Embed Size (px)
Citation preview
2. Factorial Designs• Analysis of variance of a factorial experiments
• Two-level factorial designs
• Factor effects estimation
• Assessing the effects’ significance
• Normal plot of effects
E. Barrios Design and Analysis of Engineering Experiments 2–1E. Barrios Design and Analysis of Engineering Experiments 2–1
Session 2 Factorial Designs 2Session 2 Factorial Designs 2
Analysis of Variance of Factorial DesignsYield in Paper Industrya
Problem: Percentage yield as a function of reaction time, temperature and pressure ina paper industry.
Factor Levels:
LevelsFactor 1 2 3Time (hrs.) 1 2 3Temperature (oC) 30 70 100Pressure (psi) 30 70 100
Data Set:
======================================================obs time Temp Press yield obs time Temp Press yield------------------------------------------------------
1 1 1 1 68.5 15 3 2 2 84.52 2 1 1 74.5 16 1 3 2 72.53 3 1 1 70.5 17 2 3 2 70.04 1 2 1 72.8 18 3 3 2 66.55 2 2 1 72.0 19 1 1 3 68.76 3 2 1 69.5 20 2 1 3 74.67 1 3 1 72.5 21 3 1 3 74.78 2 3 1 75.5 22 1 2 3 72.09 3 3 1 65.0 23 2 2 3 76.0
10 1 1 2 73.0 24 3 2 3 76.011 2 1 2 75.0 25 1 3 3 73.112 3 1 2 72.5 26 2 3 3 76.013 1 2 2 80.1 27 3 3 3 70.514 2 2 2 81.5======================================================a
aDevore (1995). Exercise 11.29.
E. Barrios Design and Analysis of Engineering Experiments 2–2E. Barrios Design and Analysis of Engineering Experiments 2–2
Session 2 Factorial Designs 3Session 2 Factorial Designs 3
Analysis of Variance of Factorial DesignYield in Paper IndustryIf we now include in the model the factor time (A) and all second order interactions.
Model:y = µ + τA + τB + τC + τAB + τAC + τBC + ε
ANOVA Table:
Response: yieldDf Sum Sq Mean Sq F value Pr(>F)
time 2 42.112 21.056 8.7628 0.0096484Temp 2 110.732 55.366 23.0416 0.0004788Press 2 68.136 34.068 14.1781 0.0023445time:Temp 4 67.761 16.940 7.0501 0.0098173time:Press 4 35.184 8.796 3.6606 0.0558947Temp:Press 4 136.437 34.109 14.1952 0.0010480Residuals 8 19.223 2.403
Conclusion: Main factor effects and second order interactions are significant.
E. Barrios Design and Analysis of Engineering Experiments 2–3E. Barrios Design and Analysis of Engineering Experiments 2–3
Session 2 Factorial Designs 4Session 2 Factorial Designs 4
Analysis of Variance of Factorial DesignYield in Paper IndustryFinally, if we decompose further the sum of squares of the residuals by including thirdorder interactions. The resulting model is
y = µ + τA + τB + τC + τAB + τAC + τBC + τABC + ε
ANOVA Table:
Response: yieldDf Sum Sq Mean Sq F value Pr(>F)
time 2 42.112 21.056Temp 2 110.732 55.366Press 2 68.136 34.068time:Temp 4 67.761 16.940time:Press 4 35.184 8.796Temp:Press 4 136.437 34.109time:Temp:Press 8 19.223 2.403Residuals 0 0.000
Conclusion: All main, second and third order interactions are estimable but there areno degrees of freedom left to estimate the error variance. Thus, it is not possible toassess statistically the significance of such effects.
E. Barrios Design and Analysis of Engineering Experiments 2–4E. Barrios Design and Analysis of Engineering Experiments 2–4
Session 2 Factorial Designs 5Session 2 Factorial Designs 5
Two-Level Factorial Designsa
To perform a factorial design, you select a fixed number of levels of each of a num-ber of factors (variables) and then run experiments in all possible combinations. Thefactors can be quantitative (e.g. various temperatures) or qualitative (e.g. differentmethods).
Two-level factorial designs are of special importance because of the following:
1. They require relatively few runs per factor studied.
2. The interpretation of the observations produced by the designs can proceed largelyby using common sense and computer graphics.
3. When the factors are quantitative, although unable to fully explore a wide regionin the factor space, they often determine a promising direction for further experi-mentation.
4. Design can be suitably augmented when a more thorough local exploration isneeded—a process called sequential assembly.
