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
• 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 process development

Used extensively in manufacturing industries
Part of basic training programs such as Six-sigma
Design and Analysis of Experiments
       A 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, …
                          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)
        (Full) Factorial Designs

• All possible combinations

• General: I x J x K …

• Two-level designs: 2 x 2, 2 x 2 x 2, … 
      (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
    Will focus on
  two-level designs

OK in screening phase
   i.e., identifying
  important factors
      (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
Full Factorial Design
9.5




      5.5
  Algebra
-1 x -1 = +1
     …
 Design Matrix




Full Factorial Design
                                  7

                                  9

                                  9

                                  9

                                  8

                                  3

                                  8

                                  3




9+9+3+3             7+9+8+8
          6                   8


              6 – 8 = -2
     Fractional Factorial Designs
•   Why?
•   What?
•   How?
•   Properties
                        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)
                              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 runs
   from a         -run design?
Many possible “fractional” designs
Here’s one choice
     Here’s another …




Need a principled approach!
          Regular Fractional Factorial Designs




                                            Balanced design
                               All factors occur and low and high levels
                             same number of times; Same for interactions.
Wow!
                                Columns are orthogonal. Projections …
                                      selecting FFD’s
       Need a principled approach for Good statistical properties
     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
Notion of “resolution”  coming soon to theaters near you …
       Regular Fractional Factorial Designs




                       Half fraction of a   design =           design
                            3 factors studied -- 1-half fraction
                                       8/2 = 4 runs

Need a principled approach for selecting FFD’s (later)
                                  Resolution III
                  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
    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
                   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?)
X5 = X1*X2*X3; X6 = X2*X3*X4  X5*X6 = X1*X4 (yes, better)
                Design Generators
                 and 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 …
                       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
         ¼ fraction of




X5 = X2*X3*X4; X6 = X1*X2*X3*X4;    X5*X6 = X1

or I = 2345 = 12346 = 156  Resolution III design
X5 = X1*X2*X3; X6 = X2*X3*X4  X5*X6 = X1*X4

or I = 1235 = 2346 = 1456    Resolution IV design
                  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
                      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  …
       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.
         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
                                 Project 1: 2^(7-2) design
                  PATTERN        OE-        DOSE   TESTIMO   FRAMING   EE-DEPTH   SOURCE     SOURCE-
                                DEPTH               NIALS                                     DEPTH

                  +----+-        LO          1       HI       Gain        HI       Team        HI

                  --+-++-        HI          1       LO       Gain       LO        Team        HI

                  ++----+        LO          5       HI       Gain        HI       HMO         LO

                  +---+++        LO          1       HI       Gain       LO        Team        LO

                  ++-++-+        LO          5       HI        Loss      LO        HMO         LO

                  --+--++        HI          1       LO       Gain        HI       Team        LO

 32 trx           +--+++-        LO          1       HI        Loss      LO        Team        HI


combos            -++----        HI          5       LO       Gain        HI       HMO         HI

                  -++-+-+        HI          5       LO       Gain       LO        HMO         LO

                  -++++--        HI          5       LO        Loss      LO        HMO         HI

                  ----+--        HI          1       HI       Gain       LO        HMO         HI

                  -+-+++-        HI          5       HI        Loss      LO        Team        HI



   Factors            Source     Source-Depth
                                                          Effects                         Aliases
   OE-Depth                 X           X
   Dose                     X           X                 OE-Depth*Dose                    = Testimonials*Source
   Testimonials             X                             OEDepth*Testimonials             = Dose*Source
   Framing                              X
                                                          OE-Depth*Source                  = Dose*Testimonials
   EE-Depth                             X
   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