Experimental Design

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Experimental Design

• Paul Geladi

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Example

• T,P
• A B ---gt C
• C reponse (want much)
• T, P factors (can regulate)

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What kind of function could describe the contour
plot?
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Expert method (Stan Deming)
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Shotgun method
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Method
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Classical method
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The best method factorial design
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Why experimental design?

• Reduce number of experiments (cost, time)
• Extract maximal information
• Understand what happens
• Predict future behaviour

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Response surfaces

• Surface in 3D
• Contour plot

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Response surfaces and functions

• Large surface NO function
• Small surface simple function
• Therefore try to start close to an expected
  optimum

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Problems

• Many factors - many levels
• 6 factors at 4 levels 212 runs
• Reduce number of factors
• Only 2 levels
• Discard factors
• This is called SCREENING

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2
45
75
30
P
20
40
25
35
10
1
80
120
T
Average temperature effect 20 Average pressure
effect 30 Interaction effect 10 Important
possible to have diagonal moves
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1
45
75
30
P
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Coded levels -1 and 1
40
25
35
10
-1
-1
1
T
This is the smallest FULL FACTORIAL DESIGN
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1
45
75
30
P
20
40
25
35
10
-1
-1
1
T
Design table or design matrix
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New terms to remember

• Factor
• Level
• Coded level
• Run
• Response
• Effect
• Interaction
• Response surface plot - contour plot
• Response surface function
• Design table or matrix
• Full factorial design
• Screening

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3 important things

• Equation
• Picture
• Worksheet (table, matrix)

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Chemical reaction

• AB --gt C
• Temperature (T)
• Pressure (P)
• Catalyst (K)
• Selection of levels

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Chemical reaction

• AB ---gt C
• Temperature (T)
• Pressure (P)
• Catalyst (K)
• Response
• 3 factors
• Each factor at 2 levels

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Chemical reaction

• Table design matrix
• Picture
• Equation

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Number of runs

• Levels Factors
• 23 8
• 32 9
• 24 16
• 33 27
• 25 32
• etc

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1
K
-1
1
P
-1
-1
1
T
y b0 b1x1 b2x2 b3x3 e
y response xi factor bi coefficient e
residual b0 average level
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Chemical reaction

• Table design matrix

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1
K
80
45
-1
1
68
54
P
52
83
72
60
-1
-1
1
T
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1
K
-1
1
P
-1
-1
1
T
y b0 b1x1 b2x2 b3x3 b12x1x2 b13x1x3
b23x2x3 b123x1x2x3 e
y response xi factor bi coefficient e
residual b0 average level
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Bread baking
Randomization!
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Bread baking
Randomization!
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1
K
-1
1
P
-1
-1
1
T
y b0 b1x1 b2x2 b3x3 b12x1x2 b13x1x3
b23x2x3 b123x1x2x3 e
xi are known, y can be measured bi to be found
while keeping the residual e small expectations
for e?
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How to interpret?

• Equation coefficients
• b0 b1 b2 b3 b12 b13 b23
  b123
• 64.3 11.5 -2.5 0.75 0.75 5.0 0 0.25
• VERY rarely b0
• Large coefficient important factor
• Zero or small coefficient ignore
• Interaction usually present
• What does negative mean?

T P K TP TK PK TPK
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How to interpret?

• b0 ignore
• b123 ignore
• main coefficients b1 b2 b3
• two factor interactions b12 b13 b23
• if a two factor interaction is important, then
  the main coefficients can not be removed

T P K TP TK PK TPK
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1
K
80
45
-1
1
68
54
P
52
83
72
60
-1
-1
1
T
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How to interpret?Thanks to coding, the
coefficients are comparable
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More things to look at

• Normal distribution of coefficients
• Normal distribution of residuals
• Residuals (histogram?)
• Residuals versus time
• ANOVA
• An estimate of standard deviation is needed

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Standard deviation

• 3 sources
• repeat center points
• duplicate runs
• residual

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Nomenclature

• Replicate
• Duplicate
• Center point
• Randomization
• Run order