5. The form the basis for two-level fractional factorial designs. Particularly usefulfor factor screening and building blocks in the sequential assembly of experimentaldesigns.
aBHH2e Chapter 5E. Barrios Design and Analysis of Engineering Experiments 2–5E. Barrios Design and Analysis of Engineering Experiments 2–5
Session 2 Factorial Designs 6Session 2 Factorial Designs 6
Two-Level Factorial DesignsPilot Plant Investigationa
23 Full Factorial Design
The next table shows a 23 factorial design with 2 quantitative factors, temperature Tand concentration C, and one qualitative factor, catalyst K. The response y is thechemical yield.
Factor Levels
Factor Label low (−) high (+)Temperature (oC) T 160 180
Concentration (%) C 20 40
Catalyst (type) K A B
Experimental Design and Response
Factors Responserun T C K y
1 − − − 60
2 + − − 72
3 − + − 54
4 + + − 68
5 − − + 52
6 + − + 83
7 − + + 45
8 + + + 80
a
aBHH2e, Chapter 5; BHH Chapter 10.
E. Barrios Design and Analysis of Engineering Experiments 2–6E. Barrios Design and Analysis of Engineering Experiments 2–6
Session 2 Factorial Designs 7Session 2 Factorial Designs 7
Two-Level Factorial DesignsPilot Plant InvestigationDisplay the data: Any pattern?
E. Barrios Design and Analysis of Engineering Experiments 2–7E. Barrios Design and Analysis of Engineering Experiments 2–7
Session 2 Factorial Designs 8Session 2 Factorial Designs 8
Two-Level Factorial DesignsPilot Plant InvestigationMain Effects:
Temperature T main effect:
τ T = yT+ − yT−
=72 + 68 + 83 + 80
4− 60 + 54 + 52 + 45
4
= 75.75− 52.75 = 23
=
∑cTiyi
n/2
E. Barrios Design and Analysis of Engineering Experiments 2–8E. Barrios Design and Analysis of Engineering Experiments 2–8
Session 2 Factorial Designs 9Session 2 Factorial Designs 9
Two-Level Factorial DesignsPilot Plant InvestigationSecond Order Interaction Effects:
Temperature–Catalyst Interaction Effect TK: T ×K = T ∗K = T : K
Using the +/− coding:cTK = cT ∗ cK
Interaction Effect
τ TK =
∑cTKiyi
n/2= 10
E. Barrios Design and Analysis of Engineering Experiments 2–9E. Barrios Design and Analysis of Engineering Experiments 2–9
Session 2 Factorial Designs 10Session 2 Factorial Designs 10
Two-Level Factorial DesignsPilot Plant InvestigationThird Order Interaction Effects:
Temperature–Concentration–Catalyst Interaction Effect TCK: T×K = T ∗C ∗K =T : C : K
Using the +/− coding:cTK = cT ∗ cC ∗ cK
Interaction Effect
τ TCK =
∑cTKiyi
n/2= 10
E. Barrios Design and Analysis of Engineering Experiments 2–10E. Barrios Design and Analysis of Engineering Experiments 2–10
Session 2 Factorial Designs 11Session 2 Factorial Designs 11
Two-Level Factorial DesignsPilot Plant Investigation
Temperature–Catalyst Interaction:
[TK] Projected Space
Temperature (T)
Cat
alys
t (K
)
57 70
48.5 81.5
− +−
+
5060
7080
Interaction Plot
Temparature (T)
yiel
d (y
)
−1 1
Catalyst (K)
1−1
E. Barrios Design and Analysis of Engineering Experiments 2–11E. Barrios Design and Analysis of Engineering Experiments 2–11
Session 2 Factorial Designs 12Session 2 Factorial Designs 12
Two-Level Factorial DesignsPilot Plant Investigation
Estimation of Effects:
Factor Effect E =
∑cEiyi
n/2
run b0 T C K TC TK CK TCK
1 1 −1 −1 −1 +1 +1 +1 −1
2 1 +1 −1 −1 −1 −1 +1 +1
3 1 −1 +1 −1 −1 +1 −1 +1
4 1 +1 +1 −1 +1 −1 −1 −1
5 1 −1 −1 +1 +1 −1 −1 +1
6 1 +1 −1 +1 −1 +1 −1 −1
7 1 −1 +1 +1 −1 −1 +1 −1
8 1 +1 +1 +1 +1 +1 +1 +1
Effect 64.3 23.0 −5.0 1.5 1.5 10.0 0.0 0.5
Std Error ±0.7 ±1.4 ±1.4 ±1.4 ±1.4 ±1.4 ±1.4 ±1.4
E. Barrios Design and Analysis of Engineering Experiments 2–12E. Barrios Design and Analysis of Engineering Experiments 2–12
Session 2 Factorial Designs 13Session 2 Factorial Designs 13
Two-Level Factorial DesignsPilot Plant InvestigationNormal Plot of Effects:
−5 0 5 10 15 20 25
−1.
