# Fractional Factorial Designs (PowerPoint)

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```					        Design and Analysis of
Multi-Factored Experiments

Fractional Factorial Designs

L. M. Lye              DOE Course          1
Design of Engineering Experiments
– The 2k-p Fractional Factorial Design
• Motivation for fractional factorials is obvious; as
the number of factors becomes large enough to be
“interesting”, the size of the designs grows very
quickly
• Emphasis is on factor screening; efficiently
identify the factors with large effects
• There may be many variables (often because we
don’t know much about the system)
• Almost always run as unreplicated factorials, but
often with center points

L. M. Lye              DOE Course                       2
Why do Fractional Factorial
Designs Work?
• The sparsity of effects principle
– There may be lots of factors, but few are important
– System is dominated by main effects, low-order
interactions
• The projection property
– Every fractional factorial contains full factorials in
fewer factors
• Sequential experimentation
– Can add runs to a fractional factorial to resolve
difficulties (or ambiguities) in interpretation
L. M. Lye                    DOE Course                         3
The One-Half Fraction of the 2k
• Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1
• Consider a really simple case, the 23-1
• Note that I =ABC

L. M. Lye                        DOE Course                                4
The One-Half
Fraction of the 23
For the principal fraction,
notice that the contrast for
estimating the main effect A
is exactly the same as the
contrast used for estimating
the BC interaction.
This phenomena is called
aliasing and it occurs in all
fractional designs
Aliases can be found
directly from the columns in
the table of + and - signs

L. M. Lye   DOE Course                                    5
The Alternate Fraction of the 23-1

• I = -ABC is the defining relation
• Implies slightly different aliases: A = -BC,
B= -AC, and C = -AB
• Both designs belong to the same family, defined
by
I   ABC
• Suppose that after running the principal fraction,
the alternate fraction was also run
• The two groups of runs can be combined to form a
full factorial – an example of sequential
experimentation
L. M. Lye                 DOE Course                6
Example: Run 4 of the 8 t.c.’s in 23: a, b, c, abc

It is clear that from the(se) 4 t.c.’s, we cannot
estimate the 7 effects (A, B, AB, C, AC, BC,
ABC) present in any 23 design, since each estimate
uses (all) 8 t.c’s.

What can be estimated from these 4 t.c.’s?

L. M. Lye                 DOE Course                     7
4A = -1 + a - b + ab - c + ac - bc + abc
4BC = 1 + a - b - ab -c - ac + bc + abc

Consider
(4A + 4BC)= 2(a - b - c + abc)
or
2(A + BC)= a - b - c + abc

Overall:
2(A + BC)= a - b - c + abc
2(B + AC)= -a + b - c + abc
2(C + AB)= -a - b + c + abc
In each case, the 4 t.c.’s NOT run cancel out.

L. M. Lye                  DOE Course            8
Had we run the other 4 t.c.’s:
A  B  AB  C     AC   BC ABC
1, ab, ac, bc,                    1  -  -   +  -      +    +  -
a  +  -   -  -      -    +  +
We would be able to estimate      b  -  +   -  -      +    -  +
ab  +  +   +  -      -    -  -
A - BC                            c
ac
-
+
-
-
+
-
+
+
-
+
-
-
+
-
bc  -  +   -  +      -    +  -
B - AC                           abc +  +   +  +      +    +  +

C - AB
(generally no better or worse than with + signs)
A   B   AB   C   AC BC ABC
1
ab
-
+
-
+
+
+
-
-
+ + -
-
NOTE: If you “know” (i.e., are
-  -
ac   +   -    -   +      willing to assume) that all
+ -   -
bc   -   +    -   +    - + -
interactions = 0, then you can say
either (1) you get 3 factors for “the price” of 2.
(2) you get 3 factors at “1/2 price.”
L. M. Lye                         DOE Course                   9
Suppose we run those 4:
1, ab, c, abc;
We would then estimate
A+B               two main effects
C + ABC           together usually
AC + BC           less desirable

In each case, we “Lose” 1 effect completely, and get the
other 6 in 3 pairs of two effects.
Members of the pair are CONFOUNDED
Members of the pair are ALIASED

L. M. Lye                  DOE Course                      10
With 4 t.c.’s, one should expect to get only
3 “estimates” (or “alias pairs”) - NOT
unrelated to “degrees of freedom being one
fewer than # of data points” or “with c
columns, we get (c - 1) df.”

