The Essentials of 2-Level Design of Experiments Part II: The Essentials of Fractional Factorial Designs Developed by Don Edwards, John Grego and James Lynch Center for Reliability and Quality Sciences Department of Statistics University of South Carolina 803-777-7800 II.3 Screening Designs in 8 runs пЃџAliasing for 4 Factors in 8 Runs пЃџ5 Factors in 8 runs пЃџ A U-Do-It Case Study пЃџFoldover of Resolution III Designs II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs пЃџ A -1 1 -1 1 -1 1 -1 1 B In an earlier exercise from II.2, four factors were studied in 8 runs by using only those runs from a 24 design for which ABCD was positive: C D AB -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 1 -1 1 AD BC 1 1 1 1 -1 -1 BD CD 1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 1 1 1 1 1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 ACD -1 -1 -1 ABD 1 -1 1 ABC -1 1 -1 1 AC -1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 1 BCD ABCD -1 1 -1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs пЃџ пЃџ We use вЂњIвЂќ to denote a column of ones and note that I=ABCD for this particular design DEFINITION: The set of effects whose levels are constant (either 1 or -1) in a design are design generators. пЃџ E.g, the design generator for the example in II.2 with 4 factors in 8 runs is I=ABCD пЃџ The alias structure for all effects can be constructed from the design generator II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs вЂў вЂў To construct the confounding structure, we need two simple rules: Rule 1: Any effect column multiplied by I is unchanged (E.g., AxI=A) A п‚ґ I пЂЅ A пѓ©пЂ пЂ 1пѓ№пЂ пѓ©пЂ 1пѓ№пЂ пѓ©пЂ пЂ1 пѓ№пЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ пЂ 1пѓєпЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ пЂ1 пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ пѓєпЂ п‚ґ пЂ1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ пЂ1 пѓЄпЂ пѓєпЂ пѓ«пЂ 1пѓ»пЂ пѓЄпЂ пѓєпЂ пЂЅ 1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ пѓ«пЂ 1пѓ»пЂ пѓЄпЂ пѓєпЂ пЂ1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ пЂ1 пѓЄпЂ пѓєпЂ пѓ«пЂ 1 пѓ»пЂ II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs пЃ® Rule 2: Any effect multiplied by itself is equal to I (E.g., AxA=I) A п‚ґ A пЂЅ I пѓ©пЂ пЂ 1пѓ№пЂ пѓ©пЂ пЂ1 пѓ№пЂ пѓ©пЂ 1 пѓ№пЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ пЂ 1пѓєпЂ пѓЄпЂ 1пѓєпЂ пѓЄпЂ пЂ1 пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ 1 пѓєпЂ пѓЄпЂ пѓєпЂ п‚ґ пЂ1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ пЂ1 пѓЄпЂ пѓєпЂ пѓ«пЂ 1пѓ»пЂ пѓЄпЂ пѓєпЂ пЂЅ пЂ1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ пЂ1 пѓЄпЂ пѓєпЂ пѓ«пЂ 1 пѓ»пЂ пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ 1 пѓЄпЂ пѓєпЂ пѓ«пЂ 1 пѓ»пЂ II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs вЂў вЂў We can now construct an alias table by multiplying both sides of the design generator by any effect. E.g., for effect A, we have the steps: вЂў вЂў вЂў вЂў AxI=AxABCD A=IxBCD (Applying Rule 1 to the left and Rule 2 to the right) A=BCD (Applying Rule 1 to the right) If we do this for each effect, we find A =BCD A B=CD BD =A C A CD =B B=A CD A C=BD C D = AB B CD =A C =A B D A D =BC A BC=D A B C D =I D =A B C BC=A D A BD =C II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs пЃџ Several of these statements are redundant. When we remove the redundant statements, we obtain the alias structure (which usually starts with the design generator): I= A B C D D =A B C A =BCD A B=CD B=A CD A C=BD C =A B D A D =BC The alias structure will be complicated for more parsimonious designs; we will add a few more guidelines for constructing alias tables later on. II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ Suppose five two-level factors A, B, C, D, E are to be examined. If using a full factorial design, there would be 25=32 runs, and 31 effects estimated вЂ“ вЂ“ вЂ“ вЂ“ вЂ“ 5 main effects 10 two-way interactions 10 three-way interactions 5 four-way interactions 1 five-way interaction пЃџ In many cases so much experimentation is impractical, and highorder interactions are probably negligible, anyway. пЃџ In the rest of section II, we will ignore three-way and higher interactions! II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ An experimenter