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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

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 2 some or all alias pairs one of the effects is KNOWN. This usually means “thought to be zero”4-1 with t.c.’s

1, ab, ac, bc, ad, bd, cd, abcd

Can estimate: A+BCD

B+ACD

C+ABD

AB+CD

AC+BD

BC+AD

D+ABC

- 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, 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 experiments1

Design and Analysis of inevitable in a many factor situation. It is generally better to study Multi-Factored Experiments

Aliasing Structure

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Construction of a one half fraction
Construction of a One-half Fraction inevitable in a many factor situation. It is generally better to study

The basic design; the design generator

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Example: 2 inevitable in a many factor situation. It is generally better to study 5-1 : 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

B=ADE AC=4 BCD=ACE

C=ABCDE AD=BE BCE=ACD

D=ABE AE=BD

E=ABD BC=4

CD=4

CE=4

- lose 1

- other 30 in 15 alias pairs of 2

- run 16 t.c.’s

15 estimates

*AxABDE=BDE

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See if they are (collectively) acceptable. inevitable in a many factor situation. It is generally better to study

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

BD=3

BE=3

CD=3

CE=3

DE=3

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Next step: Find the 2 blocks (only inevitable in a many factor situation. It is generally better to study one of which will be run)

  • Assume we choose I=ABDE

    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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1 inevitable in a many factor situation. It is generally better to study

a

b

ab

c

ac

bc

abc

e

ae

be

abe

ce

ace

bce

abce

Next: Pick which block to run.

(say, block II)

Next: Go out and collect the data.

Next: Analyze it.

a.) find a proper Yates’ order.

i.) pick a letter and for a moment

call it “DEAD.” (assume we pick “d”)

ii.) use the remaining (“live”) letters

to form the STANDARD Yates’ order:

(see right column)

Now append the dead letter as needed to form

the block chosen to be run:

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There’s only one way to do it, i.e. either adding “d” or not adding “d”; one way will work, one won’t.

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Example 2: or not adding “d”; one way will work, one won’t.

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 or not adding “d”; one way will work, one won’t.

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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Let’s find the 4 blocks: I =ABC = BCDE = ADE or not adding “d”; one way will work, one won’t.

Assume we run the Principal block (block 1)

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Run the 8 t.c.’s and analyze: Since it’s a 2 or not adding “d”; one way will work, one won’t.5-2, we designate 2 letters as “DEAD” (say b, d), write a standard Yates’ order in the other (3) (live) letters, and append the dead letters to form the t.c.’s being run:

I = ABC = BCDE = ADE

t.c.

yield

(1)

(2)

(3)

Estimate

1

.

-

.

a (

bd)

-2+5-2

A

.

c (b)

-2+3-4

C

.

ac (d)

AC - +4-3

B

e (d)

.

-4+3-2

E

D

.

ae (b)

AE -3+4-

.

ce (

bd)

CE -3+2-3

.

ace

ACE -2+3-2

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Good to know rule: or not adding “d”; one way will work, one won’t.

t.c. with even # letters in common with even-lettered effect is a + for that effect; t. c. with odd # letters in common with odd-lettered effect is a + for that effect; otherwise a - (minus)

abd in ABCDE: 3 in common with 5

ODD with ODD +

abce in ABCDFG: 3 in common with 6

ODD with EVEN -

# letters in effect

Even

Odd

# letters t.c. has in

Common with Effect

Even

Odd

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Examples
Examples or not adding “d”; one way will work, one won’t.

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Example
Example or not adding “d”; one way will work, one won’t.

Interpretation of results often relies on making some assumptions

Ockham’srazor

Confirmation experiments can be important

See the projection of this design into 3 factors

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Projection of fractional factorials
Projection of Fractional Factorials or not adding “d”; one way will work, one won’t.

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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Possible strategies for follow up experimentation following a fractional factorial design
Possible Strategies for or not adding “d”; one way will work, one won’t.Follow-Up Experimentation Following a Fractional Factorial Design

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The one quarter fraction of the 2 k
The One-Quarter Fraction of the 2 or not adding “d”; one way will work, one won’t.k

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The one quarter fraction of the 2 6 2
The One-Quarter Fraction of the 2 or not adding “d”; one way will work, one won’t.6-2

Complete defining relation: I = ABCE = BCDF = ADEF

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Analysis of fractional factorials
Analysis of Fractional Factorials or not adding “d”; one way will work, one won’t.

  • 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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