Stat 470-16 • Today: Start Chapter 4 • Assignment 4:
Stat 470-16
• Today: Start Chapter 4
• 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 B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD-1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1-1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1-1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1-1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1-1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1-1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1-1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1-1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 11 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 11 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -11 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -11 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 11 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -11 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1
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 B C p q r-1 -1 +1 -1 -1 -1
+1 -1 +1-1 +1 +1+1 +1 -1
+1 -1 -1 -1 -1 +1+1 -1 -1-1 +1 -1+1 +1 +1
-1 +1 -1 -1 -1 -1+1 -1 +1-1 +1 +1+1 +1 -1
+1 +1 +1 -1 -1 +1+1 -1 -1-1 +1 -1+1 +1 +1
Impact of the Randomization Restrictions
• Two Sources of randomization Two sources of error
– Between plot error: ew (WP error)
– Within plot error: (SP error)
• Model:
• The WP and SP error terms have mutually independent normal distributions with standard deviations σw and σs
SWeXy
s
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