The Essentials of 2-Level 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.3 The Essentials of 2-Cubed Designs

• Methodology
• Cube Plots
• Estimating Main Effects
• Estimating Interactions (Interaction Tables and
  Graphs)
• Statistical SignificanceWhen is an Effect
  Real?
• An Example With Interactions
• A U-Do-It Case Study
• Replication
• Rope Pull Exercise

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U-Do-It Case Study Ball Bearing Example

• Purpose of the Design
• Test (Under Accelerated Conditions) New Bearing
  Prototypes for Use in a Specific Application for
  Which the Current Designs Performance Was
  Unsatisfactory.
• Response of Interest y - Bearing Life (h).
• Design Factors
• Factor Levels (Lo,Hi)A Cage
  Design Current, NewB Outer Ring
  Osculation Current, New C Inner Ring Heat
  Treatment. Current, New
• The 8 Standard Runs of the 23 Design Were
  Randomly Ordered, and Each Prototype Bearing
  Tested.
• Empirical Basis for this data was motivated by
  C. Hellstrands article The necessity of modern
  quality improvement and some experiences with
  implications in the manufacture of ball bearings
  (1989, Philos. Trans. Royal Society London, A
  327, 529-537)

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U-Do-It Case Study Ball Bearing Example - A
Typical Ball Bearing
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U-Do-It Case Study Ball Bearing Example -
Operator Report Form
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U-Do-It Case Study Ball Bearing Example -
Exercise Instructions

• In Class
• Put the results of the experiment in standard
  order and enter the data into a cube plot (in
  Minitabsee handout)
• Estimate the factor effects (in Minitab)
• Construct and interpret a normal probability plot
  of the factor effects (in Minitab)

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U-Do-It Case Study Ball Bearing Example -
Exercise Instructions

• In Class
• Construct BC interaction graph in Minitab use
  table and graph to interpret BC interaction
• Determine the factor settings that maximize
  bearing life and estimate the Mean Response (EMR)
  at these settings. How close is your answer to
  the observed mean response at your optimal
  settings?
• If you would like to do hand calculations, blank
  signs tables, cube plots, etc. are provided over
  the next several slides

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U-Do-It Case Study Ball Bearing Example - Signs
Table
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U-Do-It Case Study Ball Bearing Example - Cube
Plot
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U-Do-It Case Study Ball Bearing Example - Seven
Effects Paper
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U-Do-It Case Study Ball Bearing Example -
Interaction Table
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U-Do-It Case Study Solution Ball Bearing Example
- Cube PlotBearing Lifetimes (h) Shown

• FactorA Cage DesignB Outer Ring
  OsculationC Inner Ring Heat Treatment
• Levels Lo Current, Hi New

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U-Do-It Case Study Solution Ball Bearing Example
- Signs Table
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U-Do-It Case Study Solution Ball Bearing Example
- Probability Plot
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U-Do-It Case Study Solution Ball Bearing Example
- Completed BC Interaction Table
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U-Do-It Case Study Solution Ball Bearing Example
- BC Interaction Plot

• FactorsB Outer Ring OsculationC Inner Ring
  Heat Treatment
• Levels Lo Current, Hi New

• Interpretation
• Choose the Hi Level for both B and C to Maximize
  the Bearing Life

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U-Do-It Case Study Solution Ball Bearing Example
- Expected Mean Response

• Since the BC Interaction is Significant, the Main
  Effects B and C and the BC Interaction are
  Included
• Factor A is NOT Included Since it was Not
  Significant

• For B 1, C 1, EMR 41.5
  (1)(45.5)(1)(43)(1)(39.5)/2
  105.5 (vs (99112)/2 105.5 Observed MR)

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U-Do-It Case Study Solution Ball Bearing Example
- Interpretation of the Experiment

• Unexpected Interaction Discovered (Would Not Have
  Been Discovered Using One-at-a-Time
  Experimentation). Results May Carry Over to
  Other Bearing Designs.
• Contrary to Existing Beliefs, the Two Cage
  Designs had Very Similar Lifetimes. This was
  Very Important Since Bearings Were Much Cheaper
  to Produce Under One of the Two Cage Designs.
• New Designs Performance (In the Specific
  Application Under Investigation) Far Superior to
  That of the Current Bearing Being Used.