wanted to study the effect of 5 factors on corrosion rate of iron rebar* in only 8 runs by assigning D to column AB and E to column AC in the 3-factor 8-run signs table: S ta nd ar d O rd er 1 2 3 4 5 6 7 8 A B -1 1 -1 1 -1 1 -1 1 C -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 D = AB E = AC 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 *Example based on experiment by Pankaj Arora, a student in Statistics 506 BC 1 1 -1 -1 -1 -1 1 1 AB C -1 1 1 -1 1 -1 -1 1 II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ пЃџ пЃџ пЃџ For this particular design, the experimenter used only 8 runs (1/4 fraction) of a 32 run (or 25) design (I.e., a 25-2 design). For each of these 8 runs, D=AB and E=AC. If we multiply both sides of the first equation by D, we obtain DxD=ABxD, or I=ABD. Likewise, if we multiply both sides of E=AC by E, we obtain ExE=ACxE, or I=ACE. We can say the design is comprised of the 8 runs for which both ABD and ACE are equal to one (I=ABD=ACE). II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs вЂў There are 3 other equivalent 1/4 fractions the experimenter could have used: вЂў вЂў вЂў вЂў ABD = 1, ACE = -1 (I = ABD = -ACE) ABD = -1, ACE = 1 (I = -ABD = ACE) ABD = -1, ACE = -1 (I = -ABD = -ACE) The fraction the experimenter chose is called the principal fraction II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs п‚§ п‚§ п‚§ I=ABD=ACE is the design generator If ABD and ACE are constant, then their interaction must be constant, too. Using Rule 2, their interaction is ABD x ACE = BCDE The first two rows of the confounding structure are provided below. п‚§ Line 1: I = ABD = ACE = BCDE п‚§ Line 2: п‚§ AxI=AxABD=AxACE=AxBCDE п‚§ A=BD=CE=ABCDE The shortest word in the design generator has three letters, so we call this a Resolution III design II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ U-Do-It Exercise. Complete the remaining 6 non-redundant rows of the confounding structure for the corrosion experiment. Start with the main effects and then try any two-way effects that have not yet appeared in the alias structure. II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ U-Do-It Exercise Solution. I=ABD=ACE=BCDE A=BD=CE=ABCDE B=AD=ABCE=CDE C=ABCD=AE=BDE D=AB=ACDE=BCE E=ABDE=AC=BCD BC=ACD=ABE=DE BE=ADE=ABC=CD After computing the alias structure for main effects, it may require trial and error to find the remaining rows of the alias structure II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ U-Do-It Exercise Solution. I=ABD=ACE=BCDE A=BD=CE B=AD C=AE D=AB E=AC BC=DE BE=CD We often exclude higher order terms from the alias structure (except for the design generator). II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ The corrosion experiment generated the following data: S ta nd ar d O rd er 1 2 3 4 5 6 7 8 Corro si on Ra te 2.71 0.93 4.80 2.53 4.89 3.35 12 .29 9.92 A B -1 1 -1 1 -1 1 -1 1 C -1 -1 1 1 -1 -1 1 1 D -1 -1 -1 -1 1 1 1 1 E 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs Computation of Factor Effects y 2.71 0.93 4.80 2.53 4.89 3.35 12.29 9.92 41.42 8 5.178 A+BD+ CE -1 1 -1 1 -1 1 -1 1 -7.96 4 -1.99 B+AD -1 -1 1 1 -1 -1 1 1 17.66 4 4.415 C+AE -1 -1 -1 -1 1 1 1 1 19.48 4 4.87 D+AB 1 -1 -1 1 1 -1 -1 1 -1.32 4 -.33 E+AC 1 -1 1 -1 -1 1 -1 1 .14 4 .035 BC+DE 1 1 -1 -1 -1 -1 1 1 10.28 4 2.57 BE+CD -1 1 1 -1 1 -1 -1 1 -.34 4 -.085 II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs The interaction is probably due to BC rather than DE II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃџ пЃџ Factor A at its high level reduced the corrosion rate by 1.99 units Factor B and C main effects cannot be interpreted in the presence of a significant BC interaction. C 1 2 1 2 .71 .93 1 .82 4.89 3.35 4.12 2 4 .80 2 .53 3 .67 12 .29 9.92 11 .11 B II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs пЃ® B and C at their high levels greatly increase corrosion II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs U-Do-It Exercise: What is the EMR if the experimenter wishes to minimize the corrosion rate? II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs U-Do-It Exercise Solution пЃ® A should be set high, B and C should be low and BC should be high, so our solution is: EMR=5.178+(-1.99/2)-(4.415/2)(4.87/2)+(2.57/2) EMR=.8255 