II.2 Four Factors in Eight Runs

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II.2 Four Factors in Eight Runs

• Introduction
• Confounding
• Confounding/Aliasing
• Alias Structure
• Examples and Exercises
• A Demonstration of the Effects of Confounding

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II.2 Four Factors in Eight Runs
IntroductionFigure 2 - 23 Design Signs Table
andFour Factors in Eight Runs Design Matrix

• Lets Compare
• 23 Design Signs Table
• Four Factors in Eight Runs Design Matrix

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II.2 Four Factors in Eight Runs Introduction
Exercise-Four Factors in Eight Runs Signs Table

• To compute estimates, create columns for a signs
  table by multiplying columns as before some are
  done for you.

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II.2 Four Factors in Eight Runs Introduction
Exercise Solution-Four Factors in Eight Runs
Signs Table

• The completed signs table is below

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II.2 Four Factors in Eight Runs
IntroductionExercise-Four Factors in Eight Runs
Signs Table Solution

• By plan the column for D column for ABC, so we
  say that for this design "DABC." Also, we can
  see from above
• A BCD
• B ACD
• C ABD
• AB CD
• AC BD
• BC AD
• I ABCD
• Where "I" is a column of ones.

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II.2 Four Factors in Eight Runs Confounding

• If we use the signs table to estimate D, what we
  really get is an estimate of D ABC. (Exactly
  the same estimate wed get if we had done a full
  24 Design, computed D and ABC and added them.)
• The two effects are stuck together hence, we
  say they are confounded with each other (on
  purpose here).
• Similarly, in this design,
• A is confounded with BCD
• B is confounded with ACD
• AB is confounded with CD
• ETC!

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II.2 Four Factors in Eight Runs Confounding

• To Illustrate
• We want to know if method 1 is better than method
  2 for a task. Ann does method 1, Dan does method
  2. If Anns results are better, is it because
  method 1 is better than method 2? Or, is Ann
  better than Dan? Or, is it both? The factor
  worker is confounded with the factor method . We
  cant separate their effects.
• Confounding can sometimes be a very dumb thing to
  do (but not always).

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II.2 Four Factors in Eight Runs Confounding

• When we get the data and compute D, the result
  is really an estimate of D ABC.
• So, (another new word coming - duck!) D is a
  false name for the estimate - an alias. When two
  effects are confounded, we say they are aliases
  of each other.
• The Alias Structure (also called the
  confounding structure) of the design is this
  table youve already seen (rearranged here)
• I ABCD
• A BCD
• B ACD
• C ABD
• D ABC
• AB CD
• AC BD
• BC AD

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II.2 Four Factors in Eight Runs An
ExampleRevisit Examples 2 and 4 of Part I

• Response y Throughput (KB/sec)
• The Original Experiment was a 24 Design (16 Runs)
• Four Factors A, B, C, D, performance tuning
  parameters such as
• number of buffers
• size of unix inode tables for file handling
• Two Levels
• In Example 2 an 8 Run Design with only Three
  Factors was Considered for Illustrative Purposes.
  The Numbers were Rounded Off for Ease of
  Calculation
• Original Data Was In Tenths and Involved Four
  Factors
• The Estimate of the Three-way Interaction ABC was
  also Estimating the Effect of D. (D and ABC are
  confounded/aliased.)

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II.2 Four Factors in Eight Runs An
ExampleRevisit Examples 2 and 4 of Part I

• ABCD I determines runs in half fraction
• D ABC for these runs(Complementary half
  fraction is determinedby ABCD -I, or D -ABC,
  for these runs)

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II.2 Four Factors in Eight Runs An
ExampleDesign Matrix
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II.2 Four Factors in Eight Runs An
ExampleSigns Table

• Use Eight Run Signs Table to Estimate Effects
• Factor D is Assigned to the Last Column, ABC
• Use Alias Structure to Determine What These
  Quantities Are Estimating

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II.2 Four Factors in Eight Runs An
ExampleNormal Probability Plot

• Effect ABCD is Statistically Significant

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II.2 Four Factors in Eight Runs An
ExampleInterpretation

• Response y Throughput (KB/sec)
• Assuming BCD is negligible, you should choose A
  Hi (A ) to maximize y Caution ASSUME

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II.2 Four Factors in Eight Runs U-Do-It
Exercise Violin Example

• For the Violin Data, Pretend That a Half-fraction
  of the Full 24 Was Run. For your convenience,
  the violin data and signs table is on the next
  slide as well as an eight run signs table with
  the aliasing structure that determines the
  half-fraction
• Find the Levels of Factors A, B, C and D that
  Would Have Been Run
• Pick out the observed ys for these runs. Enter
  these into an eight-run response table and
  compute the observed effects.
• Compare these effects to those which were
  computed from the full 24

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II.2 Four Factors in Eight RunsU-Do-It Exercise
Violin Example - Signs Tables