Fractional Factorial Designs:A TutorialVijay NairDepartments of Statistics and Industrial & Operations Engineeringvnn@umich.edu

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