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Fractional Factorial Designs:A TutorialVijay NairDepartments of Statistics and Industrial & Operations [email protected]
Design of Experiments (DOE)in Manufacturing IndustriesStatistical 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 designAchieve robust performance
Key technology in product and process development
Used extensively in manufacturing industriesPart of basic training programs such as Six-sigma
Design and Analysis of ExperimentsA Historical OverviewFactorial 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,
OverviewFactorial ExperimentsFractional Factorial DesignsWhat?Why?How?Aliasing, Resolution, etc.PropertiesSoftwareApplication to behavioral intervention researchFFDs for screening experimentsMultiphase optimization strategy (MOST)
(Full) Factorial DesignsAll possible combinations
General: I x J x K
Two-level designs: 2 x 2, 2 x 2 x 2,
(Full) Factorial DesignsAll possible combinations of the factor settings
Two-level designs: 2 x 2 x 2
General: I x J x K combinations
Will focus on two-level designs
OK in screening phasei.e., identifyingimportant factors
(Full) Factorial DesignsAll possible combinations of the factor settings
Two-level designs: 2 x 2 x 2
General: I x J x K combinations
Full Factorial Design
9.55.5
Algebra-1 x -1 = +1
Full Factorial DesignDesign Matrix
9 + 9 + 3 + 367 + 9 + 8 + 886 8 = -27
9
9
9
8
3
8
3
Fractional Factorial DesignsWhy?What?How?Properties
Treatment combinationsIn 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) Why Fractional Factorials?Full FactorialsNo. of combinationsThis is only fortwo-levels
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
How to select a subset of 4 runsfrom a -run design?Many possible fractional designs
Heres one choice
Need a principled approach!Heres another
Need a principled approach for selecting FFDsRegular Fractional Factorial DesignsWow!Balanced designAll factors occur and low and high levels same number of times; Same for interactions.Columns are orthogonal. Projections Good statistical properties
Need a principled approach for selecting FFDs
What is the principled approach?
Notion of exploiting redundancy in interactions Set X3 column equal to the X1X2 interaction column
Notion of resolution coming soon to theaters near you
Need a principled approach for selecting FFDsRegular Fractional Factorial DesignsHalf fraction of a design = design3 factors studied -- 1-half fraction 8/2 = 4 runs
Resolution III (later)
X3 = X1X2 X1X3 = X2 and X2X3 = X1 (main effects aliased with two-factor interactions) Resolution III design
Confounding or Aliasing NO FREE LUNCH!!!
X3=X1X2 ??aliased
For half-fractions, always best to alias the new (additional) factor with the highest-order interaction term
Want to study 5 factors (1,2,3,4,5) using a 2^4 = 16-run designi.e., construct half-fraction of a 2^5 design = 2^{5-1} design
X5 = X2*X3*X4; X6 = X1*X2*X3*X4; X5*X6 = X1 (can we do better?)
What about bigger fractions?Studying 6 factors with 16 runs? fraction of
X5 = X1*X2*X3; X6 = X2*X3*X4 X5*X6 = X1*X4 (yes, better)
Design Generatorsand ResolutionX5 = 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
ResolutionResolution III: (1+2)Main effect aliased with 2-order interactions
Resolution IV: (1+3 or 2+2)Main effect aliased with 3-order interactions and2-factor interactions aliased with other 2-factor
Resolution V: (1+4 or 2+3)Main effect aliased with 4-order interactions and2-factor interactions aliased with 3-factor interactions
X5 = X2*X3*X4; X6 = X1*X2*X3*X4; X5*X6 = X1
or I = 2345 = 12346 = 156 Resolution III design
fraction of
X5 = X1*X2*X3; X6 = X2*X3*X4 X5*X6 = X1*X4
or I = 1235 = 2346 = 1456 Resolution IV design
Aliasing RelationshipsI = 1235 = 2346 = 1456
Main-effects:1=235=456=2346; 2=135=346=1456; 3=125=246=1456; 4=
15-possible 2-factor interactions:12=3513=2514=5615=23=4616=4524=3626=34
Balanced designs Factors occur equal number of times at low and high levels; interactions sample size for main effect = of total. sample size for 2-factor interactions = of total.Columns are orthogonal Properties of FFDs
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 interactionsResolution 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.
Selecting Resolution IV designsConsider 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=3513=2514=5615=23=4616=4524=3626=34
Latest design for Project 1
Project 1: 2^(7-2) design32 trxcombos
PATTERNOE-DEPTHDOSETESTIMONIALSFRAMINGEE-DEPTHSOURCESOURCE-DEPTH+----+-LO1HIGainHITeamHI--+-++-HI1LOGainLOTeamHI++----+LO5HIGainHIHMOLO+---+++LO1HIGainLOTeamLO++-++-+LO5HILossLOHMOLO--+--++HI1LOGainHITeamLO+--+++-LO1HILossLOTeamHI-++----HI5LOGainHIHMOHI-++-+-+HI5LOGainLOHMOLO-++++--HI5LOLossLOHMOHI----+--HI1HIGainLOHMOHI-+-+++-HI5HILossLOTeamHI
FactorsSourceSource-DepthOE-DepthXXDoseXXTestimonialsX Framing XEE-Depth X
EffectsAliasesOE-Depth*Dose = Testimonials*SourceOEDepth*Testimonials = Dose*SourceOE-Depth*Source = Dose*Testimonials
Role of FFDs in Prevention ResearchTraditional approach: randomized clinical trials of control vs proposed programNeed 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 RefinementConfirmation