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Fractional Factorial Designs: A Tutorial

Fractional Factorial Designs: A Tutorial. Vijay Nair Departments of Statistics and Industrial & Operations Engineering vnn@umich.edu. Design of Experiments (DOE) in Manufacturing Industries.

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Fractional Factorial Designs: A Tutorial

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  1. Fractional Factorial Designs:A Tutorial Vijay Nair Departments of Statistics and Industrial & Operations Engineering vnn@umich.edu

  2. Design of Experiments (DOE)in Manufacturing Industries • Statistical methodology for systematically investigating a system's input-output relationship to achieve one of several goals: • Identify important design variables (screening) • Optimize product or process design • Achieve robust performance • Key technology in product and processdevelopment Used extensively in manufacturing industries Part of basic training programs such as Six-sigma

  3. Design and Analysis of ExperimentsA Historical Overview • Factorial and fractional factorial designs (1920+)  Agriculture • Sequential designs (1940+)  Defense • Response surface designs for process optimization (1950+)  Chemical • Robust parameter design for variation reduction (1970+)  Manufacturing and Quality Improvement • Virtual (computer) experiments using computational models (1990+)  Automotive, Semiconductor, Aircraft, …

  4. Overview • Factorial Experiments • Fractional Factorial Designs • What? • Why? • How? • Aliasing, Resolution, etc. • Properties • Software • Application to behavioral intervention research • FFDs for screening experiments • Multiphase optimization strategy (MOST)

  5. (Full) Factorial Designs • All possible combinations • General: I x J x K … • Two-level designs: 2 x 2, 2 x 2 x 2, … 

  6. (Full) Factorial Designs • All possible combinations of the factor settings • Two-level designs: 2 x 2 x 2 … • General: I x J x K … combinations

  7. Will focus on two-level designs OK in screening phase i.e., identifying important factors

  8. (Full) Factorial Designs • All possible combinations of the factor settings • Two-level designs: 2 x 2 x 2 … • General: I x J x K … combinations

  9. Full Factorial Design

  10. 9.5 5.5

  11. Algebra -1 x -1 = +1 …

  12. Design Matrix Full Factorial Design

  13. 7 9 9 9 8 3 8 3 7 + 9 + 8 + 8 9 + 9 + 3 + 3 6 8 6 – 8 = -2

  14. Fractional Factorial Designs • Why? • What? • How? • Properties

  15. Why Fractional Factorials? Treatment combinations Full Factorials No. of combinations  This is only for two-levels In engineering, this is the sample size -- no. of prototypes to be built. In prevention research, this is the no. of treatment combos (vs number of subjects)

  16. How? Box et al. (1978) “There tends to be a redundancy in [full factorial designs] – redundancy in terms of an excess number of interactions that can be estimated … Fractional factorial designs exploit this redundancy …” philosophy

  17. How to select a subset of 4 runs from a -run design? Many possible “fractional” designs

  18. Here’s one choice

  19. Here’s another … Need a principled approach!

  20. Regular Fractional Factorial Designs Balanced design All factors occur and low and high levels same number of times; Same for interactions. Columns are orthogonal. Projections …  Good statistical properties Wow! Need a principled approach for selecting FFD’s

  21. What is the principled approach? Notion of exploiting redundancy in interactions  Set X3 column equal to the X1X2 interaction column Need a principled approach for selecting FFD’s

  22. Notion of “resolution”  coming soon to theaters near you …

  23. Regular Fractional Factorial Designs Half fraction of a design = design 3 factors studied -- 1-half fraction  8/2 = 4 runs Resolution III (later) Need a principled approach for selecting FFD’s

  24. Confounding or Aliasing  NO FREE LUNCH!!! X3=X1X2  ?? aliased X3 = X1X2  X1X3 = X2 and X2X3 = X1 (main effects aliased with two-factor interactions) – Resolution III design

  25. Want to study 5 factors (1,2,3,4,5) using a 2^4 = 16-run design i.e., construct half-fraction of a 2^5 design = 2^{5-1} design For half-fractions, always best to alias the new (additional) factor with the highest-order interaction term

  26. What about bigger fractions? Studying 6 factors with 16 runs? ¼ fraction of X5 = X2*X3*X4; X6 = X1*X2*X3*X4;  X5*X6 = X1 (can we do better?)

  27. X5 = X1*X2*X3; X6 = X2*X3*X4  X5*X6 = X1*X4 (yes, better)

  28. Design Generatorsand Resolution X5 = X1*X2*X3; X6 = X2*X3*X4  X5*X6 = X1*X4 5 = 123; 6 = 234; 56 = 14  Generators: I = 1235 = 2346 = 1456 Resolution: Length of the shortest “word” in the generator set  resolution IV here So …

  29. Resolution Resolution III: (1+2) Main effect aliased with 2-order interactions Resolution IV: (1+3 or 2+2) Main effect aliased with 3-order interactions and 2-factor interactions aliased with other 2-factor … Resolution V: (1+4 or 2+3) Main effect aliased with 4-order interactions and 2-factor interactions aliased with 3-factor interactions

  30. ¼ fraction of X5 = X2*X3*X4; X6 = X1*X2*X3*X4;  X5*X6 = X1 or I = 2345 = 12346 = 156  Resolution III design

  31. X5 = X1*X2*X3; X6 = X2*X3*X4  X5*X6 = X1*X4 or I = 1235 = 2346 = 1456  Resolution IV design

  32. Aliasing Relationships I = 1235 = 2346 = 1456 Main-effects: 1=235=456=2346; 2=135=346=1456; 3=125=246=1456; 4=… 15-possible 2-factor interactions: 12=35 13=25 14=56 15=23=46 16=45 24=36 26=34

  33. Properties of FFDs Balanced designs Factors occur equal number of times at low and high levels; interactions … sample size for main effect = ½ of total. sample size for 2-factor interactions = ¼ of total. Columns are orthogonal  …

  34. How to choose appropriate design? Software  for a given set of generators, will give design, resolution, and aliasing relationships • SAS, JMP, Minitab, … Resolution III designs  easy to construct but main effects are aliased with 2-factor interactions Resolution V designs  also easy but not as economical (for example, 6 factors  need 32 runs) Resolution IV designs  most useful but some two-factor interactions are aliased with others.

  35. Selecting Resolution IV designs Consider an example with 6 factors in 16 runs (or 1/4 fraction) Suppose 12, 13, and 14 are important and factors 5 and 6 have no interactions with any others Set 12=35, 13=25, 14= 56 (for example)  I = 1235 = 2346 = 1456  Resolution IV design All possible 2-factor interactions: 12=35 13=25 14=56 15=23=46 16=45 24=36 26=34

  36. Latest design for Project 1 Project 1: 2^(7-2) design 32 trx combos

  37. Role of FFDs in Prevention Research • Traditional approach: randomized clinical trials of control vs proposed program • Need to go beyond answering if a program is effective  inform theory and design of prevention programs  “opening the black box” … • A multiphase optimization strategy (MOST)  center projects (see also Collins, Murphy, Nair, and Strecher) • Phases: • Screening (FFDs) – relies critically on subject-matter knowledge • Refinement • Confirmation

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