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Sizing Mixture (RSM) Designsfor Adequate Precision via
Fraction of Design Space (FDS)
Pat WhitcombStat-Ease, Inc.612 746 2036
Gary W. OehlertSchool of Statistics
U i it f Mi t
1
612.746.2036fax 612.746.2056
University of Minnesota612.625.1557
Sizing Mixture (RSM) Designs
Sizing Mixture (RSM) Designs
Review – Power to size factorial designs
Precision in place of powerPrecision in place of powerIntroduce FDS
Sizing designs for precisionTwo component mixtureThree component constrained mixture
2
Sizing designs to detect a difference
Summary
Sizing Mixture (RSM) Designs
Copyright 2009 Stat-Ease, Inc.
2
Sizing Factorial Designs
KnownFactors
UnknownFactors
T i i l
Screening KnownFactors
UnknownFactors
T i i l
ScreeningDuring screening and characterization the emphasis is on identifying factor effects
yes
Factor effectsand interactions
Response
Curvature?
Screening
no
Trivialmany
Vital fewCharacterization
Optimizationyes
Factor effectsand interactions
Response
Curvature?
Screening
no
Trivialmany
Vital fewCharacterization
Optimization
is on identifying factor effects.
What are the important design factors?
For this purpose power is an ideal metric to evaluate design suitability.
3
Surfacemethods
Confirm? Backup
Celebrate!
no
yes
Verification
Surfacemethods
Confirm? Backup
Celebrate!
no
yes
Verification
Sizing Mixture (RSM) Designs
12
One Factor
Factorial Design – Power23 Full Factorial Δ=2 and σ=1
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12
9
10
11
R1
-1.00
-0.50
0.00
0.50
1.00
-1.00
-0.50
0.00
0.50
1.00
8
9
10
R
1
A: A B: B Δ
4
-1.00 -0.50 0.00 0.50 1.00
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A: ASizing Mixture (RSM) Designs
Copyright 2009 Stat-Ease, Inc.
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Factorial Design – Power23 Full Factorial Δ=2 and σ=1
Power is reported at a 5.0% alpha level to detect the specified signal/noise ratio.Recommended power is at least 80%.
R1Signal (delta) = 2.00 Noise (sigma) = 1.00 Signal/Noise (delta/sigma) = 2.00
5
A B C57.2 % 57.2 % 57.2 %
The following three slides dig into how power is computed.
Sizing Mixture (RSM) Designs
One Replicate of 23 Full Factorial C = (XTX)-1 matrix
The design determines the standard error of the coefficient:
0.125 0 0 0 0 0 0 0⎛⎜
⎞⎟
C
0
0
0
0
0
0
0.125
0
0
0
0
0
0
0.125
0
0
0
0
0
0
0.125
0
0
0
0
0
0
0.125
0
0
0
0
0
0
0.125
0
0
0
0
0
0
0.125
0
0
0
0
0
0
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎟⎟⎟⎟⎟⎟⎟⎟⎟
=
6
( ) ( )i i i
i 2 2i ii
t-value = SE c ˆ 0.125 ˆβ β β
= =β σ σ
Sizing Mixture (RSM) Designs
0 0 0 0 0 0 0 0.125⎜⎝
⎟⎠
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NonCentrality Parameter23 Full Factorial Δ=2 and σ=1
The reference t distribution assumes the null hypothesis of Δ = 0. The noncentrality parameter (2.828) defines the t distribution under the alternate hypothesis of Δ = 2
( )( )
ii
i 2 2ii ii
2
2noncentrality = c ˆ c ˆ
1
0 125 1
Δβ
=σ σ
=
distribution under the alternate hypothesis of Δ = 2.
7
( )( )20.125 1
1 2.8280.3536
= =
Sizing Mixture (RSM) Designs
Factorial Design – Power23 Full Factorial Δ=2 and σ=1
noncentral tα=0.05,df=4 with noncentrality parameter of 2.828
Power = 57.2%
2.828
8Sizing Mixture (RSM) Designs
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Factorial Design – PowerTwo Replicates of 23 Full Factorial Δ=2 and σ=1
Power is reported at a 5.0% alpha level to detect the specified signal/noise ratio.Recommended power is at least 80%.
