Design and Analysis

of Experiments

Prof. Dr. Anselmo E de Oliveira

anselmo.quimica.ufg.br

anselmo.disciplinas@gmail.com

Part VII: Fractional Factorial Designs

• 2k: increasing k the number of runs required for a complete replicate of the design outgrows the resources of most experimenters

• Design redundancy and the number of effects – Binomial coefficient (combinations without

repetition)

𝑛𝑘

=𝑛!

𝑘! 𝑛 − 𝑘 !

where 𝑛 is the number of things to choose from, and we choose 𝑘 of them (no repetition, order doesn't matter)

27 full factorial design:

mean = 1

main effects (n = 7, k = 1)

two-factor interactions (n = 7, k = 2)

three-factor interactions (n = 7, k = 3)

n = 7, k = 4

n = 7, k = 5

n = 7, k = 6

n = 7, k = 7

128 effects

71

= 7

72

= 21

73

= 35

74

= 35

.

.

. 77

= 1 There are only 28 (7+21) degrees of freedom associated with effects that are likely to be of major interest. The remaining 99 are associated with three-factor and higher interactions.

• If the experimenter can reasonably assume that certain high-order interactions are negligible: fractional factorial design

• Screening experiments

• The successful use of fractional factorial designs is based on three key ideas: – The sparsity of effects principle: when there are

several variables, the system or process is likely to be driven primarily by some of the main effects and low-order interactions

– The projection property: fractional factorial designs can be projected into larger designs in the subset of significant factors

– Sequential experimentation: it is possible to combine the runs of two (or more) fractional factorials to construct sequentially a larger design to estimate the factor effects and interactions of interest

• 𝑘 = 3

• Two levels

• Four runs

• One-half fraction of a 23 design

23−1 = 4 treatment combinations

𝟐𝟑−𝟏 design

run

1

2

3

4

5

6

7

8

A

-

+

-

+

-

+

-

+

B

-

-

+

+

-

-

+

+

C

-

-

-

-

+

+

+

+

ABC

-

+

+

-

+

-

-

+

I

+

+

+

+

+

+

+

+

run ABC

1 -

4 -

6 -

7 -

run ABC

2 +

3 +

5 +

8 +

generator

• 𝐴𝐵𝐶 = generator

• A 23−1 design is formed by selecting

only those treatment combinations

that have a plus in the 𝐴𝐵𝐶 column

• 𝐼 is also always plus

• 𝐼 = 𝐴𝐵𝐶 is the defining relation

main effects

run

2 a

3 b

5 c

8 abc

I

+

+

+

+

A

+

-

-

+

B

-

+

-

+

C

-

-

+

+

AB

-

-

+

+

AC

-

+

-

+

BC

+

-

-

+

ABC

+

+

+

+

interaction effects

It is impossible to differentiate between

A and BC

B and AC

C and AB

A = 𝓁A + 𝓁BC

B = 𝓁B + 𝓁AC

C = 𝓁C + 𝓁AB

𝓁A A + BC

𝓁B B + AC

𝓁C C + AB aliases

𝐼 = 𝐴𝐵𝐶

𝐴 ∙ 𝐼 = 𝐴 ∙ 𝐴𝐵𝐶 = 𝐴2𝐵𝐶

como 𝐴2 = 𝐼

𝐴 = 𝐵𝐶

Similarly,

𝐵 ∙ 𝐼 = 𝐵 ∙ 𝐴𝐵𝐶 = 𝐴𝐵2𝐶

𝐵 = 𝐴𝐶

and

𝐶 ∙ 𝐼 = 𝐶 ∙ 𝐴𝐵𝐶 = 𝐴𝐵𝐶2

𝐶 = 𝐴𝐵

The one-half

fraction with

𝐼 = +𝐴𝐵𝐶 is the

principal fraction

• Using the other half-fraction: 𝐼 = −𝐴𝐵𝐶

run

1 (1)

