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

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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
  Lynch Center for Reliability and Quality
  SciencesDepartment of StatisticsUniversity of
  South Carolina803-777-7800

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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

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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

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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 IABCD 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 IABCD
• The alias structure for all effects can be
  constructed from the design generator

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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., AxIA)

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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., AxAI)

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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
• AxIAxABCD
• AIxBCD (Applying Rule 1 to the left and Rule 2
  to the right)
• ABCD (Applying Rule 1 to the right)
• If we do this for each effect, we find

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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.
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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 2532 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!

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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
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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, DAB and EAC. If we
  multiply both sides of the first equation by D,
  we obtain DxDABxD, or IABD.
• Likewise, if we multiply both sides of EAC by E,
  we obtain ExEACxE, or IACE.
• We can say the design is comprised of the 8 runs
  for which both ABD and ACE are equal to one
  (IABDACE).

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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

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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs

• IABDACE 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
• AxIAxABDAxACEAxBCDE
• ABDCEABCDE

The shortest word in the design generator has
three letters, so we call this a Resolution III
design
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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.

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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs

• U-Do-It Exercise Solution.
• IABDACEBCDE
• ABDCEABCDE
• BADABCECDE
• CABCDAEBDE
• DABACDEBCE
• EABDEACBCD
• BCACDABEDE
• BEADEABCCD

After computing the alias structure for main
effects, it may require trial and error to find
the remaining rows of the alias structure
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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs

• U-Do-It Exercise Solution.
• IABDACEBCDE
• ABDCE
• BAD
• CAE
• DAB
• EAC
• BCDE
• BECD

We often exclude higher order terms from the
alias structure (except for the design generator).
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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs

• The corrosion experiment generated the following
  data

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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs

• Computation of Factor Effects

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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs
The interaction is probably due to BC rather than
DE
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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.

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II.3 Screening Designs in Eight Runs Five
Factors in 8 Runs

• B and C at their high levels greatly increase
  corrosion

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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?

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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
• EMR5.178(-1.99/2)-(4.415/2)-(4.87/2)(2.57/2)
• EMR.8255

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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., IABCD 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?

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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
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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.

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

• ALIAS STRUCTURE
• The design generator was computed as follows.
  Since DBC, when we multiply each side of the
  equation by D, we obtain DxDBCD or IBCD. Also,
  since EABC, when we mulitply each side of this
  equation by E, we obtain IABCE. The interaction
  of BCD and ABCE will also be constant (and
  positive in this case), so we have
  IBCDxABCEAxBxBxCxCxDxEADE
• The design generator is IBCDABCEADE

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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
• ADEBCEABCD
• BCDACEABDE
• CBDABEACDE
• DBCABCDEAE
• EBCDEABCAD
• ABACDCEBDE
• ACABDBECDE

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

• ALIAS STRUCTURE
• Eliminating higher order interactions, the alias
  structure is
• IBCDABCEADE
• ADE
• BCD
• CBD
• DBCAE
• EAD
• ABCE
• ACBE

Main effects are confounded with two way effects,
making this a Resolution III design.
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II.3 Screening Designs in Eight Runs U-Do-It
Solution

• ANALYSIS--Computation of Factor Effects

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

• ANALYSIS--Plot of Factor Effects

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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!