R1Signal (delta) = 2.00 Noise (sigma) = 1.00 Signal/Noise (delta/sigma) = 2.00
9
A B C95.6 % 95.6 % 95.6 %
Sizing Mixture (RSM) Designs
Two Replicates of 23 Full FactorialC = (XTX)-1 matrix
The design determines the standard error of the coefficient:0.0625 0 0 0 0 0 0 0⎛
⎜⎞⎟
C
0
0
0
0
0
0
0.0625
0
0
0
0
0
0
0.0625
0
0
0
0
0
0
0.0625
0
0
0
0
0
0
0.0625
0
0
0
0
0
0
0.0625
0
0
0
0
0
0
0.0625
0
0
0
0
0
0
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎟⎟⎟⎟⎟⎟⎟⎟⎟
=
10
( ) ( )i i i
i 2 2i ii
t-value = SE c ˆ 0.0625 ˆ
β β β= =
β σ σ
Sizing Mixture (RSM) Designs
0 0 0 0 0 0 0 0.0625⎜⎝
⎟⎠
Copyright 2009 Stat-Ease, Inc.
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NonCentrality ParameterTwo Replicates of 23 Full Factorial Δ=2 and σ=1
ii 2
Δβ
( )( )
ii 2 2
ii ii
2
2noncentrality = c ˆ c ˆ
1
0.0625 1
1 4 0
β=
σ σ
=
11
4.00.25
= =
Sizing Mixture (RSM) Designs
Factorial Design – PowerTwo Replicates of 23 Full Factorial Δ=2 and σ=1
noncentral tα=0.05,df=12 with noncentrality parameter of 4.0
Power = 95.6%
4.0
12Sizing Mixture (RSM) Designs
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Conclusions
Factorial DOE
D i i dDuring screening and characterization (factorials) emphasis is on identifying factor effects.
What are the important design factors?
For this purpose power is an
13
For this purpose power is an ideal metric to evaluate design suitability.
Sizing Mixture (RSM) Designs
Sizing Mixture (RSM) Designs
Review – Power to size factorial designs
Precision in place of powerPrecision in place of powerIntroduce FDS
Sizing designs for precisionTwo component mixtureThree component constrained mixture
14
Sizing designs to detect a difference
Summary
Sizing Mixture (RSM) Designs
Copyright 2009 Stat-Ease, Inc.
8
Sizing Response Surface Designs
KnownFactors
UnknownFactors
T i i l
Screening KnownFactors
UnknownFactors
T i i l
Screening
yes
Factor effectsand interactions
Response
Curvature?
Screening
no
Trivialmany
Vital fewCharacterization
Optimizationyes
Factor effectsand interactions
Response
Curvature?
Screening
no
Trivialmany
Vital fewCharacterization
Optimization
When the goal is optimization emphasis is the fitted surface.
How well does the surface represent true behavior?
F hi i i
15
Surfacemethods
Confirm? Backup
Celebrate!
no
yes
Verification
Surfacemethods
Confirm? Backup
Celebrate!
no
yes
Verification
For this purpose precision (FDS) is an good metric to evaluate design suitability.(Assuming model adequacy; i.e. insignificant lack of fit.)
Sizing Mixture (RSM) Designs
1. Estimate the designed for polynomial well.
2. Give sufficient information to allow a test for lack of fit.H i d i i t th ffi i t i th d l
Design Properties
Have more unique design points than coefficients in the model.Provide an estimate of “pure” error.
3. Remain insensitive to outliers, influential values and bias from model misspecification.
4. Be robust to errors in control of the component levels.
5. Provide a check on model assumptions, e.g., normality of errors.
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6. Generate useful information throughout the region of interest,i.e., provide a good distribution of .
7. Do not contain an excessively large number of trials.( ) 2ˆVar Y σ
Sizing Mixture (RSM) Designs
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FDSFraction of Design Space
Fraction of Design Space:Calculates the volume of the design space havingCalculates the volume of the design space having a prediction variance (PV) less than or equal to a specified value.The ratio of this volume to the total volume of the design volume is the fraction of design space.Produces a single plot showing the cumulative f ti f th d i th i (f
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fraction of the design space on the x-axis (from zero to one) versus the PV on the y-axis.