0−
0.5
0.0
0.5
1.0
effects
norm
al s
core
T
C
K
T:C
T:K
C:K
T:C:K
E. Barrios Design and Analysis of Engineering Experiments 2–13E. Barrios Design and Analysis of Engineering Experiments 2–13
Session 2 Factorial Designs 14Session 2 Factorial Designs 14
Two-Level Factorial DesignsProcess Development Studya
24 Full Factorial DesignA process development experiment in which 4 factors were studied in a 24 factorialdesign: amount of catalyst charge (x1), temperature (x2), pressure (x3), and concen-tration of one of the reactants.
Factor Levels
Factor Label low (−) high (+)Catalyst charge (lb) x1 10 15
Temperature (oC) x2 220 240
Pressure (psi) x3 50 80
Concentration (%) x4 10 12
Experimental Design and Response
run Factors conversionorder x1 x2 x3 x4 y
8 − − − − 702 + − − − 60
10 − + − − 894 + + − − 81
15 − − + − 699 + − + − 621 − + + − 88
13 + + + − 8116 − − − + 605 + − − + 49
11 − + − + 8814 + + − + 823 − − + + 60
12 + − + + 526 − + + + 867 + + + + 79
a
aBHH2e, Chapter 5; BHH Chapter 10.
E. Barrios Design and Analysis of Engineering Experiments 2–14E. Barrios Design and Analysis of Engineering Experiments 2–14
Session 2 Factorial Designs 15Session 2 Factorial Designs 15
Two-Level Factorial DesignsProcess Development StudyContrast coefficients
I x1 x2 x3 x4 x12 x13 x14 x23 x24 x34 x123 x124 x134 x234 x1234 conversion1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 701 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 601 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 891 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 811 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 691 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 621 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 881 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 811 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 601 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 491 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 881 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1 821 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 601 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 521 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 861 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79
E. Barrios Design and Analysis of Engineering Experiments 2–15E. Barrios Design and Analysis of Engineering Experiments 2–15
Session 2 Factorial Designs 16Session 2 Factorial Designs 16
Two-Level Factorial DesignsProcess Development StudyEstimated Effects and Standard Error
Estimated StandardEffect effect errorIntercept 72.25 ±0.27x1 −8.00 ±0.55x2 24.00 ±0.55x3 −0.25 ±0.55x4 −5.50 ±0.55x12 1.00 ±0.55x13 0.75 ±0.55x14 0.00 ±0.55x23 −1.25 ±0.55x24 4.50 ±0.55x34 −0.25 ±0.55x123 −0.75x124 0.50x134 −0.25x234 −0.75x1234 −0.25
Assuming third and higher order interactions are neg-ligible:
se2 =1
5
[(−0.75)2 + (+0.50)2 + (−0.25)2 + (−0.75)2 + (−0.25)2] = 0.30
se =√
0.30 = 0.55
E. Barrios Design and Analysis of Engineering Experiments 2–16E. Barrios Design and Analysis of Engineering Experiments 2–16
Session 2 Factorial Designs 17Session 2 Factorial Designs 17
Two-Level Factorial DesignsProcess Development StudyNormal Plot of Effects
0 10 20 30
−1
01
effects
norm
al s
core
x1
x2
x3
x4
x12 x13
x14
x23
x24
x34
x123
x124
x134
x234 x1234
E. Barrios Design and Analysis of Engineering Experiments 2–17E. Barrios Design and Analysis of Engineering Experiments 2–17
Session 2 Factorial Designs 18Session 2 Factorial Designs 18
Normal PlotNormal Probability Density
a) Normal probability density
0.682
2σ
µ − 3σµ − 2σ µ − σ µ µ + σ µ + 2σµ + 3σ
−3 −2 −1 0 1 2 3
E. Barrios Design and Analysis of Engineering Experiments 2–18E. Barrios Design and Analysis of Engineering Experiments 2–18
Session 2 Factorial Designs 19Session 2 Factorial Designs 19
