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New Results about Randomization and Split-Plotting

New Results about Randomization and Split-Plotting. by James M. Lucas 2003 Quality & Productivity Research Conference Yorktown Heights, New York May 21-23, 2003. Contact Information. James M. Lucas J. M. Lucas and Associates 5120 New Kent Road

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New Results about Randomization and Split-Plotting

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  1. New Results about Randomization and Split-Plotting by James M. Lucas 2003 Quality & Productivity Research Conference Yorktown Heights, New York May 21-23, 2003

  2. Contact Information James M. Lucas J. M. Lucas and Associates 5120 New Kent Road Wilmington, DE 19808 (302) 368-1214 JamesM.Lucas@worldnet.att.net J. M. Lucas and Associates

  3. Huey Ju Jeetu Ganju Frank Anbari Peter Goos Malcolm Hazel Derek Webb John Borkowski Research Team J. M. Lucas and Associates

  4. PRELIMINARIES How do you run Experiments?

  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? J. M. Lucas and Associates

  6. 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? J. M. Lucas and Associates

  7. 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. J. M. Lucas and Associates

  8. 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) J. M. Lucas and Associates

  9. 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] J. M. Lucas and Associates

  10. RNR EXPERIMENTS • (Random Run Order Without Resetting Factors) • OFTEN USED BY EXPERIMENTERS • NEVER EXPLICITLY RECOMMENDED • ADVANTAGES • Often achieves successful results • Can be cost-effective • DISADVANTAGES • 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 J. M. Lucas and Associates

  11. 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 J. M. Lucas and Associates

  12. 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) J. M. Lucas and Associates

  13. 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 J. M. Lucas and Associates

  14. 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 J. M. Lucas and Associates

  15. 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 J. M. Lucas and Associates

  16. Computer Generated Design Arrays Different criteria give different “n” point designs Do not pick a single “n” Some “n” values may achieve an excellent design Check 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” J. M. Lucas and Associates

  17. 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 =w2/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) J. M. Lucas and Associates

  18. 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 J. M. Lucas and Associates

  19. 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 J. M. Lucas and Associates

  20. 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(w2+ 2)/ 2k • Blocking Observation to achieve Super Efficiency J. M. Lucas and Associates

  21. 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=ABCDE Use 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 + 12/4 +22 /8 • B, AB and CF have 02/32 + 22 /8 • Other terms have variance 02/32 • G-efficiency = 22(02+12+22)/(2202+1612+2022 ) >1.0 • Drop 22 terms for one h-t-c factor results J. M. Lucas and Associates

  22. Observations • Does not use Maximum Resolution or Minimum Abberation • Similar results for most 2k factorials J. M. Lucas and Associates

  23. Super Efficient Experiments are not always Optimal 26-1 Main effects plus 2FI model G-optimum design has 12 blocks when d gets large

  24. Conclusions • Showed K-W Equivalence theorem does not hold for Split-Plot Experiments • Discussed Implications • Exciting research area • Much more to do J. M. Lucas and Associates

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