# Fractional Factorial Designs A Tutorial

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```					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
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