In any event, clearly, there are BETTER
and WORSE sets of 4 t.c.’s out of a 23.
(Better & worse 23-1 designs)

L. M. Lye             DOE Course                   11
Prospect in fractional factorial designs is
attractive if in some or all alias pairs one
of the effects is KNOWN. This usually
means “thought to be zero”

L. M. Lye              DOE Course                12
Consider a 24-1 with t.c.’s
1, ab, ac, bc, ad, bd, cd, abcd

Can estimate: A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
D+ABC

- 8 t.c.’s
Note:       -Lose 1 effect
-Estimate other 14 in 7 alias pairs of 2
L. M. Lye                     DOE Course               13
“Clean” estimates of the remaining member of the pair can

For those who believe, by conviction or via selected
empirical evidence, that the world is relatively simple, 3
and higher order interactions (such as ABC, ABCD, etc.)
may be announced as zero in advance of the inquiry. In
this case, in the 24-1 above, all main effects are CLEAN.
Without any such belief, fractional factorials are of
uncertain value. After all, you could get A + BCD = 0, yet
A could be large +, BCD large -; or the reverse; or both
zero.

L. M. Lye                   DOE Course                       14
Despite these reservations fractional factorials are
almost inevitable in a many factor situation. It is
generally better to study 5 factors with a quarter
replicate (25-2 = 8) than 3 factors completely (23 =
8). Whatever else the real world is, it’s Multi-
factored.

The best way to learn “how” is to work (and
discuss) some examples:

L. M. Lye                DOE Course                    15
Design and Analysis of
Multi-Factored Experiments

Aliasing Structure and constructing a FFD

L. M. Lye                    DOE Course                 16
Example:           25-1 : A, B, C, D, E
Step 1: In a 2k-p, we “lose” 2p-1.
Here we lose 1. Choose the effect to lose. Write it as a
“Defining relation” or “Defining contrast.”
I = ABDE
Step 2: Find the resulting alias pairs:
*A=BDE           AB=DE     ABC=CDE
- lose 1          B=ADE           AC=4      BCD=ACE
- other 30 in 15 C=ABCDE          AD=BE     BCE=ACD
alias pairs of 2
D=ABE           AE=BD
- run 16 t.c.’s
E=ABD           BC=4
15 estimates
CD=4
CE=4 *AxABDE=BDE
L. M. Lye                  DOE Course                          17
See if they are (collectively) acceptable.
Another option (among many others):

I = ABCDE

A=4        AB=3
B=4        AC=3
D=4        AE=3
E=4        BC=3
BD=3
BE=3
CD=3
CE=3
DE=3
L. M. Lye                  DOE Course        18
Next step: Find the 2 blocks (only one of which will be run)
• Assume we choose I=ABDE

I                   II
1       c             a      ac
ab      abc           b      bc
Same process    de      cde           ade    acde
as a            abde    abcde         bde    bcde
Scheme          bd      bcd           abd    abcd
ae      ace           e      ce
be      bce           abe    abce

L. M. Lye                DOE Course                       19
Example 2:

In a 25 , there
25-2 A, B, C, D, E
are 31 effects;
Must “lose” 3; other 28
with 8 t.c.,
in 7 alias groups of 4
there are 7 df &
7 estimates
available

L. M. Lye                      DOE Course                        20
Choose the 3: Like in confounding schemes, 3rd
must be product of first 2:
I = ABC = BCDE = ADE
A = BC          = 5 = DE
B = AC          = 3 = 4
C = AB          = 3 = 4
Find alias D = 4           = 3 = AE
groups: E = 4              = 3 = AD
BD = 3          = CE = 3
BE = 3          = CD = 3

