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Design and Analysis of Multi-Factored Experiments Fractional Factorial Designs L. M. Lye DOE Course 1 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 factor screening; 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 L. M. Lye DOE Course 2 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 L. M. Lye DOE Course 3 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 L. M. Lye DOE Course 4 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 L. M. Lye DOE Course 5 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 I ABC • 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 L. M. Lye DOE Course 6 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? L. M. Lye DOE Course 7 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. L. M. Lye DOE Course 8 Had we run the other 4 t.c.’s: A B AB C AC BC ABC 1, ab, ac, bc, 1 - - + - + + - a + - - - - + + We would be able to estimate b - + - - + - + ab + + + - - - - A - BC c ac - + - - + - + + - + - - + - bc - + - + - + - B - AC abc + + + + + + + C - AB (generally no better or worse than with + signs) A B AB C AC BC ABC 1 ab - + - + + + - - + + - - NOTE: If you “know” (i.e., are - - ac + - - + willing to assume) that all + - - bc - + - + - + - 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.” L. M. Lye DOE Course 9 Suppose we run those 4: 1, ab, c, abc; We would then estimate A+B two main effects C + ABC together usually AC + BC less desirable 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 L. M. Lye DOE Course 10 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) L. M. Lye DOE Course 11 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” L. M. Lye DOE Course 12 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 Note: -Lose 1 effect -Estimate other 14 in 7 alias pairs of 2 L. M. Lye DOE Course 13 “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. L. M. Lye DOE Course 14 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: L. M. Lye DOE Course 15 Design and Analysis of Multi-Factored Experiments Aliasing Structure and constructing a FFD L. M. Lye DOE Course 16 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 - lose 1 B=ADE AC=4 BCD=ACE - other 30 in 15 C=ABCDE AD=BE BCE=ACD alias pairs of 2 D=ABE AE=BD - run 16 t.c.’s E=ABD BC=4 15 estimates CD=4 CE=4 *AxABDE=BDE L. M. Lye DOE Course 17 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 L. M. Lye DOE Course 18 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 Same process de cde ade acde as a abde abcde bde bcde Confounding ad acd d cd Scheme bd bcd abd abcd ae ace e ce be bce abe abce L. M. Lye DOE Course 19 Example 2: In a 25 , there 25-2 A, B, C, D, E are 31 effects; Must “lose” 3; other 28 with 8 t.c., in 7 alias groups of 4 there are 7 df & 7 estimates available L. M. Lye DOE Course 20 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 Find alias D = 4 = 3 = AE groups: E = 4 = 3 = AD BD = 3 = CE = 3 BE = 3 = CD = 3 Assume we use this design. L. M. Lye DOE Course 21 Let’s find the 4 blocks: I =ABC = BCDE = ADE 1 2 3 4 1 a b d Abd a bd ad ab Bc b abc c bcd Acd a cd abcd ac de ade bde e abe be ae abde bcde abcde cde bce ace ce abce acde Assume we run the Principal block (block 1) L. M. Lye DOE Course 22 An easier way to construct a one-half fraction The basic design; the design generator L. M. Lye DOE Course 23 Examples L. M. Lye DOE Course 24 Example Interpretation of results often relies on making some assumptions Ockham’s razor Confirmation experiments can be important See the projection of this design into 3 factors L. M. Lye DOE Course 25 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 L. M. Lye DOE Course 26 The One-Quarter Fraction of the 2k L. M. Lye DOE Course 27 The One-Quarter Fraction of the 26-2 Complete defining relation: I = ABCE = BCDF = ADEF L. M. Lye DOE Course 28 Possible Strategies for Follow-Up Experimentation Following a Fractional Factorial Design L. M. Lye DOE Course 29 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. L. M. Lye DOE Course 30 Design and Analysis of Multi-Factored Experiments Design Resolution and Minimal-Run Designs