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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.

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The Essentials of 2-Level Design of ExperimentsPart II: The Essentials of Fractional Factorial Designs

Developed by Don Edwards, John Grego and James LynchCenter for Reliability and Quality SciencesDepartment of StatisticsUniversity of South Carolina803-777-7800

ii 3 screening designs in 8 runs
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
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 aliasing for 4 factors in 8 runs4
II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs
  • 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 aliasing for 4 factors in 8 runs5
II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs
  • 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 aliasing for 4 factors in 8 runs6
II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs
  • Rule 2: Any effect multiplied by itself is equal to I (E.g., AxA=I)
ii 3 screening designs in eight runs aliasing for 4 factors in 8 runs7
II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs
  • 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 aliasing for 4 factors in 8 runs8
II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs
  • 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
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 five factors in 8 runs10
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 five factors in 8 runs11
II.3 Screening Designs in Eight Runs: Five 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 five factors in 8 runs12
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 five factors in 8 runs13
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 five factors in 8 runs14
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 runs15
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 runs16
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 runs17
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • The corrosion experiment generated the following data:
ii 3 screening designs in eight runs five factors in 8 runs19
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 runs20
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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 five factors in 8 runs21
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • B and C at their high levels greatly increase corrosion
ii 3 screening designs in eight runs five factors in 8 runs22
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 runs23
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs

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 five factors in 8 runs24
II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs
  • 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
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 case study26
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.

ii 3 screening designs in eight runs u do it solution
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 solution28
II.3 Screening Designs in Eight Runs: U-Do-It Solution

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 solution29
II.3 Screening Designs in Eight Runs: U-Do-It Solution

ALIAS STRUCTURE

  • 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 solution30
II.3 Screening Designs in Eight Runs: U-Do-It Solution

ANALYSIS--Computation of Factor Effects

ii 3 screening designs in eight runs u do it solution32
II.3 Screening Designs in Eight Runs: U-Do-It Solution

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!