Today: 4.2

1
Lecture 11

• Today 4.2
• Next day 4.3-4.6

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Analysis of Unreplicated 2k Factorial Designs

• For cost reasons, 2k factorial experiments are
  frequently unreplicated
• Can assess significance of the factorial effects
  using a normal or half-normal probability plot
• May prefer a formal significance test procedure
• Cannot use an F-test or t-test because there are
  no degrees of freedom for error

3
Lenths Method

• Situation
• have performed an unreplicated 2k factorial
  experiment
• have 2k-1 factorial effects
• want to see which effects are significantly
  different from 0
• If none of the effects is important of the
  factorial effects is an independent realization
  of a N( , )
• Can use this fact to develop an estimator of the
  effect variance based on the median of the
  absolute effects

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Lenths Method

• s0
• PSE
• tPSE,i

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Example Original Growth Layer Experiment

• Effect Estimates and QQ-Plot

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Lenths Method

• s0
• PSE
• tPSE,i

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Fractional Factorial Designs at 2-Levels

• 2k factorial experiments can be very useful in
  exploring a relatively large number of factors in
  relatively few trials
• When k is large, the number of trials is large
• Suppose have enough resources to run only a
  fraction of the 2k unique treatments
• Which sub-set of the 2k treatments should one
  choose?

8
Example

• Suppose have 5 factors, each at 2-levels, but
  only enough resources to run 16 trials
• Can use a 16-run full factorial to design the
  experiment
• Use the 16 unique treatments for 4 factors to set
  the levels of the first 4 factors (A-D)
• Use an interaction column from the first 4
  factors to set the levels of the 5th factor

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Example
10
Fractional Factorial Designs at 2-Levels

• Use a 2k-p fractional factorial design to explore
  k factors in 2k-p trials
• In general, can construct a 2k-p fractional
  factorial design from the full factorial design
  with 2k-p trials
• Set the levels of the first (k-p) factors similar
  to the full factorial design with 2k-p trials
• Next, use the interaction columns between the
  first (k-p) factors to set levels of the
  remaining factors

11
Fractional Factorial Designs at 2-Levels

• Use a 2k-p fractional factorial design to explore
  k factors in 2k-p trials
• In general, can construct a 2k-p fractional
  factorial design from the full factorial design
  with 2k-p trials
• Set the levels of the first (k-p) factors similar
  to the full factorial design with 2k-p trials
• Next, use the interaction columns between the
  first (k-p) factors to set levels of the
  remaining factors

12
Example

• Suppose have 7 factors, each at 2-levels, but
  only enough resources to run 16 trials
• Can use a 16-run full factorial to design the
  experiment
• Use the 16 unique treatments for 4 factors to set
  the levels of the first 4 factors (A-D)
• Use interaction columns from the first 4 factors
  to set the levels of the remaining 3 factors

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Example

• The 3 relations imply other relations
• Defining contrast sub-group
• Word-length pattern

14

• How can we compare designs?
• Resolution
• Minimum aberration

15
Example

• Suppose have 7 factors, each at 2-levels, but
  only enough resources to run 32 trials
• Can use a 27-2 fractional factorial design
• Which one is better?
• D1 IABCDFABCEGDEFG
• D2 IABCFADEGBCDEG