Sizing Mixture (RSM) Designs
FDS – PVFraction of Design Space
Prediction Variance:
( )ˆvar y
PV is a function of:
x0 – the location in the design space (i.e. the x coordinates for all model terms).
( )( ) ( ) 10
0 0 02
var −= = T T
yPV x x X X x
s
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X – the experimental design (i.e. where the runs are in the design space).
Sizing Mixture (RSM) Designs
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FDS – StdErrFraction of Design Space
Prediction standard error of the expected value:
( )ˆ( )
( ) ( )
( ) ( )0
100 0 02
ˆ0 0
ˆvar −= =
= =
T T
y
yPV x x X X x
s
sStdErr x PV x
s
19Sizing Mixture (RSM) Designs
FDS PlotFraction of Design Space
1. Pick random points in the design space.
2 Calculate the standard error of the expected value2. Calculate the standard error of the expected value
3. Plot the standard error as a fraction of the design space.
( )01y T T
0 0
SEx X X x
s−
=
FDS Graph
0.800.901.00
20Fraction of Design Space
StdE
rr M
ean
0.00 0.25 0.50 0.75 1.00
0.000.100.200.300.400.500.600.70
Sizing Mixture (RSM) Designs
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Simplex-LatticeAugmented and 4 Replicates
A: A1.00
0.572
FDS Graph1.00
0.00 0.00
0.47
0.57 0.57
0.57
2
StdE
rr M
ean
0.25
0.50
0.75
0.47
0.57
Mix section 2 21
B: B1.00
C: C1.000.00
StdErr of Design
0.470.470.570.57
0.572 2
Fraction of Design Space
0.00 0.25 0.50 0.75 1.00
0.00
Sizing Mixture (RSM) Designs
Review – Power to size factorial designs
Precision in place of powerPrecision in place of powerIntroduce FDS
Sizing designs for precisionTwo component mixtureThree component constrained mixture
22
Sizing designs to detect a difference
Summary
Sizing Mixture (RSM) Designs
Copyright 2009 Stat-Ease, Inc.
12
Two Component MixtureLinear Model
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Two Component Mix
4
5
6
R1
2
2
Sizing Mixture (RSM) Designs 23
3
0
1
0.25
0.75
0.5
0.5
0.75
0.25
1
0
Actual A
Actual B
2
Two Component MixtureLinear Model
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Two Component MixThe solid center line is the fitted model;is the expected value or mean
prediction.y
4
5
6
R1
The curved dotted lines are the computer generated confidence limits, or the actual precision.
d Is the half-width of the desired confidence interval, or the desired precision. It is used to create the outer straight lines.
ˆ ±y d
Sizing Mixture (RSM) Designs 24
Note: The actual precision of the fitted value depends on where we are predicting.
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0
1
0.25
0.75
0.5
0.5
0.75
0.25
1
0
Actual A
Actual B
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Two Component MixtureLinear Model
Evaluate the standard error as a fraction of design space for this design.
4
5
6
7
R1
Two Component Mix
2
2
Sizing Mixture (RSM) Designs
25
3
01
0.250.75
0.50.5
0.750.25
10
Actual AActual B
2
Two Component MixtureLinear Model
Click on “Evaluate” and then “Graphs”:
Sizing Mixture (RSM) Designs 26
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Two Component MixtureLinear Model
Want a linear surface to represent the true response value within ± 0.64 with 95% confidence.
The overall standard deviation for this response is 0.55.
Enter:
d = 0 64
Sizing Mixture (RSM) Designs 27
d = 0.64
s = 0.55
a (α) = 1 – 0.95 = 0.05
Two Component MixtureLinear Model
Only 53% of the design space is precise enough to predict the mean within ± 0.64.
