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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
• 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
• 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
                    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
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

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

     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
                                 bc  -  +   -  +      -    +  -
B - AC                           abc +  +   +  +      +    +  +

C - AB
(generally no better or worse than with + signs)
     A   B   AB   C   AC BC ABC
                       + + -
                         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

            - 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
    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

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-

    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
                                   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
    C=4        AD=3
    D=4        AE=3
    E=4        BC=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
     Confounding     ad      acd           d      cd
     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

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
             a      bd               ad     ab
              b     abc               c    bcd
             a      cd              abcd     ac
             de     ade             bde      e
             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
            The basic design; the design generator

L. M. Lye                   DOE Course               23

L. M. Lye    DOE Course   24
                          Interpretation of
                          results often relies on
                          making some
                          Ockham’s razor
                          experiments can be
                          See the projection of
                          this design into 3

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

                                Every fractional
                                factorial contains
                                full factorials in
                                fewer factors
                                The “flashlight”
                                A one-half fraction
                                will project into a
                                full factorial in any
                                k – 1 of the original
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
                          Strategies for
                           Following a
                         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
  “clean” estimates cannot be made.
L. M. Lye            DOE Course                 30
        Design and Analysis of
       Multi-Factored Experiments
            Design Resolution and Minimal-Run

L. M. Lye                DOE Course             31
    Design Resolution for Fractional Factorial
• 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
• 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
 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
  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                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

                  A: A

• 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

  shown.                                                         70.00

                                                                 60.00                                B




                                                                         0.0000   0.4937    0.9875    1.481   1.975


  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
• [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                  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
L. M. Lye            DOE Course                49
            A very scary thought
• Could a positive effect be cancelled by an “anti-
• 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
• 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
• It works on any Resolution III design. It is
  especially popular with Plackett-Burman
  designs, such as the 11 factors in 12-run
• 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


Is it really AD or
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
• 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
• 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

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
• 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

• 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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