design of experiments a short introduction
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M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
6M Meeting Turin
Seminar
Design of Experiments – a short Introduction
STZ Prozesskontrolle und Datenanalyse
Prof. Dipl. Phys. Waltraud Kessler
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Contents
Basic Statistics for Design of Experiments (DoE)
Sample Size, Power of a Design Analysis of Variance (ANOVA)
Screening Designs
Full Factorial Designs Fractional Factorial Designs
Designs for Response Surface Analysis or Optimization
Central Composite Designs (CCD) Optimization for Several Responses – Find the "Sweet Spot"
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Design of Experiments needs Statistics
We measure parts (very small) of a population (we run only some
experiments of the many possible ones)
Uncertainty: We measure the sample and describe the population
Questions that arise:
sample size - how many experiments do we really need?
extreme values - is this an outliers? Or is it an effect?
calculate mean, median - when are the different?
calculate standard deviation, interquartile range – what range is acceptable?
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Design of Experiments needs Statistics
By experiments you can measure the mean and standard
deviation of a sample.
This is used to estimate the true but unknown mean
and standard deviation of the underlying population.
„All models are wrong, some are helpful“ cit. G. Box
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Calculating Effects - Error of mean
1
1
2
2
n
xx
s
n
i
i 2ss Standard deviation: Variance:
The SEM quantifies the precision of the mean.
It is a measure of how accurately you estimate the true mean of the population.
A bigger n gives a more accurate estimation of the true population mean.
Standard error of the mean:
n
sSEM
degree of freedom
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Power of a Design is important
Δµ=µ-µ0 = 1
α=5%, n = 4
=20%
Power = 80%
φ φ
Question: What sample size n is necessary to measure
a mean 0 with a given accuracy of
Δµ=µ-µ0 = 1
α=5%, n = 9
=5%
Power = 95%
SEM = n
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Analysis of Variance (ANOVA)
Comparison of several means – find significant effects
Calculate the error and calculate the effects for experiments
One-way ANOVA
Changing only one factor
Two-way ANOVA
Changing two factors
Multi-way ANOVA
Changing several factors
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
DoE - Design of Experiments
General Factorial Designs
Find significant main effects and interactions
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Changing two factors
• Bioprocess for bioethanol conversion from cellulose
• Factor A: Time of treatment
• Factor B: Treatment
Response:
Conversion cellulose %
Significant effects:
Main effects:Treatment and Time
Non linear effect: Time
Interaction: Treatment*Time
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
DoE - Design of Experiments
Two level full factorial designs
Find significant main effects and interactions
Do the minimum number of experiments
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Two level full factorial designs
Task: Determine yield of a product depending on 3 factors: a) time b) temperature CPP (Critical Process Parameters) c) concentration Classical approach: One variable at a time (OVAT) - keep temperature and concentration constant, vary time - keep time and concentration constant, vary temperature - keep time and temperature constant, vary concentration Disadvantage: - Keep factors constant on which level? - You need many experiments. - You will not find interactions.
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Building a Full Factorial Design
Each factor is varied on 2 levels: low and high (-1 and +1)
Factors are varied independently of each other.
A:Time
C: Concentration
B: Temperature
B- low A+ high
B+ high
A- low
C- low
C+ high
2N experiments
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Few Experiments – many Effects
To investigate the influence of N factors
2N experiments have to be run
For 3 factors there are 7 possible effects:
Main effects: A, B, C
Two way interactions: AB, AC, BC
Three way interaction: ABC
• With only 8 experiments we can calculate all 7 effects.
• Additional experiments needed to determine the error.
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Do the Experiments in Random Order
The 2N-factorial design is the most simple
complete experimental design,
to examine the influence of N factors
and their interactions on a response.
Important:
Do whenever possible the runs in random order!
This avoids systematic errors to be regarded as effects.
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Graphical representation of regression equation
Main effects
AC interaction (Time*Concentration)
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Graphical Representation of Interactions
Lines are parallel
no interaction
Lines depart from parallelity
positive interaction
Lines depart from parallelity
negative interaction
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
DoE - Design of Experiments
Screening Designs: Two-level Factorial Designs
Full Factorial Designs
Fractional Factorial Designs
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Fractional Factorial Designs with two levels
A
C
B
(- - -) (+ - -)
(+ + +)
Problem: • The number of factors to investigate is high • E.g. 5 factors requires 32 experiments • Too many experiments are necessary
Solution: • Only part of the experiments are run • E.g. for 5 factors only 16 experiments
Disadvantage: • It is not possible to calculate all interactions
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Fractional Factorials - Resolution
The detailed overview can be found in: Design Expert – Software for Experimental Design – www.statease.com
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Design Resolution The risk of misinterpretation because of the influence of confounding
on the factors depends on the degree of confounding. We speak of different resolution for the designs.
.
Work with resolution V (or higher) designs, if possible!