Assume we use this design.
L. M. Lye                  DOE Course            21
Let’s find the 4 blocks: I =ABC = BCDE = ADE
1      2                3      4
1      a                b      d
Abd
Bc
b     abc               c    bcd
Acd
a      cd              abcd     ac
abe    be                ae   abde
bcde   abcde             cde    bce
ace     ce             abce   acde

Assume we run the Principal block (block 1)

L. M. Lye              DOE Course                  22
An easier way to construct a one-half
fraction
The basic design; the design generator

L. M. Lye                   DOE Course               23
Examples

L. M. Lye    DOE Course   24
Example
Interpretation of
results often relies on
making some
assumptions
Ockham’s razor
Confirmation
experiments can be
important
See the projection of
factors

L. M. Lye    DOE Course                       25
Projection of Fractional Factorials

Every fractional
factorial contains
full factorials in
fewer factors
The “flashlight”
analogy
A one-half fraction
will project into a
full factorial in any
k – 1 of the original
factors
L. M. Lye          DOE Course                        26
The One-Quarter Fraction of the 2k

L. M. Lye         DOE Course             27
The One-Quarter Fraction of the 26-2
Complete defining relation: I = ABCE = BCDF = ADEF

L. M. Lye               DOE Course                      28
Possible
Strategies for
Follow-Up
Experimentation
Following a
Fractional
Factorial Design

L. M. Lye   DOE Course               29
Analysis of Fractional Factorials
• Easily done by computer
• Same method as full factorial except that
effects are aliased
• All other steps same as full factorial e.g.
ANOVA, normal plots, etc.
• Important not to use highly fractionated
designs - waste of resources because
L. M. Lye            DOE Course                 30
Design and Analysis of
Multi-Factored Experiments
Design Resolution and Minimal-Run
Designs

L. M. Lye                DOE Course             31
Design Resolution for Fractional Factorial
Designs
• The concept of design resolution is a useful way to
catalog fractional factorial designs according to
the alias patterns they produce.
• Designs of resolution III, IV, and V are
particularly important.
• The definitions of these terms and an example of
each follow.

L. M. Lye              DOE Course                  32
1. Resolution III designs

• These designs have no main effect aliased with
any other main effects, but main effects are aliased
with 2-factor interactions and some two-factor
interactions may be aliased with each other.
• The 23-1 design with I=ABC is a resolution III
design or 2III3-1.
• It is mainly used for screening. More on this
design later.

L. M. Lye              DOE Course                   33
2. Resolution IV designs

• These designs have no main effect aliased with
any other main effect or two-factor interactions,
but two-factor interactions are aliased with each
other.
• The 24-1 design with I=ABCD is a resolution IV
design or 2IV4-1.
• It is also used mainly for screening.

L. M. Lye              DOE Course                     34
3. Resolution V designs

• These designs have no main effect or two factor
interaction aliased with any other main effect or
two-factor interaction, but two-factor interactions
are aliased with three-factor interactions.
• A 25-1 design with I=ABCDE is a resolution V
design or 2V5-1.
• Resolution V or higher designs are commonly
used in response surface methodology to limit the
number of runs.

L. M. Lye              DOE Course                   35
Guide to choice of fractional factorial designs

Factors        2         3           4            5         6            7            8
4 runs       Full    1/2 (III)      -            -          -           -            -

8         2 rep      Full      1/2 (IV)   1/4 (III)   1/8 (III)   1/16 (III)       -

16         4 rep     2 rep        Full     1/2 (V)     1/4 (IV)    1/8 (IV)     1/16 (IV)

32         8 rep     4 rep       2 rep       Full      1/2 (VI)    1/4 (IV)     1/8 (IV)

64         16 rep    8 rep       4 rep      2 rep        Full      1/2 (VII)     1/4 (V)

128        32 rep   16 rep       8 rep      4 rep       2 rep        Full       1/2 (VIII)

L. M. Lye                           DOE Course                                     36
Guide (continued)

Factors        9           10           11            12            13           14            15
4 runs       -            -             -             -            -             -            -

8          -            -             -             -            -             -            -