II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs п‚§ п‚§ п‚§ D and E could have been assigned to any of the last 4 columns (AB, AC, BC or ABC) in the 3-factor 8-run signs table. All of the resulting designs would be Resolution III, which means that at least one main effect would be aliased with at least one two-way effect. For a Resolution IV design (e.g., 4 factors in 8 runs) п‚§ The shortest word in the design generator has 4 letters (e.g., I=ABCD for 4 factors in 8 runs) п‚§ No main effects are aliased with two-way effects, but at least one two-way effect is aliased with another two-way effect п‚§ What qualities would a Resolution V design have? II.3 Screening Designs in Eight Runs: U-Do-It Case Study A statistically-minded vegetarian* studied 5 factors that would affect the growth of alfalfa sprouts. Factors included measures such as presoak time and watering regimen. The response was biomass measured in grams after 48 hours. Factor D was assigned to the BC column and factor E was assigned to the ABC column in the 3factor 8-run signs table. S ta nd ar d O rd er 1 2 3 4 5 6 7 8 A B -1 1 -1 1 -1 1 -1 1 C -1 -1 1 1 -1 -1 1 1 AB -1 -1 -1 -1 1 1 1 1 *Suggested by a STAT 506 project, Spring 2000 1 -1 -1 1 1 -1 -1 1 AC 1 -1 1 -1 -1 1 -1 1 D = BC E = ABC 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 II.3 Screening Designs in Eight Runs: U-Do-It Case Study The runs table appears below. Find the alias structure for this data and analyze the data. S ta nd ar d O rd er 1 2 3 4 5 6 7 8 G row th 9.7 14 .7 12 .3 12 .7 11 .2 13 .1 10 .1 15 .0 A B -1 1 -1 1 -1 1 -1 1 C -1 -1 1 1 -1 -1 1 1 D -1 -1 -1 -1 1 1 1 1 E 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 II.3 Screening Designs in Eight Runs: U-Do-It Solution пЃџ пЃџ ALIAS STRUCTURE The design generator was computed as follows. Since D=BC, when we multiply each side of the equation by D, we obtain DxD=BCD or I=BCD. Also, since E=ABC, when we mulitply each side of this equation by E, we obtain I=ABCE. The interaction of BCD and ABCE will also be constant (and positive in this case), so we have I=BCDxABCE=AxBxBxCxCxDxE=ADE The design generator is I=BCD=ABCE=ADE II.3 Screening Designs in Eight Runs: U-Do-It Solution ALIAS STRUCTURE пЃџ Working from the design generator, the remaining rows of the design structure will be: A=DE=BCE=ABCD B=CD=ACE=ABDE C=BD=ABE=ACDE The first two interaction D=BC=ABCDE=AE terms we would normally E=BCDE=ABC=AD try (AB and AC) had not AB=ACD=CE=BDE yet appeared in the alias AC=ABD=BE=CDE structure, which made the last two rows of the table easy to obtain. II.3 Screening Designs in Eight Runs: U-Do-It Solution ALIAS STRUCTURE пЃџ Eliminating higher order interactions, the alias structure is I=BCD=ABCE=ADE A=DE B=CD Main effects are C=BD confounded with two way D=BC=AE effects, making this a E=AD Resolution III design. AB=CE AC=BE II.3 Screening Designs in Eight Runs: U-Do-It Solution ANALYSIS--Computation of Factor Effects Grams 9.7 14.7 12.3 12.7 11.2 13.1 10.1 15.0 98.8 8 12.35 A+DE -1 1 -1 1 -1 1 -1 1 12.2 4 3.05 B+CD -1 -1 1 1 -1 -1 1 1 1.40 4 .35 C+BD -1 -1 -1 -1 1 1 1 1 0.0 4 0.0 AB+CE 1 -1 -1 1 1 -1 -1 1 -1.60 4 -.40 AC+BE 1 -1 1 -1 -1 1 -1 1 1.40 4 .35 D+BC+ AE 1 1 -1 -1 -1 -1 1 1 .20 4 .05 E+AD -1 1 1 -1 1 -1 -1 1 7.60 4 1.90 II.3 Screening Designs in Eight Runs: U-Do-It Solution ANALYSIS--Plot of Factor Effects N o r ma l P lo t o f the E ffe c ts (r e s po ns e is Gr o w th, A lpha = .05 ) 99 E ffec t T y p e N o t S ig n ific an t 95 S ig n ific an t A Pe r c e nt 90 F a cto r N am e A A 80 B B 70 C C 60 D D E E 50 40 30 20 10 5 1 -1 0 1 Effe c t Le nth's PS E = 0 .52 5 2 3 II.3 Screening Designs in Eight Runs: U-Do-It Solution ANALYSIS--Interpretation пЃџ пЃџ пЃџ пЃџ Factor A at its high level increases the yield by 3.05 grams Factor E at its high level increases the yield by 1.90 grams Both of these effects are confounded with two way interactions, but we have used the simplest possible explanation for the significant effects we observed Note: the most important result in the actual experiment was an insignificant main effect. The experimenter found that the recommended presoak time for the alfalfa seeds could be lowered from 16 hours to 4 hours with no deleterious effect on the yield--a significant time savings!

1/--страниц