Experimental design (2)
Contents: Example of full factorial with diagnostics Fractional (reduced) factorial Plackett-Burman Composite Box-Behnken Blocking Optimization: steepest ascent and simplex Robustness Transformation of the responses (Box-Cox) Paul Geladi
New example
The composition of an HPLC elution liquid determines the efficiency of peak separation in the chromatogram.
3 factors at 2 levels, 23 full factorial. Response = peak separation
Effects = 2x coefficients
Hypotheses
• H0 : all coefficients are zero • H1 : some coefficients are not zero
• Coefficients that are not zero can be used to optimize the response • Others are just noise
Quantile or normal probability plot
ANOVA table
A nice interpretation of the coefficients is via contour plots
Repetition
• Up to now we have seen the full factorial for K factors at A levels giving AK runs • We have also seen the use of center points (replicates) and duplicates to 1 find a standard deviation for K -1 hypothesis testing 1 54 • We have seen ways of interpreting P the results
-1 -1 60
45
80
68
52 72 83
T
1
Fractional Factorial designs
• There is a problem with many factors • 6 factors at 2 levels is 64 runs • Are all these runs needed? • Do we need more than 1+6+15 coefficients? • There are clever ways of removing runs • Center points would still be added
Fractional Factorial designs
• 2(4-1) 8 runs • 2(5-1) 16 runs • 2(5-2) 8 runs • 2(6-1) 32 runs • 2(6-2) 16 runs • What is lost is higher-order interactions
Fractional Factorial designs
• Confounding • Aliases -1
1 1
K
54
45 68
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P
60
52 72
83
-1
-1
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1
Linearity, quadratic and cubic terms
• Factorial design gives main coefficients and two-way interactions • Flat surfaces and saddle points • Looking for a maximum needs more • Central composite design
Linearity, quadratic and cubic terms
y = b0 + b1x1 + b2x2 + b12 x1 x2 + b11x12 + b22x22 + e y = b0 + b1x1 + b2x2 + b12 x1 x2 + b11x12 + b22x22 + b111x13 + b222x23 + e
Central composite design
• Factorial design (may be reduced) 2K-A • Center points I • Star points 2K • Distance of star points
Star point
Factorial design
Rotatable star = 2k/4
Box-Behnken sides of cube + CP
What is best: composite, Box Behnken or 33?
Types of factors
Quantitative Qualitative Controlled Uncontrolled
Blocking
• When randomization fails • Two or more operators (shifts) carry out the design • A chromatogrpahic column has to be replaced halfway • Two (identical?) reactors used in parallel • Runs over a long time • Randomize inside blocks
Blocking variable
y = b0 + b1x1 + b2x2 + bbxb + e
The blocking variable may be an unused column in some designs
Important to remember
• Randomization • Blocking • Independent experiments • Experiments to determine standard deviation • Balanced designs are easier to understand • Uncontrolled variables
Selection of levels
• Physical limitations • Control of levels • Uncontrolled factors
How to continue?
• Extra points • New design in optimal region • Steepest ascent • Simplex optimization
The Design Sequence
1. Screening of as many variables as possible on 2 meaningful levels
2. Further optimization by steepest ascent (descent), simplex or extra runs 3. Response surface model on more levels at the optimum found in 2 4. Study of response surfaces, optima, models
Robustness simplified: robust min 60 50 50 40 30 60 20 50 °C 100 110 120 130 140 150 160 170 180 190 60 70 72 40
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Toyota
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Robustness simplified: a narrow maximum min 60 30 50 40 30 40 20 30 °C 100 110 120 130 140 150 160 170 180 190 40 50 60 80 70 Formula 1 20
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min 60 50 40 30 20
Robustness simplified: falling off a cliff 30 40 5 0 60 80
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°C 100 110 120 130 140 150 160 170 180 190
The model
• • • • • • • y = Xb + e may work better with: y2 y3 ln(y) 1/y y1/2
The model
• • • • y = Xb + e may work better with: y2, y3 ,ln(y) ,1/y ,y1/2 how do you know this?
The model
• • • • y = Xb + e general transformation ya try for a = -1, -0.5, 0.5, 2, 3 etc
(ya-1)/a
SS
• Box-Cox:
• Box-Cox plot
-2 -1 0 1 2
a