TwoLevel Factorial Designs

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Two-Level Factorial Designs

• Engineering Experimental Design
• Winter 2003

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Standard 23 Design

• Where do the factors (independent variables)
  appear in this table?
• Where do the responses (dependent variable)
  appear in this table?
• What do the 1 and 1 mean?
• Should these experimental runs be made in the
  order they are shown?

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Standard 23 Design

• Factors are A, B, C
• Responses do not appear in this table?
• Choose a high and a low value for each factor.
• -1 means set factor to low level in this run
• 1 means set factor to high level in this run
• Run order should be randomized
• Failure to randomize very risky for factor C,
  since it has runs 1-4 at low level and 5-8 at
  high level

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Maximize Reaction Yield23 Factorial Design

• Objective maximize reaction yield
• Factors
• A catalyst weight percent (1,2)
• B reaction time, hours (1,2)
• C temperature, F (200,250)
• Response Reaction yield,

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Maximize Reaction Yield
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Now What?

• Calculate effects of each factor and interaction
• Decide which effects are important
• Plan another, multilevel experiment focusing on
  the important variables

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Interactions
-1 ? -1 1
Note that each factor is tested at each level 4
times.
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Investigating Interactions

• You set the value for each factor in each
  experiment
• The interactions happen naturally
• You do not set some level of AB interaction it
  happens automatically because of the levels you
  set for A and B individually
• Interactions are a physical reality of the
  system, and will happen whether you calculate an
  effect for them or not

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How to Calculate Effects

• High Total sum of all response values when the
  factor is at the 1 level
• Low Total sum of all response values when the
  factor is at the 1 level
• Difference (High Total) (Low Total)
• Note that you can also calculate the difference
  by multiplying each 1 or 1 by the response for
  its row, then summing all the values in the
  column. That is what your book says.
• Effect Difference / ( runs at each level)

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Effects
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Effects
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Scree Plot to Identify Order of Importance
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Conclusions

• Increasing catalyst weight or increasing
  temperature will increase the yield
• Increasing catalyst is most effective
• Increasing reaction time itself has little effect
  on yield, but in combination with increased
  temperature multiplies the effect of temperature

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Comparison with OFAT

• OFAT would reveal the effect of catalyst and
  temperature.
• OFAT would not reveal the time-temperature
  interaction.
• OFAT would not reveal the lack of time-catalyst
  and temperature-catalyst interaction.

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Adding a Factor

• Adding a factor to a full factorial design
  doubles the number of experimental runs
• 3 factors 23 8 runs
• 4 factors 24 16 runs
• If you are confident that an interaction is
  unimportant, you can substitute a new factor for
  that interaction term in the test matrix
• 3-way interaction least likely to be important
• Substitution of a factor for an interaction makes
  an unsaturated design

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Unsaturated Designs and Aliasing

• If a factor replaces an interaction in the
  design,
• You cannot tell the difference between the effect
  of the factor and the effect of the interaction
• Interaction is an innate property of the system.
  You do not control whether or not it happens by
  deciding whether or not to study it.
• Some or all of the effect you calculate for the
  new factor could be due to the interaction
  between other factors.
• You cannot study how the new factor interacts
  with others