L. M. Lye DOE Course 31 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. L. M. Lye DOE Course 32 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. L. M. Lye DOE Course 33 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. L. M. Lye DOE Course 34 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. L. M. Lye DOE Course 35 Guide to choice of fractional factorial designs Factors 2 3 4 5 6 7 8 4 runs Full 1/2 (III) - - - - - 8 2 rep Full 1/2 (IV) 1/4 (III) 1/8 (III) 1/16 (III) - 16 4 rep 2 rep Full 1/2 (V) 1/4 (IV) 1/8 (IV) 1/16 (IV) 32 8 rep 4 rep 2 rep Full 1/2 (VI) 1/4 (IV) 1/8 (IV) 64 16 rep 8 rep 4 rep 2 rep Full 1/2 (VII) 1/4 (V) 128 32 rep 16 rep 8 rep 4 rep 2 rep Full 1/2 (VIII) L. M. Lye DOE Course 36 Guide (continued) Factors 9 10 11 12 13 14 15 4 runs - - - - - - - 8 - - - - - - - 16 1/32 (III) 1/64 (III) 1/128 (III) 1/256 (III) 1/512 (III) 1/1024 (III) 1/2048 (III) 32 1/16 (IV) 1/32 (IV) 1/64 (IV) 1/128 (IV) 1/256 (IV) 1/512 (IV) 1/1024 (IV) 64 1/8 (IV) 1/16 (IV) 1/32 (IV) 1/64 (IV) 1/128 (IV) 1/256 (IV) 1/512 (IV) 128 1/4 (VI) 1/8 (V) 1/16 (V) 1/128 (IV) 1/64 (IV) 1/128 (IV) 1/128 (IV) L. M. Lye DOE Course 37 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. L. M. Lye DOE Course 38 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 $$$). L. M. Lye DOE Course 39 • 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. L. M. Lye DOE Course 40 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. L. M. Lye DOE Course 41 • 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. L. M. Lye DOE Course 42 Factors for initial DOE on dancing objects Factor Name Low Level (-) High Level (+) A Material of container Plastic Glass B Size of container Small Large C Liquid Club Soda Lemon Lime D Temperature Room Ice Cold E Cap on container No Yes F Type of object Popcorn Raisin G Age of object Fresh Stale L. M. Lye DOE Course 43 • 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. L. M. Lye DOE Course 44 Results from initial dancing-raisin experiment DESIGN-EXPERT Plot Half Normal plot Rating A: A 99.00 • The half- B: B C: C D: D 97.00 E: E normal plot F: F 95.00 Half Norm al % probability G: G E of effects is 90.00 85.00 80.00 shown. 70.00 G 60.00 B 40.00 20.00 0.0000 0.0000 0.4937 0.9875 1.481 1.975 |Effect| L. M. Lye DOE Course 45 • 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. L. M. Lye DOE Course 46 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. L. M. Lye DOE Course 47 • 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. L. M. Lye DOE Course 48 • 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. L. M. Lye DOE Course 49 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! L. M. Lye DOE Course 50 Complete Fold-Over of Resolution III Design • You can break the aliases between main effects and two-factor interactions by using a complete fold-over of the Resolution III design. • It works on any Resolution III design. It is especially popular with Plackett-Burman designs, such as the 11 factors in 12-run experiment. • Let’s see how the fold-over works on the dancing raisin experiments with all signs reversed on the control factors. L. M. Lye DOE Course 51 Complete Fold-Over of Raisin Experiment • See handout for the augmented design. The second block of experiments has all signs reversed on the factors A to F. • Notice that the signs of the two-factor interactions do not change from block 1 to block 2. • For example, in block 1 the signs of column B and EG are identical, but in block 2 they differ; thus the combined design no longer aliases B with EG. • If B is really the active effect, it should come out on the plot of effects for the combined design. L. M. Lye DOE Course 52 Augmented Design Factor B has DESIGN-EXPERT Plot Interaction Graph disappeared and AD Response 1 5.000 D: D has taken its place. X = A: A Y = D: D D- -1.000 3.875 D+ 1.000 Actual Factors B: B = 0.0000 What happened to Res pons e 1 C: C = 0.0000 E: E = 0.0000 2.750 family unity? F: F = 0.0000 G: G = 0.0000 1.625 Is it really AD or something else, since 0.5000 AD is aliased with CF -1.000 -0.5000 0.0000 0.5000 1.000 and EG? A: A L. M. Lye DOE Course 