4 ab

6 ac

7 bc

I

+

+

+

+

A

-

+

+

-

B

-

+

-

+

C

-

-

+

+

AB

+

+

-

-

AC

+

-

+

-

BC

+

-

-

+

ABC

-

-

-

-

𝓁A 𝐴 − 𝐵𝐶

𝓁B 𝐵 − 𝐴𝐶

𝓁C 𝐶 − 𝐴𝐵

Thus, when we estimate A, B, and C

with this particular fraction, we are

really estimating 𝐴 − 𝐵𝐶, 𝐵 − 𝐴𝐶, and

𝐶 − 𝐴𝐵

In practice, it does not matter which

fraction is actually used. Both

factions belong to the same family

Construction of the Half-Fraction

1. Write down a basic design consisting of the runs for a full

2k-1 design

22

run A B

1 - -

2 + -

3 - +

4 + +

23-1 ; 𝑰 = +𝑨𝑩𝑪

A B 𝑪 = 𝑨𝑩

- - +

+ - -

- + -

+ + +

23-1 ; 𝑰 = −𝑨𝑩𝑪

A B 𝑪 = −𝑨𝑩

- - -

+ - +

- + +

+ + -

2. Add the kth factor by identifying its

plus and minus levels with the plus

and minus signs of the highest order

interaction 𝐴𝐵𝐶 … 𝐾 − 1

Design Resolution

• In general, the resolution of a design is one more than the smallest order interaction that some main effect is confounded (aliased) with.

– If some main effects are confounded with some 2-level interactions, the resolution is 3. • The one-half fraction of the 23 design with the defining

relation 𝐼 = 𝐴𝐵𝐶 (or 𝐼 = −𝐴𝐵𝐶) is a 2𝐼𝐼𝐼3−1 design

– For most practical purposes, a resolution 5 design is excellent and a resolution 4 design may be adequate.

– Resolution 3 designs are useful as economical screening designs.

http://www.itl.nist.gov/div898/handbook/pri/section7/pri7.htm

Projection of Fractions into Factorials

• Any fractional design of

resolution R contains

complete factorial designs in

any subset of 𝑅 − 1 factors

• If an experimenter has

several factors of potential

interest but believes that only

𝑅 − 1 of them have important

efects, then a fractional

factorial design of resolution

𝑅 is the appropriate choice of

design

Example: Pilot Plant Filtration Rate Experiment

A chemical product is produced in a pressure

vessel. A factorial experiment is carried out in

the pilot plant to study the factors thought to

influence the filtration rate of this product.

A = temperature

B = pressure

C = concentration of formaldehyde

D = stirring rate

response: filtration rate in gal/h

24 full factorial design (16 runs)

run 𝒚

(1) 45

a 71

b 48

ab 65

c 68

ac 60

bc 80

abc 65

d 43

ad 100

bd 45

abd 104

cd 75

acd 86

bcd 70

abcd 96

A = 21.625

C = 9.875

D = 14.625

AC = -18.125

AD = 16.625

run A B C

1 - - -

2 + - -

3 - + -

4 + + -

5 - - +

6 + - +

7 - + +

8 + + +

𝑫 = 𝑨𝑩𝑪 y

- 45 (1)

+ 100 ad

+ 45 bd

- 65 ab

+ 75 cd

- 60 ac

- 80 bc

+ 96 abcd

Main effects:

𝐴. 𝐼 = 𝐴. 𝐴𝐵𝐶𝐷

A = A2BCD

A = BCD

B.I = B.ABCD

B = AB2CD

B = ACD

C.I = C.ABCD

C = ABC2D

C = ABD

D.I = D.ABCD

D = ABCD2

D = ABC

24-1 design with 𝐼 = 𝐴𝐵𝐶𝐷, 2IV4−1

• Each main effect is aliased with a three-factor interaction

two-factor interactions:

AB.I = AB.ABCD

AB = A2B2CD

AB = CD

AC.I = AC.ABCD

AC = A2BC2D

AC = BD

AD.I = AD.ABCD

AD = A2BCD2

AD = BC

23 design = 7 effects

o 3 main

o 3 second-order

o 1 third-order

24-1 design = 7 effects

o 4 main

o 3 second-order

• Every two-factor interaction is aliased with another two-

factor interaction

main effect: A

Two-factor interaction: AB

𝑦

45 (1)

100 ad

45 bd

65 ab

75 cd

60 ac

80 bc

96 abcd

𝓁A = 19

𝓁B = 1.5

𝓁C = 14

𝓁D = 16.5

𝓁AB = -1

𝓁AC = -18.5

𝓁AD = 19

Because factor B is not signficant (𝓁B), we drop it from

consideration.

