fractional factorial designs: a tutorial vijay nair departments of statistics and industrial &...
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Fractional Factorial Designs:A Tutorial
Vijay NairDepartments of Statistics and
Industrial & Operations [email protected]
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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 industriesPart 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 phasei.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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Full Factorial Design
Design Matrix
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9 + 9 + 3 + 3 67 + 9 + 8 + 8
8
6 – 8 = -2
7
9
9
9
8
3
8
3
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Fractional Factorial Designs
• Why?
• What?
• How?
• Properties
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Treatment combinations
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)
Why Fractional Factorials?
Full FactorialsNo. of combinations
This is only for
two-levels
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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 runsfrom a -run design?
Many possible “fractional” designs
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Here’s one choice
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Need a principled approach!
Here’s another …
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Need a principled approach for selecting FFD’s
Regular Fractional Factorial Designs
Wow!
Balanced designAll factors occur and low and high levels
same number of times; Same for interactions.Columns are orthogonal. Projections …
Good statistical properties
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Need a principled approach for selecting FFD’s
What is the principled approach?
Notion of exploiting redundancy in interactions Set X3 column equal to
the X1X2 interaction column
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Notion of “resolution” coming soon to theaters near you …
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Need a principled approach for selecting FFD’s
Regular Fractional Factorial Designs
Half fraction of a design = design3 factors studied -- 1-half fraction
8/2 = 4 runs
Resolution III (later)
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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
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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
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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?¼ fraction of
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X5 = X1*X2*X3; X6 = X2*X3*X4 X5*X6 = X1*X4 (yes, better)
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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 …
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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
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X5 = X2*X3*X4; X6 = X1*X2*X3*X4; X5*X6 = X1
or I = 2345 = 12346 = 156 Resolution III design
¼ fraction of
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X5 = X1*X2*X3; X6 = X2*X3*X4 X5*X6 = X1*X4
or I = 1235 = 2346 = 1456 Resolution IV design
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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
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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 …
Properties of FFDs
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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 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
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PATTERN OE-DEPTH DOSE TESTIMONIALS
FRAMING EE-DEPTH SOURCE SOURCE-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
+--+++- LO 1 HI Loss LO Team HI
-++---- 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
OE-Depth X X
Dose X X
Testimonials X
Framing X
EE-Depth X
Effects Aliases
OE-Depth*Dose = Testimonials*Source
OEDepth*Testimonials = Dose*Source
OE-Depth*Source = Dose*Testimonials
Project 1: 2^(7-2) design
32 trxcombos
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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