Fractional Factorial Designs:A Tutorial

Vijay NairDepartments of Statistics and

Industrial & Operations Engineeringvnn@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 industriesPart of basic training programs such as Six-sigma

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, …

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 phasei.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

…

Full Factorial Design

Design Matrix

9 + 9 + 3 + 3 67 + 9 + 8 + 8

8

6 – 8 = -2

7

9

9

9

8

3

8

3

Fractional Factorial Designs

• Why?

• What?

• How?

• Properties

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

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 runsfrom a -run design?

Many possible “fractional” designs

Here’s one choice

Need a principled approach!

Here’s another …

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

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

Notion of “resolution” coming soon to theaters near you …

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)

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

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

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

X5 = X1*X2*X3; X6 = X2*X3*X4 X5*X6 = X1*X4 (yes, better)

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 …

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

X5 = X2*X3*X4; X6 = X1*X2*X3*X4; X5*X6 = X1

or I = 2345 = 12346 = 156 Resolution III design

¼ fraction of

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

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

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=3513=2514=5615=23=4616=4524=3626=34

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

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