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Design and Analysis of Multi-Factored Experiments

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

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Design and Analysis of Multi-Factored Experiments

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

  2. 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

  3. 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

  4. 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

  5. 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

  6. Guide to choice of fractional factorial designs DOE Course

  7. Guide (continued) DOE Course

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. Factors for initial DOE on dancing objects DOE Course

  14. 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

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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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. DOE Course

  22. 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. DOE Course

  23. Augmented Design Factor B has disappeared and AD has taken its place. What happened to family unity? Is it really AD or something else, since AD is aliased with CF and EG? DOE Course

  24. 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)? DOE Course

  25. 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. DOE Course

  26. Interaction plots of CF and EG DOE Course

  27. 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. DOE Course

  28. 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!! DOE Course

  29. 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! DOE Course

  30. 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 sire that you check the aliases and see whether it is worth doing. DOE Course

  31. 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. DOE Course

  32. 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! DOE Course

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