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Design and Analysis of Multi-Factored Experiments. Fractional Factorial Designs. Design of Engineering Experiments – The 2 k-p Fractional Factorial Design.

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design and analysis of multi factored experiments

Design and Analysis ofMulti-Factored Experiments

Fractional Factorial Designs

DOE Course

design of engineering experiments the 2 k p fractional factorial design
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 factorscreening; 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

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why do fractional factorial designs work
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 2 k
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

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the one half fraction of the 2 3
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

DOE Course

the alternate fraction of the 2 3 1
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.c’s.

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


(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.

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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



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: A+BCD







- 8 t.c.’s

-Lose 1 effect

-Estimate other 14 in 7 alias pairs of 2


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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, it’s Multi-factored.

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

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design and analysis of multi factored experiments16

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.”


Step 2: Find the resulting alias pairs:








- lose 1

- other 30 in 15 alias pairs of 2

- run 16 t.c.’s

15 estimates


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See if they are (collectively) acceptable.

Another option (among many others):


A=4 AB=3

B=4 AC=3

C=4 AD=3

D=4 AE=3

E=4 BC=3






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Next step: Find the 2 blocks (only one of which will be run)
  • Assume we choose I=ABDE


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



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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

DOE Course

Choose the 3: Like in confounding schemes, 3rd

must be product of first 2:


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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Let’s find the 4 blocks: I =ABC = BCDE = ADE




Assume we run the Principal block (block 1)

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an easier way to construct a one half fraction
An easier way to construct a one-half fraction

The basic design; the design generator

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DOE Course


Interpretation of results often relies on making some assumptions


Confirmation experiments can be important

See the projection of this design into 3 factors

DOE Course

projection of fractional factorials
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

DOE Course

the one quarter fraction of the 2 6 2
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
Possible Strategies for Follow-Up Experimentation Following a Fractional Factorial Design

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analysis of fractional factorials
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.

DOE Course

design and analysis of multi factored experiments31

Design and Analysis of Multi-Factored Experiments

Design Resolution and Minimal-Run Designs

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design resolution for fractional factorial designs
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.

DOE Course

1 resolution iii designs
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.

DOE Course

2 resolution iv designs
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.

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3 resolution v designs
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.

DOE Course

guide continued38
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.

DOE Course

more on minimal run designs
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 $$$).

DOE Course

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.

DOE Course

case study dancing raisin experiment
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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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
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.

DOE Course

alias structure
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.

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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
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
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.

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complete fold over of raisin experiment
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
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
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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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
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!!

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single factor fold over
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
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
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.

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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!

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