16     1/32 (III)   1/64 (III)   1/128 (III)   1/256 (III)   1/512 (III) 1/1024 (III) 1/2048 (III)

32     1/16 (IV)    1/32 (IV)    1/64 (IV)     1/128 (IV) 1/256 (IV) 1/512 (IV) 1/1024 (IV)

64     1/8 (IV)     1/16 (IV)    1/32 (IV)     1/64 (IV)     1/128 (IV) 1/256 (IV) 1/512 (IV)

128     1/4 (VI)      1/8 (V)      1/16 (V)     1/128 (IV)    1/64 (IV)    1/128 (IV) 1/128 (IV)

L. M. Lye                                 DOE Course                                                   37
Guide (continued)
• Resolution V and higher  safe to use (main and
two-factor interactions OK)
• Resolution IV  think carefully before
proceeding (main OK, two factor interactions are
aliased with other two factor interactions)
• Resolution III  Stop and reconsider (main
effects aliased with two-factor interactions).
• See design generators for selected designs in the
attached table.

L. M. Lye             DOE Course                      38
More on Minimal-Run Designs
• In this section, we explore minimal designs with
one few factor than the number of runs; for
example, 7 factors in 8 runs.
• These are called “saturated” designs.
• These Resolution III designs confound main
effects with two-factor interactions – a major
weakness (unless there is no interaction).
• However, they may be the best you can do when
confronted with a lack of time or other resources
(like \$\$\$).

L. M. Lye              DOE Course                     39
• If nothing is significant, the effects and
interactions may have cancelled itself out.
• However, if the results exhibit significance, you
must take a big leap of faith to assume that the
reported effects are correct.
• To be safe, you need to do further experimentation
– known as “design augmentation” - to de-alias
(break the bond) the main effects and/or two-
factor interactions.
• The most popular method of design augmentation
is called the fold-over.

L. M. Lye             DOE Course                   40
Case Study: Dancing Raisin Experiment
• The dancing raisin experiment provides a vivid
demo of the power of interactions. It normally
involves just 2 factors:
– Liquid: tap water versus carbonated
– Solid: a peanut versus a raisin
• Only one out of the four possible combinations
produces an effect. Peanuts will generally float,
and raisins usually sink in water.
• Peanuts are even more likely to float in carbonated
liquid. However, when you drop in a raisin, they
drop to the bottom, become coated with bubbles,
which lift the raisin back to the surface. The
bubbles pop and the up-and-down process
continues.
L. M. Lye                  DOE Course              41
• BIG PROBLEM – no guarantee of success
• A number of factors have been suggested as
causes for failure, e.g., the freshness of the
raisins, brand of carbonated water, popcorn
• These and other factors became the subject
of a two-level factorial design.
• See table on next page.

L. M. Lye            DOE Course                42
Factors for initial DOE on dancing objects
Factor Name                              Low Level (-) High Level (+)
A               Material of container    Plastic       Glass
B               Size of container        Small         Large
C               Liquid                   Club Soda     Lemon Lime
D               Temperature              Room          Ice Cold
E               Cap on container         No            Yes
F               Type of object           Popcorn       Raisin
G               Age of object            Fresh         Stale
L. M. Lye                           DOE Course                43
• The full factorial for seven factors would
require 128 runs. To save time, we run only
1/16 of 128 or a 27-4 fractional factorial
design which requires only 8 runs.
• This is a minimal design with Resolution
III. At each set of conditions, the dancing
performance was rated on a scale of 1 to 10.
• The results from this experiment is shown
in the handout.

L. M. Lye           DOE Course               44
Results from initial dancing-raisin experiment
DESIGN-EXPERT Plot                                              Half Normal plot
Rating

A: A
99.00

• The half-
B: B
C: C
D: D                                           97.00
E: E
normal plot     F: F                                           95.00

Half Norm al % probability
G: G                                                                                           E
of effects is                                                  90.00

85.00
80.00
shown.                                                         70.00
G

60.00                                B

40.00

20.00

0.0000

0.0000   0.4937    0.9875    1.481   1.975

|Effect|

L. M. Lye                    DOE Course                                                                         45
• Three effects stood out: cap (E), age of object (G),
and size of container (B).
• The ANOVA on the resulting model revealed
highly significant statistics.
• Factors G+ (stale) and E+ (capped liquid) have a
negative impact, which sort of make sense.
However, the effect of size (B) does not make
much sense.
• Could this be an alias for the real culprit (effect),
perhaps an interaction?
• Take a look at the alias structure in the handout.

