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New Results about Randomization and Split-Plotting
byJames M. Lucas
2003 Quality & Productivity Research Conference
Yorktown Heights, New YorkMay 21-23, 2003
J. M. Lucas and Associates 2
Contact Information
James M. LucasJ. M. Lucas and Associates 5120 New Kent Road Wilmington, DE 19808 (302) 368-1214 [email protected]
J. M. Lucas and Associates 3
Research Team Huey Ju Jeetu Ganju Frank Anbari Peter Goos
Malcolm Hazel Derek Webb John Borkowski
PRELIMINARIES
How do you run Experiments?
J. M. Lucas and Associates 5
QUESTIONS How many of you are involved with
running experiments? How many of you “randomize” to guard
against trends or other unexpected events?
If the same level of a factor such as temperature is required on successive runs, how many of you set that factor to a neutral level and reset it?
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ADDITIONAL QUESTIONS
How many of you have conducted experiments on the same process on which you have implemented a Quality Control Procedure?
What did you find?
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COMPARING RESIDUAL STANDARD DEVIATION FROM AN EXPERIMENT WITHRESIDUAL STANDARD DEVIATION FROM AN IN-CONTROL PROCES
MY OBSERVATIONS
EXPERIMENTAL STANDARD DEVIATION IS LARGER. 1.5X TO 3X IS COMMON.
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HOW SHOULD EXPERIMENTS BE CONDUCTED?
•“COMPLETE RANDOMIZATION” (and the completely randomized design)
•RANDOMIZED NOT RESET (Also Called Random Run Order (RRO) Experiments) (Often Achieved When Complete Randomization is Assumed)
•SPLIT PLOT BLOCKING (Especially When There are Hard-to-Change Factors)
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Randomized Not Reset (RNR) Experiments
A large fraction (perhaps a large majority) of industrial experiments are Randomized not Reset (RNR) experiments
Properties of RNR experiments and a discussion of how experiments should be conducted: “Lk Factorial Experiments with Hard-to-Change and
Easy-to-Change Factors” Ju and Lucas, 2002, JQT 34, 411-421 [studies one H-T-C factor and uses Random Run Order (RRO) rather than RNR]
“Factorial Experiments when Factor Levels Are Not Necessarily Reset” Webb, Lucas and Borkowski, 2003, JQT, to appear [studies >1 HTC Factor]
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RNR EXPERIMENTS (Random Run Order Without Resetting Factors)
OFTEN USED BY EXPERIMENTERS NEVER EXPLICITLY RECOMMENDED
ADVANTAGES•Often achieves successful results•Can be cost-effectiveDISADVANTAGES•Often can not be detected after experiment is conducted (Ganju and Lucas 99)•Biased tests of hypothesis (Ganju and Lucas 97, 02)•Can often be improved upon•Can miss significant control factors
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Results for Experiments with Hard-to-Change and Easy-to-Change Factors
One H-T-C or E-T-C Factor: use split-plot blocking
Two H-T-C Factors: may split-plot Three or more H-T-C Factors:
consider RNR or Low Cost Options Consider “Diccon’s Rule”: Design
for the H-T-C Factor
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New Results Joint work with Peter Goos Builds on the Kiefer-Wolfowitz
Equivalence Theorem Implications about Computer
generated designs (especially when there are Hard-to-Change Factors)
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Kiefer-Wolfowitz Equivalence Theorem is the design probability measure M() = X’X/n (kxk matrix for a n point
design) d(x, ) = x’(M())-1x (normalized variance) So called Approximate Theory The following are equivalent: maximizes det M() minimizes d(x, ) Max (d(x, ) = k
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Very Important Theorem Helps find Optimum Designs Basis for much computer aided design
work Justifies using |X’X| Criterion
Shows “Classical Designs” are great “Which Response Surface Design is Best”
Technometrics (1976) 16, 411-417 Computer generated designs not
needed for “standard” situations
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Optimality Criteria Determinant (D-optimality)
Maximize |X’X| D-efficiency = {|X’X/n|/ |X*’X*/n*|}1/k where
X* is an optimum n* point design Global (G-optimality)
Minimize the maximum variance G-efficiency = k/Max d(x, )
G-efficiency < D-efficiency No bad designs with high G-efficiency
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Computer Generated Design Arrays
Different criteria give different “n” point designs
Do not pick a single “n”Some “n” values may achieve an excellent
designCheck other criteria (especially G-)
Lucas (1978) “Discussion of: D-Optimal Fractions of Three Level Factorial Designs”
Borkowski (2003) “Using A Genetic Algorithm to Generate Small Exact Response Surface Designs”
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Equivalence Theorem does not hold for Split-Plot Experiments D- and G- criteria converge to different
designs Example: r reps of a 23 Factorial (linear
terms model) Optimum design depends on d =w
2/2 where w is the whole-plot and is the split-plot error
For large values of d: D-optimal design has 4 r blocks with I = A = BC G-optimal design has 8r – 2 blocks (Number of
observations minus number of split-plot terms)
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Computer Generated Split-Plot Experiments Useful Research Recent publications:
Trinca and Gilmour (2001) “Multi-stratum Response Surface Designs” Technometrics 43: 25-33
Goos and Vandebroek (2001) “Optimal Split-Plot Designs” JQT 33: 436-450
Goos and Vandebroek (2003) “Outperforming Completely Randomized Designs” JQT to appear
All use |X’X| Criterion
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RELATED SPLIT-PLOT FINDINGS
SUPER EFFICIENT EXPERIMENTS (With One or Two Hard-to-Change Factor) SPLIT PLOT BLOCKING GIVES HIGHER PRECISION AND LOWER COSTS THAN COMPLETELY RANDOMIZED EXPERIMENTS
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Design Precision: Calculating Maximum Variance Simplifications for 2k factorials Sum Variances of individual terms Whole plot terms:
w2/ number blocks + 2/ 2k
Split plot terms: 2/2k
Completely randomized design has variance: k(w
2+ 2)/ 2k
Blocking Observation to achieve Super Efficiency
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26-1 with one or two Hard-to-Change Factors
Main Effects plus interaction Model 22 Terms = (1 + 6 + 15)
Use Resolution V, not VI with I=ABCDEUse four blocks I=A=BCF=ABCF=BCDE=ADEF=DEF
Nest Factor B within each A block giving a split-split-plot with 8 Blocks =B2=AB2=CF2=ACF2=CDE2=ABDEF2=BDEF2
I and A have variance 02/32 + 1
2/4 +22 /8
B, AB and CF have 02/32 + 2
2 /8 Other terms have variance 0
2/32 G-efficiency = 22(0
2+12+2
2)/(2202+161
2+2022 )
>1.0 Drop 2
2 terms for one h-t-c factor results
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Observations Does not use Maximum Resolution
or Minimum Abberation Similar results for most 2k
factorials
Super Efficient Experiments are not always Optimal
26-1 Main effects plus 2FI model
G-optimum design has 12 blocks when d gets large
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Conclusions Showed K-W Equivalence theorem
does not hold for Split-Plot Experiments
Discussed Implications Exciting research area Much more to do