Normal PlotNormal Probability Cumulative Distribution
Cum
ulat
ed P
roba
bilit
y
b) Normal probability distribution
0.0
0.2
0.4
0.6
0.8
1.0
0.159
0.841
2σ
0.682
µ − 3σµ − 2σ µ − σ µ µ + σ µ + 2σµ + 3σ
−3 −2 −1 0 1 2 3
E. Barrios Design and Analysis of Engineering Experiments 2–19E. Barrios Design and Analysis of Engineering Experiments 2–19
Session 2 Factorial Designs 20Session 2 Factorial Designs 20
Normal PlotNormal Probability Paper
Pro
babi
lity
c) Normal probability plot
0.001
0.01
0.05
0.1
0.2
0.3
0.5
0.7
0.8
0.9
0.95
0.99
0.999
−3
−2
−1
0
1
2
3
norm
al s
core
s
0.159
0.841
2σ
0.682
µ − 3σµ − 2σ µ − σ µ µ + σ µ + 2σµ + 3σ
−3 −2 −1 0 1 2 3
E. Barrios Design and Analysis of Engineering Experiments 2–20E. Barrios Design and Analysis of Engineering Experiments 2–20
Session 2 Factorial Designs 21Session 2 Factorial Designs 21
Normal Plota) Normal probability density
0.682
2σ
µ − 3σ µ − 2σ µ − σ µ µ + σ µ + 2σ µ + 3σ
−3 −2 −1 0 1 2 3
Cum
ulat
ed P
roba
bilit
y
b) Normal probability distribution
0.0
0.2
0.4
0.6
0.8
1.0
0.159
0.841
2σ
0.682
µ − 3σ µ − 2σ µ − σ µ µ + σ µ + 2σ µ + 3σ
−3 −2 −1 0 1 2 3
z: standarized normal scale
Pro
babi
lity
c) Normal probability plot
0.001
0.01
0.050.1
0.20.3
0.5
0.70.8
0.90.95
0.99
0.999
−3
−2
−1
0
1
2
3
norm
al s
core
s
0.159
0.841
2σ
0.682
µ − 3σ µ − 2σ µ − σ µ µ + σ µ + 2σ µ + 3σ
−3 −2 −1 0 1 2 3
E. Barrios Design and Analysis of Engineering Experiments 2–21E. Barrios Design and Analysis of Engineering Experiments 2–21
Session 2 Factorial Designs 22Session 2 Factorial Designs 22
Normal PlotSimulated Normal Samples
y
−3 −2 −1 0 1 2 3
−3 −2 −1 0 1 2 3
−2
−1
0
1
2
y
norm
al s
core
0.023
0.159
0.5
0.841
0.977
y
−3 −2 −1 0 1 2 3
−3 −2 −1 0 1 2 3
−2
−1
0
1
2
y
norm
al s
core
0.023
0.159
0.5
0.841
0.977
y
−3 −2 −1 0 1 2 3
−3 −2 −1 0 1 2 3
−2
−1
0
1
2
y
norm
al s
core
0.023
0.159
0.5
0.841
0.977
E. Barrios Design and Analysis of Engineering Experiments 2–22E. Barrios Design and Analysis of Engineering Experiments 2–22
Session 2 Factorial Designs 23Session 2 Factorial Designs 23
Normal PlotPilot Plant InvestigationNormal Plot of Effects
−5 0 5 10 15 20 25
−1.
0−
0.5
0.0
0.5
1.0
effects
norm
al s
core
T
C
K
T:C
T:K
C:K
T:C:K
E. Barrios Design and Analysis of Engineering Experiments 2–23E. Barrios Design and Analysis of Engineering Experiments 2–23
Session 2 Factorial Designs 24Session 2 Factorial Designs 24
Normal PlotProcess Development StudyNormal Probability Paper
order effect factor Probx1 1 -8.00 x1 3.3x4 2 -5.50 x4 10.0x23 3 -1.25 x23 16.7x123 4 -0.75 x123 23.3x234 5 -0.75 x234 30.0x1234 6 -0.25 x1234 36.7x34 7 -0.25 x34 43.3x3 8 -0.25 x3 50.0x134 9 -0.25 x134 56.7x14 10 0.00 x14 63.3x124 11 0.50 x124 70.0x13 12 0.75 x13 76.7x12 13 1.00 x12 83.3x24 14 4.50 x24 90.0x2 15 24.00 x2 96.7
Prob = 100(i− 1/2)/15
0.5
1
2
5
10
20
30
40
50
60
70
80
90
95
98
99
99.5
0.5
1
2
5
10
20
30
40
50
60
70
80
90
95
98
99
99.5N=15 N=31 N=63 N=16 N=32 N=64
E. Barrios Design and Analysis of Engineering Experiments 2–24E. Barrios Design and Analysis of Engineering Experiments 2–24
Session 2 Factorial Designs 25Session 2 Factorial Designs 25
E. Barrios Design and Analysis of Engineering Experiments 2–25E. Barrios Design and Analysis of Engineering Experiments 2–25
Session 2 Factorial Designs 26Session 2 Factorial Designs 26
Two-Level Factorial DesignsProcess Development StudyNormal Plot of Effects
0 10 20 30
−1
01
effects
norm
al s
core
x1
x2
x3
x4
x12 x13
x14
x23
x24
x34
x123
x124
x134
x234 x1234
E. Barrios Design and Analysis of Engineering Experiments 2–26E. Barrios Design and Analysis of Engineering Experiments 2–26
Session 2 Factorial Designs 27Session 2 Factorial Designs 27
Two-Level Factorial DesignsProcess Development StudyMinitab Normal Plot of Effects
E. Barrios Design and Analysis of Engineering Experiments 2–27E. Barrios Design and Analysis of Engineering Experiments 2–27