L. M. Lye               DOE Course                    46
Alias Structure
• Each main effect is actually aliased with 15 other
effects. To simplify, we will not list 3 factor
interactions and above.
• [A] = A+BD+CE+FG
• [C] = C+AE+BF+DG
• [D] = D+AB+CG+EF
• [E] = E+AC+BG+DF
• [F] = F+AG+BC+DE
• [G] = G+AF+BE+CD
• Can you pick out the likely suspect from the lineup for
B? The possibilities are overwhelming, but they can be
narrowed by assuming that the effects form a family.
L. M. Lye              DOE Course                  47
• The obvious alternative to B (size) is the
interaction EG. However, this is only one of
several alternative “hierarchical” models
that maintain family unity.
• E, G and EG (disguised as B)
• B, E, and BE (disguised as G)
• B, G, and BG (disguised as E)
• The three interaction graphs are shown in
the handout.

L. M. Lye           DOE Course               48
• Notice that all three interactions predict the
same maximum outcome. However, the
actual cause remains murky. The EG
interaction remains far more plausible than
the alternatives.
• Further experimentation is needed to clear
things up.
• A way of doing this is by adding a second
block of runs with signs reversed on all
factors – a complete fold-over. More on this
later.
L. M. Lye            DOE Course                49
A very scary thought
• Could a positive effect be cancelled by an “anti-
effect”?
• If you a Resolution III design, be prepared for the
possibility that a positive main effect may be
wiped out by an aliased interaction of the same
magnitude, but negative.
• The opposite could happen as well, or some
combination of the above. Therefore, if nothing
comes out significant from a Resolution III design,
you cannot be certain that there are no active
effects.
• Two or more big effects may have cancelled each
other out!
L. M. Lye              DOE Course                  50
Complete Fold-Over of Resolution III Design
• You can break the aliases between main
effects and two-factor interactions by using
a complete fold-over of the Resolution III
design.
• It works on any Resolution III design. It is
especially popular with Plackett-Burman
designs, such as the 11 factors in 12-run
experiment.
• Let’s see how the fold-over works on the
dancing raisin experiments with all signs
reversed on the control factors.
L. M. Lye           DOE Course                   51
Complete Fold-Over of Raisin Experiment
• See handout for the augmented design. The second
block of experiments has all signs reversed on the
factors A to F.
• Notice that the signs of the two-factor interactions
do not change from block 1 to block 2.
• For example, in block 1 the signs of column B and
EG are identical, but in block 2 they differ; thus
the combined design no longer aliases B with EG.
• If B is really the active effect, it should come out
on the plot of effects for the combined design.

L. M. Lye              DOE Course                   52
Augmented Design
Factor B has            DESIGN-EXPERT Plot                                Interaction Graph
disappeared and AD      Response 1                       5.000
D: D

has taken its place.    X = A: A
Y = D: D

D- -1.000                     3.875
D+ 1.000
Actual Factors
B: B = 0.0000
What happened to

Res pons e 1
C: C = 0.0000
E: E = 0.0000                    2.750
family unity?           F: F = 0.0000
G: G = 0.0000

1.625

something else, since                                   0.5000

AD is aliased with CF                                            -1.000   -0.5000   0.0000   0.5000   1.000

and EG?                                                                             A: A

L. M. Lye               DOE Course                                                                 53
• The problem is that a complete fold-over of
a Resolution III design does not break the
aliasing of the two-factor interactions.
• The listing of the effect AD – the
interaction of the container material with
beverage temperature – is done arbitrarily
by alphabetical order.
• The AD interaction makes no sense
physically. Why should the material (A)
depend on the temperature of beverage (B)?
L. M. Lye          DOE Course               54
Other possibilities
• It is not easy to discount the CF interaction:
liquid type (C) versus object type (F). A
chemical reaction is possible.
• However, the most plausible interaction is
between E and G, particularly since we now
know that these two factors are present as
main effects.
• See interaction plots of CF and EG.

