Design and Analysis of Experiments - Quimica … · Design and Analysis of Experiments Prof. Dr....
Transcript of Design and Analysis of Experiments - Quimica … · Design and Analysis of Experiments Prof. Dr....
Design and Analysis
of Experiments
Prof. Dr. Anselmo E de Oliveira
anselmo.quimica.ufg.br
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
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
• 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
Summary tables of useful fractional factorial designs
Generators