24 full design

A = 21,625

C = 9,875

D = 14,625

AC = -18,125

AD = 16,625

This 2𝐼𝑉4−1 design can be projected into a single

replicate of the 23 design in factors A, C, and D

(-) (+)

A

(+)

(-)

D (-)

(+)

C

45

80

75 96

60

100

65

45

AC interaction: A(-): concentration has a large positive effect A(+) : concentration has a very small effect AD interaction: A(-): stirring rate has a very small effect A(+) : stirring rate has a large positive effect

𝑦

45 (1)

100 ad

45 bd

65 ab

75 cd

60 ac

80 bc

96 abcd

> library(FrF2) > design<-FrF2(8, randomize = FALSE, + factor.names = c("A", "B", "C","D"), + default.levels = c(-1, +1)) > y<-c(45,100,45,65,75,60,80,96) > design<-add.response(design=design,response=y) > design A B C D y 1 -1 -1 -1 -1 45 2 1 -1 -1 1 100 3 -1 1 -1 1 45 4 1 1 -1 -1 65 5 -1 -1 1 1 75 6 1 -1 1 -1 60 7 -1 1 1 -1 80 8 1 1 1 1 96 class=design, type= FrF2 > design.lm <- lm(y~A*B*C*D,data=design) > design.mean<-design.lm$coefficients[1] > design.effects<-design.lm$coefficients[-1]*2 > design.mean (Intercept) 70.75 > design.effects A1 B1 C1 D1 A1:B1 A1:C1 19.0 1.5 14.0 16.5 -1.0 -18.5 B1:C1 A1:D1 B1:D1 C1:D1 A1:B1:C1 A1:B1:D1 19.0 NA NA NA NA NA A1:C1:D1 B1:C1:D1 A1:B1:C1:D1 NA NA NA

> cubePlot(design.lm, "A", "D", "C")

> MEPlot(design)

> IAPlot(design)

• Regression Model 𝑦 = 𝛽 0 + 𝛽 𝐴𝐴 + 𝛽 𝐶𝐶 + 𝛽 𝐷𝐷 + 𝛽 𝐴𝐷𝐴𝐶 + 𝛽 𝐴𝐷𝐴𝐷

𝑦 = 𝛽 0 + 𝛽 1𝑥1 + 𝛽 3𝑥3 + 𝛽 4𝑥4 + 𝛽 13𝑥1𝑥3 + 𝛽 14𝑥1𝑥4

𝑦 = 70.75 +19

2𝑥1 +

14

2𝑥3 +

16.5

2𝑥4 −

18.5

2𝑥1𝑥3 +

19

2𝑥1𝑥4

𝑦 = 70.75 + 8.5𝑥1 + 7𝑥3 + 8.25𝑥4 − 9.25𝑥1𝑥3 + 9.5𝑥1𝑥4

>> x1=-1:.1:1;

>> x3=x1; x4=x1;

>> [X1,X3,X4]=meshgrid(x1,x3,x4);

>> Y=70.75+8.5*X1+7*X3+8.25*X4-9.25*X1.*X3+9.5*X1.*X4;

>> slice(X1,X3,X4,Y,[-1. 1.],[-1. 1.],[-1. 1.])

>> xlabel("X1");

>> ylabel("X3");

>> zlabel("X4");

>> colorbar on

2k-p Fractional Design • 2𝑘−𝑝 runs =

1

2𝑝 fraction of 2𝑘 full design

𝑝 = 2: 2𝑘−2 =1

2×2=

1

4 fraction of 22

• 𝑝 independent generators

• The defining relation consists of all columns that are equal to

the identity colums, 𝐼

– Ex: 𝑘 = 6, 𝑝 = 2 26−2

• generators:

𝐼 = 𝐴𝐵𝐶𝐸 𝐸 = 𝐴𝐵𝐶

𝐼 = 𝐵𝐶𝐷F 𝐹 = 𝐵𝐶𝐷

𝐼 = 𝐴𝐷𝐸𝐹

• 2𝐼𝑉6−2

• Main effect: 𝐴 𝐴. 𝐼 = 𝐴. 𝐴𝐵𝐶𝐸 = 𝐴. 𝐵𝐷𝐹 = 𝐴. 𝐴𝐷𝐸𝐹

𝐴 = 𝐵𝐶𝐸 = 𝐴𝐵𝐷𝐹 = 𝐷𝐸𝐹

• Interaction effect: 𝐴𝐵 𝐴𝐵. 𝐼 = 𝐴𝐵. 𝐴𝐵𝐶𝐸 = 𝐴𝐵. 𝐵𝐷𝐹 = 𝐴𝐵. 𝐴𝐷𝐸𝐹

𝐴𝐵 = 𝐶𝐸 = 𝐴𝐷𝐹 = 𝐵𝐷𝐸𝐹

run A B C D

1 - - - -

2 + - - -

3 - + - -

4 + + - -

𝑬 = 𝑨𝑩𝑪 𝑭 = 𝑩𝑪𝑫

- -

+ -

+ +

- +

Summary tables of useful fractional factorial designs

Generators