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 Aliasing for 4 Factors in 8 Runs 5 Factors in 8 runs A U-Do-It Case Study Foldover of Resolution III Designs

II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs 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:

II.3 Screening Designs in Eight Runs: Design Generators We use “I” 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. E.g, the design generator for the example in II.2 with 4 factors in 8 runs is I=ABCD The alias structure for all effects can be constructed from the design generator

II.3 Screening Designs in Eight Runs: An identity To construct the confounding structure, we need two simple rules: Rule 1: Any effect column multiplied by I is unchanged (E.g., AxI=A)

II.3 Screening Designs in Eight Runs: A multiplicative property Rule 2: Any effect multiplied by itself is equal to I (E.g., AxA=I)

II.3 Screening Designs in Eight Runs: Full table of aliases 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: 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

II.3 Screening Designs in Eight Runs: Reduced table of aliases Several of these statements are redundant. When we remove the redundant statements, we obtain the alias structure (which usually starts with the design generator): 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 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 5 main effects 10 two-way interactions 10 three-way interactions 5 four-way interactions 1 five-way interaction In many cases so much experimentation is impractical, and high-order interactions are probably negligible, anyway. In the rest of section II, we will ignore three-way and higher interactions!

II.3 Screening Designs in Eight Runs: Assignment of additional factors 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: *Example based on experiment by Pankaj Arora, a student in Statistics 506

II.3 Screening Designs in Eight Runs: Design generator for 5 factors in 8 runs 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: Alternative design generators There are 3 other equivalent 1/4 fractions the experimenter could have used: 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: Resolution III design 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. Line 1: I = ABD = ACE = BCDE Line 2: AxI=AxABD=AxACE=AxBCDE 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: U-Do-It alias structure 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: U-Do-It alias structure solution 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: U-Do-It Alias Structure solution simplified 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: Corrosion responses The corrosion experiment generated the following data:

II.3 Screening Designs in Eight Runs: Computation of Factor Effects

II.3 Screening Designs in Eight Runs: Corrosion Effects Plot The interaction is probably due to BC rather than DE

II.3 Screening Designs in Eight Runs: Corrosion Interaction Table 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.

II.3 Screening Designs in Eight Runs: Corrosion Interaction Plot B and C at their high levels greatly increase corrosion

II.3 Screening Designs in Eight Runs: Corrosion EMR U-Do-It U-Do-It Exercise: What is the EMR if the experimenter wishes to minimize the corrosion rate?

II.3 Screening Designs in Eight Runs: Corrosion EMR solution U-Do-It Exercise Solution 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: Resolution III vs Resolution IV 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) The shortest word in the design generator has 4 letters (e.g., I=ABCD for 4 factors in 8 runs) No main effects are aliased with two-way effects, but at least one two-way effect is aliased with another two-way effect 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 3-factor 8-run signs table. *Suggested by a STAT 506 project, Spring 2000

II.3 Screening Designs in Eight Runs: U-Do-It Growth Case Study The runs table appears below. Find the alias structure for this data and analyze the data.

II.3 Screening Designs in Eight Runs: U-Do-It Solution 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 Growth Alias Structure 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 D=BC=ABCDE=AE E=BCDE=ABC=AD AB=ACD=CE=BDE AC=ABD=BE=CDE The first two interaction terms we would normally try (AB and AC) had not yet appeared in the alias structure, which made the last two rows of the table easy to obtain.

II.3 Screening Designs in Eight Runs: U-Do-It Growth Alias Structure Simplified Eliminating higher order interactions, the alias structure is I=BCD=ABCE=ADE A=DE B=CD C=BD D=BC=AE E=AD AB=CE AC=BE Main effects are confounded with two way effects, making this a Resolution III design.

II.3 Screening Designs in Eight Runs: U-Do-It Computation of Factor Effects

II.3 Screening Designs in Eight Runs: U-Do-It Growth Effects Plot

II.3 Screening Designs in Eight Runs: U-Do-It Growth Summary ANALYSIS--Interpretation 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!