Main effects
confounded with
2-factor interactions confounded with
Resolution III - Designs 2-factor interaction main effects
Resolution IV - Designs 3-factor interaction 2-factor interaction
Resolution V - Designs 4-factor interaction 3-factor interaction
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Example for 26-2 Fractional Design
Tablets are coated with an insoluble film in a fluid bed dryer The influence of 6 process parameters on drug release is investigated
Factor Name Units Low Actual High Actual A Inlet air quantity m3/h 110 130 B Atomisation pressure bar 0.60 1.30 C Amount film polymer mg/cm2 1.00 2.00 D Inlet air Temp. °C 45 60 E Spray rate g/Min 6 13 F Drying time min 3 6
Response variable: Drug release after 2 h
Full design: 64 runs Fractional design 16 runs + 3 center points (Resolution IV)
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Effects for 26-2 Fractional Design
All possible effects are: 6 main effects 15 two-way interactions. As resolution is IV, many 2-way interactions are confounded. Significant effects: C BC could also be AE or DF CF could also be BD All others are not significant, Because of hierarchy add B and F!
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Interpreting the Results
Drug release 2 h
The factors A, D, E and their interactions are not in the model.
Factor Estimate
Intercept 40.52
B-Atomisation pressure 0.054
C-Amount film polymer -3.21
F-Drying time 0.044
BC -1.02
CF -0.73
Actual Factor Settings
A: Inlet air quantity = 110
D: Inlet air temperature = 45
E: Spray rate = 6
F: Drying time = 3
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Response Surface Designs for
Optimization
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Next Step: Regression
Models with Main Effects and Interactions
A
B
y
- Linear dependence on factor A and B
y = b0 + b1A + b2B
0
B
A
y
- Add the interaction AB
y = b0 + b1A + b2B + b12AB
0
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Response Surface Modeling (RSM) Designs Ex
pec
ted
Res
po
nse
y
It is possible to fit nonlinear (quadratic) response surfaces with these models.
A minimum of 3 design points (levels) per factor is required.
Factor A = Factor B = Time (min)
Add squared terms
y = b0 + b1A + b2B
+ b12AB
+ b11A2 + b22B
2
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Response Surface Designs for Optimization
3 Level Factorial Designs
Central Composite Designs
Several responses: Finding "Sweet spots"
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
3 Level Full Factorial Designs
Factor levels are given in transformed coordinate system (-1, 0, +1)
The center point has the coordinate (0, 0, .., 0)
The distances -1 to 0 and 0 to +1 are the same.
Easy to set up, but for more than 2 factors no more efficient.
The number of experiments increases very fast: 2 Factors 9 experiments 3 Factors 27 experiments 4 Factors 81 experiments
Factor A
Fact
or
B
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Central Composite Designs – for 2 Factors
We add experiments to the existing design of the screening phase.
Cube points from + starpoints + centre points = Central Composite design
factorial design (α-Points, axial points) + centre points
+ centre points
= Faktor A
Faktor B
Faktor B
Faktor A
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Central Composite Designs for 3 Factors
For the ideal CC design all points are on a sphere around the center points.
Number of factors Number of experiments
2 9
3 15
4 25
5 43
Minimum number of experiments
Add at least 3 more center points 5 levels on each factor
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Face Centered CC Design for 3 Factors
• The starpoints are on the cube.
• No extreme settings
• Easier to handle
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Three Responses Find Design Space
Yield Model: A + B + AB+ A2 + B2
Viscosity Model: A + B + A2 +B2
Molecular weight Model: A + B + AB
Targets for responses: Yield > 78 % Viscosity between 62 and 68 mPas Molecular weight 3400 (g/mol)
A: Time A: Time A: Time
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Working Areas for Individual Responses
Contour plots of possible working areas
Which settings are valid for all responses?
Yield Viscosity Molecular Weight
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Optimizing Several Responses
Settings in yellow area fullfill all
requirements
→ Consider uncertainties
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Design Space und Control Space
The yellow area fulfills all demands for
all three responses This is the
Design Space
The light yellow area shows the working area
where all responses fulfill the demands with
95% probability This is the
Control Space
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Knowledge Space, Design Space, Control Space
1. The design defines the Knowledge Space
2. The responses define the Design Space
3. The uncertainties (confidence intervals) define the Control Space.
The control space will be used for production. Within control space
are the factor settings for normal operation.
Knowledge Space (Factor Space)
Control Space Normal Operation
Design Space
M06 Meeting Turin, Italy 14-15 June, 2017
Design of Experiments – a short introduction
Screening Designs vs. Response Surface Methods
Factorial Designs (Full or Fractional Designs)
Designs for Response Surface Methods
During screening emphasis is on identifying factor effects and their possible interactions. Questions: What are the important design factors? Are there interactions involved?
The goal is optimization of one or several responses. Emphasis is on the fitted surface. Questions: How well does the surface represent the true behavior? Which working area complies with all specified response values?
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