Design and Analysis of MultiFactored Experiments

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Design and Analysis ofMulti-Factored Experiments

• Fractional Factorial Designs

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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
  dont know much about the system)
• Almost always run as unreplicated factorials, but
  often with center points

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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

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The One-Half Fraction of the 2k

• Notation because the design has 2k/2 runs, its
  referred to as a 2k-1
• Consider a really simple case, the 23-1
• Note that I ABC

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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
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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
• 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

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• 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.cs.
• What can be estimated from these 4 t.c.s?

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• 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.

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• Had we run the other 4 t.c.s
• 1, ab, ac, bc,
• We would be able to estimate
• A - BC
• B - AC
• C - AB
• (generally no better or worse than with signs)
• NOTE If you know (i.e., are willing to
  assume) that all 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.

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• Suppose we run those 4
• 1, ab, c, abc
• We would then estimate
• A B
• C ABC
• AC BC
• 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

two main effects together usually less desirable
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• 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)

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• 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

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• Consider a 24-1 with t.c.s
• 1, ab, ac, bc, ad, bd, cd, abcd
• Can estimate ABCD
• BACD
• CABD
• ABCD
• ACBD
• BCAD
• DABC
• - 8 t.c.s
• -Lose 1 effect
• -Estimate other 14 in 7 alias pairs of 2

Note
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• Clean estimates of the remaining member of the
  pair can then be made.
• 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.

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• 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, its Multi-factored.
• The best way to learn how is to work (and
  discuss) some examples

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Design and Analysis ofMulti-Factored Experiments

• Aliasing Structure and constructing a FFD

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• 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
• ABDE ABDE ABCCDE
• BADE AC4 BCDACE
• CABCDE ADBE BCEACD
• DABE AEBD
• EABD BC4
• CD4
• CE4

- lose 1 - other 30 in 15 alias pairs of 2 - run
16 t.c.s 15 estimates
AxABDEBDE
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• See if they are (collectively) acceptable.
• Another option (among many others)
• I ABCDE
• A4 AB3
• B4 AC3
• C4 AD3
• D4 AE3
• E4 BC3
• BD3
• BE3
• CD3
• CE3
• DE3

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• Next step Find the 2 blocks (only one of which
  will be run)
• Assume we choose IABDE
• I II
• 1 c a ac
• ab abc b bc
• de cde ade acde
• abde abcde bde bcde
• ad acd d cd
• bd bcd abd abcd
• ae ace e ce
• be bce abe abce

Same process as a Confounding Scheme
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• Example 2

25-2 A, B, C, D, E Must lose 3 other 28
in 7 alias groups of 4
In a 25 , there are 31 effects with 8 t.c.,
there are 7 df 7 estimates available
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• 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
• D 4 3 AE
• E 4 3 AD
• BD 3 CE 3
• BE 3 CD 3
• Assume we use this design.

Find alias groups
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Lets find the 4 blocks I ABC BCDE ADE
a
b
a
Assume we run the Principal block (block 1)
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An easier way to construct a one-half fraction
The basic design the design generator
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Examples
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Example
Interpretation of results often relies on making
some assumptions Ockhams razor Confirmation
experiments can be important See the projection
of this design into 3 factors
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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
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The One-Quarter Fraction of the 2k
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The One-Quarter Fraction of the 26-2
Complete defining relation I ABCE BCDF ADEF
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Possible Strategies for Follow-Up
Experimentation Following a Fractional Factorial
Design
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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 clean estimates
  cannot be made.

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Design and Analysis of Multi-Factored Experiments

• Design Resolution and Minimal-Run Designs

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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.

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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 IABC is a resolution III
  design or 2III3-1.
• It is mainly used for screening. More on this
  design later.

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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 IABCD is a resolution IV
  design or 2IV4-1.
• It is also used mainly for screening.

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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 IABCDE 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.

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Guide to choice of fractional factorial designs
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Guide (continued)
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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.

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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 ).

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• 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.

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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.

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• 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.

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Factors for initial DOE on dancing objects
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• 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.

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Results from initial dancing-raisin experiment

• The half-normal plot of effects is shown.

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• 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.

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Alias Structure

• Each main effect is actually aliased with 15
  other effects. To simplify, we will not list 3
  factor interactions and above.
• A ABDCEFG
• B BADCFEG
• C CAEBFDG
• D DABCGEF
• E EACBGDF
• F FAGBCDE
• G GAFBECD
• 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.

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• 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.

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• 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.

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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!

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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.
• Lets see how the fold-over works on the dancing
  raisin experiments with all signs reversed on the
  control factors.

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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.

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Augmented Design
Factor B has disappeared and AD has taken its
place. What happened to family unity? Is it
really AD or something else, since AD is aliased
with CF and EG?
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• 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)?

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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.

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Interaction plots of CF and EG
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• 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.

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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.
• Dont rely on the default given by the software!!

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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!

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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.

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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
  DEFG, DFEG, and DGEF.
• The trick is to label the likely interactors
  anything but D, E, F, and G.

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• 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!