Transcript Fractional Factorial Designs : A Tutorial

Fractional Factorial Designs:
A Tutorial
Vijay Nair
Departments of Statistics and
Industrial & Operations Engineering
[email protected]
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
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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
6
7+9+8+8
8
6 – 8 = -2
Fractional Factorial Designs
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•
•
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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
Wow!
Balanced design
All factors occur and low and high levels
same number of times; Same for interactions.
Columns are orthogonal. Projections …
Need a principled approach for
selecting
FFD’s properties
Good statistical
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
Resolution
Need a principled approach for selecting
FFD’sIII (later)
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:
So …
Length of the shortest “word”
in the generator set  resolution IV here
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
32 trx
combos
Factors
OEDEPTH
DOSE
TESTIMO
NIALS
FRAMING
EE-DEPTH
SOURCE
SOURCEDEPTH
+----+-
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
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
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