53 • The problem is that a complete fold-over of a Resolution III design does not break the aliasing of the two-factor interactions. • The listing of the effect AD – the interaction of the container material with beverage temperature – is done arbitrarily by alphabetical order. • The AD interaction makes no sense physically. Why should the material (A) depend on the temperature of beverage (B)? L. M. Lye DOE Course 54 Other possibilities • It is not easy to discount the CF interaction: liquid type (C) versus object type (F). A chemical reaction is possible. • However, the most plausible interaction is between E and G, particularly since we now know that these two factors are present as main effects. • See interaction plots of CF and EG. L. M. Lye DOE Course 55 Interaction plots of CF and EG DESIGN-EXPERT Plot Interaction Graph DESIGN-EXPERT Plot Interaction Graph Response 1 F: F Response 1 G: G 5.000 5.000 X = C: C X = E: E Y = F: F Y = G: G F- -1.000 3.875 G- -1.000 3.875 F+ 1.000 G+ 1.000 Actual Factors Actual Factors A: A = 0.0000 A: A = 0.0000 Res pons e 1 Res pons e 1 B: B = 0.0000 B: B = 0.0000 D: D = 0.0000 2.750 C: C = 0.0000 2.750 E: E = 0.0000 D: D = 0.0000 G: G = 0.0000 F: F = 0.0000 1.625 1.625 0.5000 0.5000 -1.000 -0.5000 0.0000 0.5000 1.000 -1.000 -0.5000 0.0000 0.5000 1.000 C: C E: E L. M. Lye DOE Course 56 • It appears that the effect of cap (E) depends on the age of the object (G). • When the object is stale (G+ line), twisting on the bottle cap (going from E- at left to E+ at right) makes little difference. • However, when the object is fresh (the G- line at the top), the bottle cap quenches the dancing reaction. More experiments are required to confirm this interaction. • One obvious way is to do a full factorial on E and G alone. L. M. Lye DOE Course 57 An alias by any other name is not necessarily the same • You might be surprised that aliased interactions such as AD and EG do not look alike. • Their coefficients are identical, but the plots differ because they combine the interaction with their parent terms. • So you have to look through each aliased interaction term and see which one makes physical sense. • Don’t rely on the default given by the software!! L. M. Lye DOE Course 58 Single Factor Fold-Over • Another way to de-alias a Resolution III design is the “single-factor fold-over”. • Like a complete fold-over, you must do a second block of runs, but this variation of the general method, you change signs only on one factor. • This factor and all its two-factor interactions become clear of any other main effects or interactions. • However, the combined design remains a Resolution III, because with the exception of the factor chosen for de-aliasing, all others remained aliased with two-factor interactions! L. M. Lye DOE Course 59 Extra Note on Fold-Over • The complete fold-over of Resolution IV designs may do nothing more than replicate the design so that it remains Resolution IV. • This would happen if you folded the 16 runs after a complete fold-over of Resolution III done earlier in the raisin experiment. • By folding only certain columns of a Resolution IV design, you might succeed in de-aliasing some of the two-factor interactions. • So before doing fold-overs, make sure that you check the aliases and see whether it is worth doing. L. M. Lye DOE Course 60 Bottom Line • The best solution remains to run a higher resolution design by selecting fewer factors and/or bigger design. • For example, you could run seven factors in 32 runs (a quarter factorial). It is Resolution IV, but all 7 main effects and 15 of the 21 two-factor interactions are clear of other two-factor interactions. • The remaining 6 two-factor interactions are: DE+FG, DF+EG, and DG+EF. • The trick is to label the likely interactors anything but D, E, F, and G. L. M. Lye DOE Course 61 • For example, knowing now that capping and age interact in the dancing raisin experiment, we would not label these factors E and G. • If only we knew then what we know now!!!! • So it is best to use a Resolution V design, and none of the problems discussed above would occur! L. M. Lye DOE Course 62