Sizing Mixture (RSM) Designs 28
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Define PrecisionHalf Width of the Confidence Interval
Confidence interval on the expected value:
The mean response is estimated and the precision of the
Statisticaldetail
The mean response is estimated and the precision of the estimate is quantified by a confidence interval:
We will use the half-width of the confidence interval (d) to define the precision desired:
( )ydfy t s ˆ,2
ˆα±
p
Mix section 4 29
( )ˆ ˆ,2,2
= =y ydfdf
dd t s or stαα
What Precision is Needed?Mean Confidence Interval Half-Width
Statisticaldetail
InputHalf-width of confidence interval: d
( )
( )
ˆ,2
1
ˆ
−
±
= ydf
T T
y d
d t s
X X
α
Estimate of standard deviation: s
Mix section 4 30
( )
( ) ( )
ˆ 0 0
1ˆ0 0
−
=
= =
T Ty
y T T
s s x X X x
sStdErr Mean FDS x X X x
s
Copyright 2009 Stat-Ease, Inc.
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Two Component MixtureLinear Model
Want a linear surface to represent the true response value within ± 0.64 with 95% confidence.
Statisticaldetail
The overall standard deviation for this response is 0.55.
.05 ,52
ˆ
2
For 95% confidence 2.571, 0.64 & 0.55
0.64 0.252.571
= = =
= = =ydf
t d s
dstα
Mix section 4 31
( )
,2
ˆ 0.25 0.450.55
= = =
df
ysStdErr Mean FDS
s
FDS – StdErr Mean PlotFraction of Design Space
1. Pick random points in the design space.
2 Calculate the standard error of the expected value
Statisticaldetail
2. Calculate the standard error of the expected value
3. Plot the standard error as a fraction of the design space.
( )01y T T
0 0
SEx X X x
S−
=
FDS Graph
0.901.00
Mix section 4 32Fraction of Design Space
StdE
rr
0.00 0.25 0.50 0.75 1.00
0.000.100.200.300.400.500.600.700.80
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Two Component MixtureLinear Model
53% is precise enough, i.e.53% has a StdErr ≤ 0.45
Sizing Mixture (RSM) Designs 33
d = 0.64s = 0.55α = 0.05
Two Component MixtureLinear Model
53% of the design space h h d i d i i
7
Two Component Mix
53%has the desired precision, i.e. is inside the solid straight (red) lines.
4
5
6
R1 ˆ ±y d
53%
Sizing Mixture (RSM) Designs 34
3
0
1
0.25
0.75
0.5
0.5
0.75
0.25
1
0
Actual A
Actual B
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Sizing for PrecisionFDS ≥ ?
How good is good enough? Rules of thumb:• For exploration want FDS ≥ 80%• For verification want FDS ≥ 90% or higher!
What can be done to improve precision?• Manage expectations; i.e. increase d• Decrease noise; i.e. decrease s• Increase risk of Type I error; i.e. increase alphayp ; p• Increase the number of runs in the design
Sizing Mixture (RSM) Designs 35
Sizing Mixture (RSM) Designs
Review – Power to size factorial designs
Precision in place of powerPrecision in place of powerIntroduce FDS
Sizing designs for precisionTwo component mixtureThree component constrained mixture
36
Sizing designs to detect a difference
Summary
Sizing Mixture (RSM) Designs
Copyright 2009 Stat-Ease, Inc.
19
Mixture Simplex Quadratic Model
Most of the action
y 4A 4B 8C 16AC= + + +12
14
Most of the action occurs on the A-C edge because of the AC coefficient of 16. The quadratic coefficient of 16 means that the response is 4 units
A (1) B (0) C (1)
0
2
4
6
8
10
R
1
Sizing Mixture (RSM) Designs 37
response is 4 units higher at A=0.5, C=0.5 than one would expect with linear blending.
A (0)
B (1)
C (0)
Mixture Simplex Quadratic Model
14
This illustrates the
y 4A 4B 8C 16AC= + + +
4
6
8
10
12
4β
C 8β =
AC14
4β =
This illustrates the quadratic blending along the A-C edge as C increases from zero to one.
