Transcript PLACKETT BURMAN DESIGNS Factors Number of experiments

EXPERIMENTAL DESIGNS
Prof.Dr.Cevdet Demir
[email protected]
MOTIVATIONS FOR DESIGN
• Screening
•Saving time
•Quantitative modelling
•Optimisation
WHY DESIGN EXPERIMENTS?
Example : Optimisation of a reaction with pH and
temperature.
IMAGINARY RESPONSE SURFACE
•
We want to find optimum
•
Response surface unknown
•
Mathematical model may not be of interest in
its own right
•
Not necessarily interested in underlying
molecular mechanism
•
Reproducibility and flat optimum
One factor at a time strategy : may miss optima
DIFFICULTY
Interactions – the response for each factor is
not independent.
The optimum temperature at pH 5 differs
from that at pH 6.
How to be on the safe side?
• Grid search. 10 pHs, 10 temperatures, 100
experiments.
•Big grid. Then smaller grid.
PROBLEMS
•Time consuming and expensive.
•Many experiments we are almost certain are not near
at optimum so are obviously a waste of time
•Reproducibility and experimental error
WHAT DO WE DO?
We need rules!
Formal experimental design
Screening
• Factorial designs
• Partial factorials and Plackett-Burman designs
Modelling and optimisation
• Response surface designs
FACTORIAL DESIGNS
SIMPLEST IDEA
•List factors
•High and low levels
•2k experiments
• Can be used to look at how significant
each factor is
•Can be used to form a quantitative model
between response (e.g. taste, texture,
colour or even market acceptability) and
value of the factors
•Can use categorical factors e.g. supplier
or whether one process used or not
•Takes all interactions into account
PROBLEM
•For many factors a lot of experiments.
•10 factors = 1024 experiments
•Screening many factors not really good
How can we reduce the number of
experiments?
FRACTIONAL FACTORIAL DESIGNS
Remove some of the experiments
Which to remove?
Fractional factorial.
Takes out certain experiments.
Fractional factorials from 2k to 2k-f
E.g.
from 8 to 4 (half)
from 32 to 16 (half)
from 32 to 8 (quarter)
Lose information about higher order interactions.
Call this confounding.
Must have at least k + 1 experiments
But a little inefficient.
What happens if 19 factors? Do we need to do 32
experiments?
PLACKETT BURMAN DESIGNS
Factors
Number of experiments equals 4N so
number of factors equals 4N-1
Typical design
•First row 0s.
•First column experiments 2 to 13 : generators
Factors
1
2
3
4
5
6
7
8
9
10
11
1
-
-
-
-
-
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-
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-
-
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2
+
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+
-
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+
+
+
-
+
3
Exp
erim 4
ents 5
+
+
-
+
-
-
-
+
+
+
-
-
+
+
-
+
-
-
-
+
+
+
+
-
+
+
-
+
-
-
-
+
+
6
+
+
-
+
+
-
+
-
-
-
+
7
+
+
+
-
+
+
-
+
-
-
-
8
-
+
+
+
-
+
+
-
+
-
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9
-
-
+
+
+
-
+
+
-
+
-
10
-
-
-
+
+
+
-
+
+
-
+
11
+
-
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-
+
+
+
-
+
+
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12
-
+
-
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-
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+
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-
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+
Plackett-Burman designs for 4N-1 factors
•Dummy factors
•Equivalence PB and fractional factorials
NUMBER OF EXPERIMENTS
Factorials : 2k
Fractional factorials : 2k-f
Plackett-Burman : 4N
NUMBER OF EXPERIMENTS
Example : 10 factors
Factorials : 1024
Fractional factorials : 16 (or 32 etc.)
Plackett-Burman : 12
AS THE NUMBER OF EXPERIMENTS IS
REDUCED SO THE INFORMATION E.G.
ABOUT INTERACTIONS IS REDUCED
Use screening experiments to reduce the
number of factors and then settle on key
factors for detailed models.
MODELLING
•Optimisation : what factors result in the best
taste?
Sometimes several criteria e.g. taste,
texture, cost, so several optima.
•Quantitative modelling : can we relate the
manufacturing process to the %protein in
wheat?
Many factors such as the origins of the
materials as well as the production, not
looking for an optimum.
CENTRAL COMPOSITE DESIGNS
Problems with factorial designs
•Two levels and so no squared terms.
•No replicates.
Relatively detailed models between response e.g.
taste and factors e.g. ingredients.
Replicate errors can be calculated.
Optima can be calculated.
Interactions can be calculated.
Number of experiments : 2k + 2k + 6
3 factors : 8 + 6 + 6 = 12
4 factors : 16 + 8 + 6 = 30
Characteristics
•Number of replicates in the centre
•Position of star or axial points (1 is not the only choice)
Reduce further by fractional factorials, for
example, 5 factors
•Quarter factorial : 8 experiments
•Axial points : 10 experiments
•Central points : 6 experiments
(Not good for interactions, half factorial
better, but might screen for interactions).