L. M. Lye            DOE Course                55
Interaction plots of CF and EG
DESIGN-EXPERT Plot                                  Interaction Graph                   DESIGN-EXPERT Plot                                Interaction Graph
Response 1                                                    F: F                      Response 1                                                  G: G
5.000                                                                                 5.000

X = C: C                                                                                X = E: E
Y = F: F                                                                                Y = G: G

F- -1.000                       3.875                                                   G- -1.000                     3.875
F+ 1.000                                                                                G+ 1.000
Actual Factors                                                                          Actual Factors
A: A = 0.0000                                                                           A: A = 0.0000
Res pons e 1

Res pons e 1
B: B = 0.0000                                                                           B: B = 0.0000
D: D = 0.0000                      2.750                                                C: C = 0.0000                    2.750
E: E = 0.0000                                                                           D: D = 0.0000
G: G = 0.0000                                                                           F: F = 0.0000

1.625                                                                                 1.625

0.5000                                                                                0.5000

-1.000   -0.5000   0.0000   0.5000   1.000                                            -1.000   -0.5000   0.0000   0.5000    1.000

C: C                                                                                  E: E

L. M. Lye                                                                DOE Course                                                                            56
• It appears that the effect of cap (E) depends
on the age of the object (G).
• When the object is stale (G+ line), twisting
on the bottle cap (going from E- at left to
E+ at right) makes little difference.
• However, when the object is fresh (the G-
line at the top), the bottle cap quenches the
dancing reaction. More experiments are
required to confirm this interaction.
• One obvious way is to do a full factorial on
E and G alone.
L. M. Lye           DOE Course                57
An alias by any other name is not necessarily the same

• You might be surprised that aliased interactions
such as AD and EG do not look alike.
• Their coefficients are identical, but the plots differ
because they combine the interaction with their
parent terms.
• So you have to look through each aliased
interaction term and see which one makes physical
sense.
• Don’t rely on the default given by the software!!

L. M. Lye               DOE Course                    58
Single Factor Fold-Over
• Another way to de-alias a Resolution III design is
the “single-factor fold-over”.
• Like a complete fold-over, you must do a second
block of runs, but this variation of the general
method, you change signs only on one factor.
• This factor and all its two-factor interactions
become clear of any other main effects or
interactions.
• However, the combined design remains a
Resolution III, because with the exception of the
factor chosen for de-aliasing, all others remained
aliased with two-factor interactions!

L. M. Lye              DOE Course                      59
Extra Note on Fold-Over
• The complete fold-over of Resolution IV designs
may do nothing more than replicate the design so
that it remains Resolution IV.
• This would happen if you folded the 16 runs after
a complete fold-over of Resolution III done earlier
in the raisin experiment.
• By folding only certain columns of a Resolution
IV design, you might succeed in de-aliasing some
of the two-factor interactions.
• So before doing fold-overs, make sure that you
check the aliases and see whether it is worth
doing.

L. M. Lye              DOE Course                   60
Bottom Line
• The best solution remains to run a higher
resolution design by selecting fewer factors and/or
bigger design.
• For example, you could run seven factors in 32
runs (a quarter factorial). It is Resolution IV, but
all 7 main effects and 15 of the 21 two-factor
interactions are clear of other two-factor
interactions.
• The remaining 6 two-factor interactions are:
DE+FG, DF+EG, and DG+EF.
• The trick is to label the likely interactors anything
but D, E, F, and G.

L. M. Lye               DOE Course                    61
• For example, knowing now that capping
and age interact in the dancing raisin
experiment, we would not label these
factors E and G.
• If only we knew then what we know
now!!!!

• So it is best to use a Resolution V design,
and none of the problems discussed above
would occur!

L. M. Lye           DOE Course                  62

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