Sizing Mixture (RSM) Designs 38
0
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C
A 4β =
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Mixture ConstrainedDesign for Quadratic Model
0.0 ≤ A ≤ 1.00.0 ≤ B ≤ 1.0
A: A1.000
0.0 ≤ C ≤ 0.1Total = 1.0
0.000 0.000
Sizing Mixture (RSM) Designs 39
B: B1.000
C: C1.0000.000
StdErr of Design
Mixture ConstrainedQuadratic Model (1 replicate)
Power at 5 % alpha level for effect ofTerm StdErr* VIF Ri-Squared 1 s 2 s 3 s
% % %A 0.87 3.44 0.7094 5.0 % 5.0 % 5.0 %B 0.87 3.44 0.7094 5.0 % 5.0 % 5.0 %C 233.34 2722.29 0.9996 5.0 % 5.0 % 5.0 %
AB 2.86 2.19 0.5442 22.9 % 67.3 % 94.7 %AC 258.98 1092.98 0.9991 5.0 % 5.0 % 5.0 %BC 258.98 1092.98 0.9991 5.0 % 5.0 % 5.0 %
*Basis Std. Dev. = 1.0
Sizing Mixture (RSM) Designs 40
Power is bottomed out at alpha!
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Define PrecisionHalf Width of the Confidence Interval
6. Generate useful information throughout the region of interest.
Question: Will predictions, using the quadratic model from this design, be precise enough for our purposes?
To know the truth requires an infinite number of runs; most likely this will exceed our budget.
So the question is how precisely do we need to estimate the response?
Sizing Mixture (RSM) Designs 41
estimate the response?
The trade off is more precision requires more runs.
Define PrecisionHalf Width of the Confidence Interval
Confidence interval on the expected value:
The mean response is estimated and the precision of theThe mean response is estimated and the precision of the estimate is quantified by a confidence interval:
We will use half-width of the confidence interval (d) to define the precision desired:
( )ydfy t s ˆ,2
ˆα±
Sizing Mixture (RSM) Designs 42
p
( )ˆ ˆ,2,2
= =y ydfdf
dd t s or stαα
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InputHalf-width of confidence interval: d
What Precision is Needed?Confidence Interval Half-Width
Estimate of standard deviation: s
( )
ˆ
,2
1−
=
=
ydf
T T
dst
s s x X X x
α
Sizing Mixture (RSM) Designs 43
( )
( ) ( )
ˆ 0 0
1ˆ0 0
−
=
= =
y
y T T
s s x X X x
sStdErr Mean FDS x X X x
s
Mixture ConstrainedQuadratic Model (1 replicate)
Want quadratic surface to represent the true response value within ±10 with 95% confidence.p
The standard deviation for this response is 7.8.
.05 ,72
ˆ
,2
For 95% confidence 2.365, 10 & 7.8
10 4.232.365
= = =
= = =ydf
t d s
dstα
Sizing Mixture (RSM) Designs 44
( )
,2
ˆ 4.23 0.547.8
= = =
df
ysStdErr Mean FDS
s
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Mixture ConstrainedQuadratic Model (1 replicate)
Only 58% of the design space has StdErr ≤ 0.54
d = 10
Sizing Mixture (RSM) Designs 45
d 0s = 7.8α = 0.05
Mixture ConstrainedQuadratic Model (2 replicates)
Power at 5 % alpha level for effect ofTerm StdErr* VIF Ri-Squared 1 s 2 s 3 s
% % %A 0.62 3.44 0.7094 5.0 % 5.0 % 5.0 %B 0.62 3.44 0.7094 5.0 % 5.0 % 5.0 %C 164.99 2722.29 0.9996 5.0 % 5.0 % 5.0 %
AB 2.02 2.19 0.5442 47.1 % 96.5 % 99.9 %AC 183.12 1092.98 0.9991 5.0 % 5.0 % 5.0 %BC 183.12 1092.98 0.9991 5.0 % 5.0 % 5.0 %
*Basis Std. Dev. = 1.0
Sizing Mixture (RSM) Designs 46
Adding a replicate does little to increase power!
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Mixture ConstrainedWhy Does Power Not Increase?
Quadratic, linear or combinations of them model response. Model Coefficients are Correlated!
14
4
6
8
10
12
14
Linear
Quadratic
Sizing Mixture (RSM) Designs 47
Individual coefficients can not be resolved!
