Transcript Statistical Data Analysis: Primer

Stat 470-16
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Today: Start Chapter 4
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Assignment 4:
Techniques for Resolving Ambiguities
• Suppose the experiment in the previous example was performed and
the AC=BE interaction was identified as significant (in addition to the
A and E main effects)
• Which is the important interaction AC or BE or both?
• Prior knowledge may indicate that one of the effects is not important
• Can conduct a follow-up experiment
Method of Orthogonal Runs
Fold-Over
Assignment Question
• Suppose in the cable shrinkage example, effects A, E and AC=BE are
identified as significant
• To resolve the aliasing of the interaction effects, a follow-up
experiment is to be performed
• Find a follow-up design to address this issue
Additional Features of a Fractional Factorial
• Main effect or two-factor interactions (2fi) is clear if it is not aliased
with other main effects or 2fi’s
• Main effect or 2fi is strongly clear if it is not aliased with other main
effects, 2fi’s or 3fi’s
Blocking Fractional Factorial Designs
• Can perform a 2k-p fractional factorial design in 2q blocks
• That is, k factors are investigated in 2k-p runs with 2q blocks
• The design is constructed by assigning p treatment factors and q
blocking factors to interactions between (k-p) of the factors
Example
• An experimenter wishes to explore the impact of 6 factors (A-F) on the
response of a system
• There exists enough resources to run 16 experiment trials in 4 blocks
• A 26-2 fraction factorial design in 22 blocks is required
Example
• Design:
– Fractional factorial: E=ABC; F=ABD
– Blocking: b1=ACD; b2=BCD
• Defining Contrast sub-group:
Example
A
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B
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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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Comment
• Must be careful when choosing the interactions to assign the factors
– Fractional factorial: E=AB; F=ABD
– Blocking: b1=ACD; b2=BCD
• Defining Contrast sub-group:
Additional Features
• Main effect or two-factor interactions (2fi) is clear if it is not aliased
with other main effects, 2fi’s or block effects
• Main effect or 2fi is strongly clear if it is not aliased with other main
effects, 2fi’s, 3fi’s or block effects
• As before, block by factor interactions are negligible
• Analysis is same as before
• Appendix 4 has blocked fractional factorial designs ranked by number
of clear effects
Fractional Factorial Split-Plot Designs
• It is frequently impractical to perform the fractional factorial design in
a completely randomized manner
• Can run groups of treatments in blocks
• Sometimes the restrictions on randomization take place because some
factors are hard to change or the process takes place in multiple stages
• Fractional factorial split-plot (FFSP) design may be a practical option
Performing FFSP Designs
• Randomization of FFSP designs different from fractional factorial
designs
• Have hard to change factors (whole-plot or WP factors) and easy to
change factors (sub-plot or SP factors)
• Experiment performed by:
– selecting WP level setting, at random.
– performing experimental trials by varying SP factors, while
keeping the WP factors fixed.
Example
• Would like to explore the impact of 6 factors in 16 trials
• The experiment cannot be run in a completely random order because 3
of the factors (A,B,C) are very expensive to change
• Instead, several experiment trials are performed with A, B, and C
fixed…varying the levels of the other factors
Design Matrix
A
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B
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C
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p
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q
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r
-1
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Impact of the Randomization Restrictions
• Two Sources of randomization  Two sources of error
– Between plot error: ew (WP error)
– Within plot error:  s (SP error)
• Model:
y  X  eW   S
• The WP and SP error terms have mutually independent normal
distributions with standard deviations σw and σs
The Design
• Situation:
– Have k factors: k1 WP factors and k2 SP factors
– Wish to explore impact in 2k-p trials
– Have a 2 k1-p1 fractional factorial for the WP factors
– Require p=p1+p2 generators
– Called a 2(k1+ k2)-(p1+ p2) FFSP design
Constructing the Design
• For a 2(k1+ k2)-(p1+ p2) FFSP design, have generators for WP and SP
designs
• Rules:
– WP generators (e.g., I=ABC ) contain ONLY WP factors
– SP generators (e.g., I=Apqr ) must contain AT LEAST 2 SP factors
• Previous design: I=ABC=Apqr=BCpqr
Analysis of FFSP Designs
• Two Sources of randomization  Two sources of error
– Between plot error: σw (WP error).
– Within plot error: σs (SP error).
• WP Effects compared to: aσs2 + bσs2
• SP effects compared to : bσs2
• df for SP > df for WP.
• Get more power for SP effects!!!
WP Effect or SP Effect?
• Effects aliased with WP main effects or interactions involving only WP
factors tested as a WP effect.
• E.g., pq=ABCD tested as a WP effect.
• Effects aliased only with SP main effects or interactions involving at
least one SP factors tested as a SP effect .
• E.g., pq=ABr tested as a SP effect.
Ranking the Designs
• Use minimum aberration (MA) criterion