Fractional Factorial Designs: A Tutorial

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Fractional Factorial DesignsA Tutorial

• Vijay Nair
• Departments of Statistics and
• Industrial Operations Engineering
• vnn_at_umich.edu

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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

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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,

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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)

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(Full) Factorial Designs

• All possible combinations
• General I x J x K
• Two-level designs 2 x 2, 2 x 2 x 2, ?

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(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

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

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Full Factorial Design
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9.5
5.5
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Algebra -1 x -1 1
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Design Matrix
Full Factorial Design
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7 9 9 9 8 3 8 3
7 9 8 8
9 9 3 3
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6 8 -2
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Fractional Factorial Designs

• Why?
• What?
• How?
• Properties

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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)
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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
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How to select a subset of 4 runs from a
-run design? Many possible fractional
designs
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Heres one choice
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Heres another
Need a principled approach!
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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 FFDs
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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 FFDs
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Notion of resolution ? coming soon to theaters
near you
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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 FFDs
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Confounding or Aliasing ? NO FREE LUNCH!!!
X3X1X2 ? ??
aliased
X3 X1X2 ? X1X3 X2 and X2X3 X1 (main
effects aliased with two-factor interactions)
Resolution III design
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Want to study 5 factors (1,2,3,4,5) using a 24
16-run design i.e., construct half-fraction of a
25 design 25-1 design
For half-fractions, always best to alias the new
(additional) factor with the highest-order
interaction term
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What about bigger fractions? Studying 6 factors
with 16 runs? ¼ fraction of

X5 X2X3X4 X6 X1X2X3X4 ? X5X6 X1
(can we do better?)
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X5 X1X2X3 X6 X2X3X4 ? X5X6 X1X4
(yes, better)
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Design Generatorsand Resolution

• X5 X1X2X3 X6 X2X3X4 ? X5X6 X1X4
• 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

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Resolution

• Resolution III (12)
• Main effect aliased with 2-order interactions
• Resolution IV (13 or 22)
• Main effect aliased with 3-order interactions
  and
• 2-factor interactions aliased with other
  2-factor
• Resolution V (14 or 23)
• Main effect aliased with 4-order interactions
  and
• 2-factor interactions aliased with 3-factor
  interactions

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¼ fraction of
X5 X2X3X4 X6 X1X2X3X4 ? X5X6 X1
or I 2345 12346 156 ? Resolution III
design
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X5 X1X2X3 X6 X2X3X4 ? X5X6 X1X4
or I 1235 2346 1456 ? Resolution IV
design
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Aliasing Relationships

• I 1235 2346 1456
• Main-effects
• 12354562346 21353461456 31252461456
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• 15-possible 2-factor interactions
• 1235
• 1325
• 1456
• 152346
• 1645
• 2436
• 2634

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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 ?
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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.

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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 1235, 1325, 14 56 (for example) ?
• I 1235 2346 1456 ? Resolution IV design
• All possible 2-factor interactions
• 1235
• 1325
• 1456
• 152346
• 1645
• 2436
• 2634

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Latest design for Project 1
Project 1 2(7-2) design
32 trx combos
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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