0
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Constraint
C
Mixture ConstrainedQuadratic Model (2 replicates)
Want quadratic surface to represent the true response value within ± 10 with 95% confidence.
The overall standard deviation for this response is 7.8.
.05 ,2
ˆ
2
,2
0For 95% confidence , 10 & 7.8
10 4.792.086
2.086= = =
= = =ydf
t d s
dstα
Sizing Mixture (RSM) Designs 48
( )
,2
ˆ 4.79 0.617.8
= = =ysStdErr Mean FDS
s
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Mixture ConstrainedQuadratic Model (2 replicates)
100% of the design space has StdErr ≤ 0.61
Sizing Mixture (RSM) Designs 49
Precision Depends On
The size of the confidence interval half-width (d):A larger half-width (d) increases the FDS.
The size of the experimental error σ:A smaller s increases the FDS.
The α risk chosen:A larger α increases the FDS.
Choose design appropriate to the problem:
Sizing Mixture (RSM) Designs 50
Choose design appropriate to the problem:Size the design for the precision required.
Copyright 2009 Stat-Ease, Inc.
26
Conclusions
Factorial DOE Mixture Design andResponse Surface Methods
During screening and characterization (factorials) emphasis is on identifying factor effects.
What are the important design factors?
When the goal is optimization (usually the case for mixture design & RSM) emphasis is on the fitted surface.
How well does the surface represent true behavior?
Sizing Mixture (RSM) Designs 51
For this purpose power is an ideal metric to evaluate design suitability.
For this purpose precision (FDS) is a good metric to evaluate design suitability.
Sizing Mixture (RSM) Designs
Review – Power to size factorial designs
Precision in place of powerPrecision in place of powerIntroduce FDS
Sizing designs for precisionTwo component mixtureThree component constrained mixture
52
Sizing designs to detect a difference
Summary
Sizing Mixture (RSM) Designs
Copyright 2009 Stat-Ease, Inc.
27
Detecting a Difference of Δwherever it may occur
Δ
Sizing Mixture (RSM) Designs 53
What is the probability of finding a difference ≥ Δif it occurs between any two points in the design space?
FPDS – StdErr Diff PlotFraction of Paired Design Space
1. Pick random pairs of points in the design space.
2 Calculate the standard error of the difference2. Calculate the standard error of the difference
3. Plot the standard error as a fraction of the design space.
( ) ( ) ( )( )1 21 1 1ˆ ˆy y T T T T T T
1 1 2 2 1 2
SEx X X x x X X x 2 x X X x
s− − −− = + −
FPDS Graph
1.60
1.80
2.00
Sizing Mixture (RSM) Designs 54Fraction of Paired Design Space
Std
Err
Diff
0.000 0.250 0.500 0.750 1.000
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
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Revisit – Mixture ConstrainedDesign for Quadratic Model
0.0 ≤ A ≤ 1.00.0 ≤ B ≤ 1.0
A: A1.000
0.0 ≤ C ≤ 0.1Total = 1.0
0.000 0.000
Sizing Mixture (RSM) Designs 55
B: B1.000
C: C1.0000.000
StdErr of Design
Mixture ConstrainedQuadratic Model (1 replicate)
Want to detect a difference of 10 on a quadratic surface with 95% confidence.
The overall standard deviation this for response is 7.8.
.05 ,72For 95% confidence 2.365, 10 & 7.8
10 4.232 365Δ
= Δ = =
Δ= = =
t s
st
Sizing Mixture (RSM) Designs 56
( )
,22.365
4.23 0.547.8
Δ= = =
dft
sStdErr Diff FPDSs
α
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Mixture ConstrainedQuadratic Model (1 replicate)
Only 37% of the design space has StdErr ≤ 0.54
Sizing Mixture (RSM) Designs 57
Δ = 10s = 7.8α = 0.05
Mixture ConstrainedQuadratic Model (2 replicates)
Want to detect a difference of 10 on a quadratic surface with 95% confidence.
The overall standard deviation this for response is 7.8.
