Transcript Statistical Data Analysis: Primer

Stat 470-14
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Today: Finish Chapter 3; Start Chapter 4
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Assignment 3: 3.14 a, b (do normal qq-plots only),c, 3.16, 3.17
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Additional questions: 3.14 b (also use the IER version of Lenth’s
method and compare to the qq-plot conclusions), 3.19
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Important Sections in Chapter 3:
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–
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3.1-3.7 (Read these)
3.8 (don’t bother reading),
3.11-3.13 (Read these)
Blocking in 2k Experiments
• The factorial experiment is an example of a completely randomized
design
• Often wish to block such experiments
• As an example, you may wish to use the same paper helicopter for
more than one trial
• But which treatments should appear together in a block?
Blocking in 2k Experiments
• Consider a 23 factorial experiment in 2 blocks
A
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B
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C
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AB
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AC
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BC
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ABC
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BLOCK
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II
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Blocking in 2k Experiments
• Presumably, blocks are important.
• Effect hierarchy suggest we sacrifice higher order interactions
• Which is better:
– b=ABC
– b=AB
• Can write as:
Blocking in 2k Experiments
• Suppose wish to run the experiment in 4 blocks
• b1=AB and b2=BC
• These imply a third relation
• Group is called the defining contrast sub-group
Blocking in 2k Experiments
• Identifies which effects are confounded with blocks
• Cannot tell difference between these effects and the blocking effects
Which design is better?
• Suppose wish to run the 23
experiment in 4 blocks
• Suppose wish to run the 23
experiment in 4 blocks
• b1=AB and b2=BC
• b1=ABC and b2=BC
• I=ABb1=BCb2=ACb1b2
• I=ABCb1=BCb2=Ab1b2
Ranking the Designs
• Let D denote a blocking design
• gi(D) is the number of i-factor interactions confounded in blocks
(i=1,2,…k)
• For any 2 blocking schemes (D1 and D2) , let r be the smallest i such
that gr (D1 )  gr (D2 )
Ranking the Designs
• Effect hierarchy suggests that the design that confounds the fewest
lower order terms is best
• So, if
g r ( D1 )  g r ( D2 )
then D1 has less aberration
• A design has minimum aberration (MA) if no design has less
aberration
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?
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
Example
A
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B
-1.00
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C
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D
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AB
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AC
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AD
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BC
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BD
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CD
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ABC
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ABD
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ACD
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BCD
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ABCD
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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
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
Example
• The 3 relations imply other relations
• Words
• Defining contrast sub-group
• Word-length pattern
Example
• Would like to have as few short words as possible
• Why?
• How can we compare designs?
• Resolution
• Minimum aberration:
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: I=ABCDF=ABCEG=DEFG
– D2: I=ABCF=ADEG=BCDEG