predict the same maximum outcome. However, the actual cause remains murky. The EG interaction remains far more plausible than the alternatives. • Further experimentation is needed to clear things up. • A way of doing this is by adding a second block of runs with signs reversed on all factors – a complete fold-over. More on this later. DOE Course

  50. A very scary thought • Could a positive effect be cancelled by an “anti-effect”? • If you a Resolution III design, be prepared for the possibility that a positive main effect may be wiped out by an aliased interaction of the same magnitude, but negative. • The opposite could happen as well, or some combination of the above. Therefore, if nothing comes out significant from a Resolution III design, you cannot be certain that there are no active effects. • Two or more big effects may have cancelled each other out! DOE Course

More Related