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 Designs Introduction

• Suppose the effects of four factors, each having
  two levels, are to be investigated.
• How many combinations of factor levels are there?
• With 16 runs, one per each treatment combination,
  we can estimate
• four main effects - (A,B,C,D)
• six two-way interactions -
  (AB,AC,AD,BC,BD,CD)
• four three-way interactions -
  (ABC,ABD,ACD,BCD)
• one four-way interaction (ABCD).

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24 Designs Analysis Tools - Design Matrix
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24 Designs Analysis Tools - Design Matrix Signs
Table
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24 Designs Analysis Tools - Signs Table
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24 Designs Analysis Tools - Fifteen Effects Paper
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24 Designs Example 4

• Response Computer throughput (kbytes/sec),
  (large ys are desirable)
• Factors A, B, C and D were various performance
  tuning parameters such as
• number of buffers
• size of unix inode tables for file
  handling Data courtesy of Greg Dobbins

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24 Designs Example 4 - Experimental Report Form
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24 Designs Example 4 - Signs TableU-Do-It

• Fill Out the Signs Table to Estimate the Factor
  Effects

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24 Designs Example 4 - Completed Signs Table
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24 Designs Example 4 - Effects Normal
Probability Plot

• Factors A and C Stand Out
• Choose Hi Settings of Both A and C since the
  response is throughput

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24 Designs Example 4 - EMR at AHi, CHi

• EMR 69.125 (5.5 2.25)/2 73

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24 Designs Examples 2 and 4 Discussion

• Examples 2 (from Lecture 6.2) and 4 are Related
• Original Data Was In Tenths
• The Numbers were Rounded Off for Ease of
  Calculation
• Example 2
• Half Fraction (24-1, 8 Runs) of the Data in
  Example 4.
• The Runs in Example 4 when ABCD1 were the runs
  used in Example 2.
• The Estimate of the Three-way Interaction ABC in
  Example 2 was also Estimating the Effect of D.
  (D and ABC are confounded/aliased. More about
  this in Part II.)

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24 Designs Examples 2 and 4 Discussion
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24 Designs Example 2 and 4 - Effects Normal
Probability Plots

• Factor A Still Stands Out
• The (Hidden) Replication in the Additional Runs
  Teased Out A Significant Effect Due to Factor C.
• The Probability Plot Suggests We Can Do An ANOVA
  of the Data Based just on Factors A and C.
  (Trick due to Cuthbert Daniel)

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24 Designs Example 4 - ANOVA Table for the
Original Data

• The ANOVA Table for the fit based on factors A
  and C and the AC interaction
• Indicates that some of the main effects are
  statistically significant but not the AC
  interaction
• The individual p-values for A and C indicate that
  both are statistically significant.