Transcript Design and Analysis of Multi

Design and Analysis of
Multi-Factored Experiments
Design Resolution and Minimal-Run
Designs
L. M. Lye
DOE Course
1
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
2
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
3
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
4
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
5
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
6
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)
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)
L. M. Lye
DOE Course
1/512 (III) 1/1024 (III) 1/2048 (III)
1/128 (IV) 1/128 (IV)
7
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
8
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
9
• 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 twofactor interactions.
• The most popular method of design augmentation
is called the fold-over.
L. M. Lye
DOE Course
10
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
11
• 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
instead of raisin, etc.
• These and other factors became the subject
of a two-level factorial design.
• See table on next page.
L. M. Lye
DOE Course
12
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
13
• 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
14
Results from initial dancing-raisin experiment
DESIGN-EXPERT Plot
Rating
99.00
97.00
95.00
Half Norm al % probability
• The halfnormal plot
of effects is
shown.
A: A
B: B
C: C
D: D
E: E
F: F
G: G
Half Normal plot
E
90.00
85.00
80.00
G
70.00
B
60.00
40.00
20.00
0.0000
0.0000
0.4937
0.9875
1.481
1.975
|Effect|
L. M. Lye
DOE Course
15
• 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
16
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
• [B] = B+AD+CF+EG
• [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
17
• 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
18
• 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
19
A very scary thought
• Could a positive effect be cancelled by an “antieffect”?
• 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
20
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
21
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
22
Augmented Design
What happened to
family unity?
DESIGN-EXPERT Plot
Response 1
Interaction Graph
D: D
5.000
X = A: A
Y = D: D
D- -1.000
D+ 1.000
Actual Factors
B: B = 0.0000
C: C = 0.0000
E: E = 0.0000
F: F = 0.0000
G: G = 0.0000
3.875
Res pons e 1
Factor B has
disappeared and AD
has taken its place.
2.750
1.625
Is it really AD or
something else, since
AD is aliased with CF
and EG?
L. M. Lye
0.5000
-1.000
-0.5000
0.0000
0.5000
1.000
A: A
DOE Course
23
• 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
24
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
25
Interaction plots of CF and EG
DESIGN-EXPERT Plot
Response 1
DESIGN-EXPERT Plot
Interaction Graph
F: F
5.000
Response 1
X = E: E
Y = G: G
G- -1.000
G+ 1.000
Actual Factors
A: A = 0.0000
B: B = 0.0000
C: C = 0.0000
D: D = 0.0000
F: F = 0.0000
3.875
Res pons e 1
F- -1.000
F+ 1.000
Actual Factors
A: A = 0.0000
B: B = 0.0000
D: D = 0.0000
E: E = 0.0000
G: G = 0.0000
2.750
3.875
2.750
1.625
1.625
0.5000
0.5000
-1.000
-0.5000
0.0000
0.5000
1.000
C: C
L. M. Lye
G: G
5.000
Res pons e 1
X = C: C
Y = F: F
Interaction Graph
-1.000
-0.5000
0.0000
0.5000
1.000
E: E
DOE Course
26
• 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 Gline 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
27
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
28
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
29
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 sire that you
check the aliases and see whether it is worth
doing.
L. M. Lye
DOE Course
30
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
31
• 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
32