The Essentials of 2Level Design of Experiments Part I: The Essentials of Full Factorial Designs

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The Essentials of 2-Level Design of
ExperimentsPart I The Essentials of Full
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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Part I. Full Factorial Designs

• 24 Designs
• Introduction
• Analysis Tools
• Example
• Violin Exercise
• 2k Designs

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24 DesignsU-Do-It - Violin Exercise
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24 DesignsU-Do-It - Violin Exercise How to Play
the Violin in 176 Easy Steps1,2

• A very scientifically-inclined violinist was
  interested in determining what factors affect the
  loudness of her instrument. She believed these
  might include
• A Pressure (Lo,Hi)
• B Bow placement (near,far)
• C Bow Angle (Lo,Hi)
• D Bow Speed (Lo,Hi)
• The precise definition of factor levels is not
  shown, but they were very rigidly defined and
  controlled in the experiment.
• Eleven replicates of the full 24 were performed,
  in completely randomized order. Analyze the
  data! 117611x16 2Data courtesy of Carla
  Padgett

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24 DesignsU-Do-It - Violin Exercise Report Form

• Responses are Averages of 11 Independent
  Replicates
• All 176 trials were randomly ordered
• Analyze and Interpret the Data

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24 DesignsU-Do-It Solution - Violin Exercise
Signs Table
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U-Do-It Exercise U-Do-It Solution - Violin
Exercise Cube Plot

• Factors
• A Pressure (Lo,Hi)
• B Bow Placement (near,far)
• C Bow Angle (Lo,Hi)
• D Bow Speed (Lo,Hi)

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24 DesignsU-Do-It Solution - Violin Exercise
Effects Normal Probability Plot

• Factors
• A Pressure (Lo,Hi)
• B Bow Placement (near,far)
• C Bow Angle (Lo,Hi)
• D Bow Speed (Lo,Hi)

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24 DesignsU-Do-It Solution - Violin Exercise
Interpretation

• The interaction between A and B is so weak that
  it is probably ignorable and will not be included
  initially. This simplifies the analysis since,
  when there are no interactions, the observed
  changes in the response will be the sum of the
  individual changes in the main effects, i.e, the
  main effects are additive.
• When the AB interaction is ignored, we expect
• A loudness increase of 3.3 decibels when
  increasing bow speed from Lo to Hi.
• A loudness increase of about 5 decibels when
  changing the bow placement from near to far.
• A loudness increase of 4.8 decibels when changing
  pressure from Lo to Hi.
• The loudness seems unaffected by the angle
  factor this non-effect is in itself
  interesting and useful.

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U-Do-It Exercise U-Do-It Solution Violin
Exercise Including the AB Interaction

• We now include the AB interaction for comparison
  purposes. Since the interaction is so weak, it
  does not appreciably change the analysis

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U-Do-It Exercise U-Do-It Solution - Violin
Exercise AB Interaction Table
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U-Do-It Exercise U-Do-It Solution - Violin
Exercise AB Interaction Table/Plot
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24 DesignsU-Do-It Solution - Violin Exercise
Interpretation

• If We Include the AB Interaction, We Expect
• Loudness to increase 3.3 when bowing speed, D,
  increases from Lo to Hi.
• Since the lines in the AB interaction are nearly
  parallel, the effect of the interaction is weak.
  This is reflected in our estimates of the EMR.

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24 DesignsU-Do-It Solution - Violin Exercise EMR

• Let us calculate the EMR if we want the response
  to be the quietest.
• If We Dont Include the AB Interaction,
• Set A, B and D at their Lo setting, -1.
• EMR 76.1 - (4.84.93.34)/2 69.58
• If We Include the AB Interaction,
• Set D at its Lo setting, -1.. The AB Interaction
  Table and Plot show that A and B still should be
  set Lo, -1. Note that when A and B are both -1,
  AB is 1..
• EMR 76.1 - (4.84.93.34)/2 (-1.3)/2 68.93

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2k DesignsIntroduction

• Suppose the effects of k factors, each having two
  levels, are to be investigated.
• How many runs (recipes) will there be with no
  replication?
• 2k runs
• How may effects are you estimating?
• There will be 2k-1 columns in the Signs Table
• Each column will be estimating an Effect
• k main effects, A, B, C,...
• k(k-1)/2 two-way interactions, AB, AC, AD,...
• k(k-1)(k-2)/3! three-way interactions
• ...
• k (k-1)-way interactions
• one k-way interaction

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2k DesignsAnalysis Tools

• Signs Table to Estimate Effects
• 2k-1 columns of signs first k estimate the k
  main effects and remaining 2k-k -1 estimate
  interactions
• 2k - 1 Effects Normal Probability Plots to
  Determine Statistically Significant Effects
• Interaction Tables/Plots to Analyze Two-Way
  Interactions
• EMR Computed as Before

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2k DesignsConcluding Comments

• Know How to Design, Analyze and Interpret Full
  Factorial Two-Level Designs
• This means that
• The design is orthogonal
• The run order is totally randomized

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2k DesignsOrthogonality

• (Hard to Explain) If a Design is Orthogonal,
  Each Factors Effect can be Estimated Without
  Interference From the Others...

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2k DesignsOrthogonality - Checking Orthogonality

• 1. Use the -1 and 1 Design Matrix.
• 2. Pick Any Pair of Columns
• 3. Create a New Column by Multiplying These Two,
  Row by Row.
• 4. Sum the New Column If the Sum is Zero, the
  Two Columns/Factors Are Orthogonal.
• 5. If Every Pair of Columns is Orthogonal, the
  Design is Orthogonal.

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2k Designs Randomization

• It is Highly Recommended That the Trials be
  Carried Out in a Randomized Order!!!

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2k Designs Randomization devices

• Slips of Paper in a Bowl
• Multi-Sided Die
• Coin Flips
• Table of Random Digits
• Pseudo-Random Numbers on a Computer

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2k Designs Randomization - Why randomize order?

• Its MAEs fault...

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2k Designs Randomization - Beware the convenient
sample!

• Randomize Run Order to Protect Against the
  Unknown Factors Which are not Either
• varied as experimental factors, or
• fixed as background effects.
• Try Hard to Determine What These Unknown Factors
  Are!

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2k Designs Randomization - Instructions for
Operators

• Having Randomized the Run Order, Present the
  Operator With Easy-to-Follow Instructions.
• Tell Him/Her Not to "Help" by Rearranging the
  Order for Convenience!

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2k Designs Randomization - Partial Randomization

• In Certain Situations It May Not Be Possible to
  Totally Randomize All the Runs
• e.g., it may be too costly to completely
  randomize the temperatures of a series of ovens
  while one may be able to totally randomize the
  other factor levels
• This Leads to Blocks of Runs Within Which The
  Factor Settings Can Be Totally Randomized
• The Analysis of Blocked Designs Will Be Discussed
  in a Later Module
• Remember An Important Goal of a DOE is to Get
  Good Data
• Randomization Protects Us From Background Sources
  of Variation Of Which We May Not Be Aware
• Blocking Allows Us to Include Known But Hard to
  Control Sources So That We Estimate Their Effect.
  We Can Then Remove Their Effect and Analyze the
  other Factor Effects