1.
Fractional Factorial Designs:
A Tutorial
Vijay Nair
Departments of Statistics and
Industrial & Operations Engineering
vnn@umich.edu
2.
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
3.
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, …
4.
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)
5.
(Full) Factorial Designs
• All possible combinations
• General: I x J x K …
• Two-level designs: 2 x 2, 2 x 2 x 2, …
6.
(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
7.
Will focus on
two-level designs
OK in screening phase
i.e., identifying
important factors
8.
(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
15.
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 Factorials
No. of combinations
This is only for
two-levels
16.
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
17.
How to select a subset of 4 runs
from a -run design?
Many possible “fractional” designs
20.
Need a principled approach for selecting FFD’s
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 …
Good statistical properties
21.
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
22.
Notion of “resolution” coming soon to theaters near you …
23.
Need a principled approach for selecting FFD’s
Regular Fractional Factorial Designs
Half fraction of a design = design
3 factors studied -- 1-half fraction
8/2 = 4 runs
Resolution III (later)
24.
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
25.
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 design
i.e., construct half-fraction of a 2^5 design
= 2^{5-1} design
26.
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
28.
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 …
29.
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
30.
X5 = X2*X3*X4; X6 = X1*X2*X3*X4; X5*X6 = X1
or I = 2345 = 12346 = 156 Resolution III design
¼ fraction of
31.
X5 = X1*X2*X3; X6 = X2*X3*X4 X5*X6 = X1*X4
or I = 1235 = 2346 = 1456 Resolution IV design
33.
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
34.
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.
35.
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
36.
PATTERN OE-DEPTH DOSE TESTIMO
NIALS
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 trx
combos
37.
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
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