.05 ,2
2
20For 95% confidence , 10 & 7.8
10 4.792.086
2.086
Δ
= Δ = =
Δ= = =
df
t s
stα
Sizing Mixture (RSM) Designs 58
( )
,2
4.79 0.8
.617
Δ= = =
df
sStdErr Diff FPDSs
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Mixture ConstrainedQuadratic Model (2 replicates)
86% of the design space has StdErr ≤ 0.61
Sizing Mixture (RSM) Designs 59
Detecting a Difference of Δ Depends On
The size of the difference (Δ):A larger difference (Δ) increases the FPDS.
The size of the experimental error σ:A smaller σ increases the FPDS
The α risk chosen:A larger α increases the FPDS.
Choose design appropriate to the problem:Si th d i t d t t diff f i t t
Sizing Mixture (RSM) Designs 60
Size the design to detect a difference of interest.
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Conclusions
Factorial DOE Mixture Design andResponse Surface Methods
During screening and characterization (factorials) emphasis is on identifying factor effects.
What are the important design factors?
When the goal is to detect a change in the response emphasis is on differences over the fitted surface.
Does a difference ≥ D occur in the design space?
Sizing Mixture (RSM) Designs 61
For this purpose power is an ideal metric to evaluate design suitability.
For this purpose precision (FPDS) is a good metric to evaluate design suitability.
Sizing Mixture (RSM) Designs
Review – Power to size factorial designs
Precision in place of powerPrecision in place of powerIntroduce FDS
Sizing designs for precisionTwo component mixtureThree component constrained mixture
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Sizing designs to detect a difference
Summary
Sizing Mixture (RSM) Designs
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Summary
Two criteria covered:
1. Precision of expected value, StdErr Mean:1. Precision of expected value, StdErr Mean:Use the StdErr Mean when the purpose of the DOE is to quantify shape; i.e. functional design (e.g. optimization).
2. Precision to detect a difference, StdErr Diff:Use the StdErr Diff when the purpose of the DOE is t d t t h i th d i
Sizing Mixture (RSM) Designs 63
to detect change in the design space.
Summary
Two criteria not covered:
1. Precision of predicted value, StdErr Pred:1. Precision of predicted value, StdErr Pred:Use the StdErr Pred when the purpose of the DOE is to develop a model for prediction.
2. Precision to form a tolerance interval, TI Multiplier:Use the TI Multiplier when the purpose of the DOE is to form a tolerance interval at a point in the d i
Sizing Mixture (RSM) Designs 64
design space.(new in Design-Expert version 8)
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References
[1] Gary W. Oehlert (2000), A First Course in Design and Analysis of Experiments, W. H. Freeman and Company.
[2] Douglas C. Montgomery (2005), 6th edition, Design and Analysis of Experiments, John Wiley and Sons IncWiley and Sons, Inc.
[3] Gary Oehlert and Patrick Whitcomb (2001), Sizing Fixed Effects for Computing Power in Experimental Designs, Quality and Reliability Engineering International, July 27, 2001
[4] Alyaa R. Zahran, Christine M. Anderson-Cook and Raymond H. Myers, Fraction of Design Space to Assess Prediction, Journal of Quality Technology, Vol. 35, No. 4, October 2003
[5] Heidi B. Goldfarb, Christine M. Anderson-Cook, Connie M. Borror and Douglas C. Montgomery, Fraction of Design Space plots for Assessing Mixture and Mixture-Process Designs, Journal of Quality Technology, Vol. 36, No. 2, October 2004.
[6] Steven DE Gryze, Ivan Langhans and Martina Vandebroek, Using the Intervals for Prediction: A Tutorial on Tolerance Intervals for Ordinary Least-Squares Regression,
65
y q g ,Chemometrics and Intelligent Laboratory Systems, 87 (2007) 147 – 154
[7] Raymond H. Myers, Douglas C. Montgomery and Christine M. Anderson-Cook (2009), 3rd
edition, Response Surface Methodology, John Wiley and Sons, Inc.[8] John A. Cornell (2002), 3rd edition, Experiments with Mixtures, John Wiley and Sons, Inc.[9] Wendell F. Smith (2005), Experimental Design for Formulation, ASA-SIAM Series on
Statistics and Applied probability.
Sizing Mixture (RSM) Designs
Sizing Mixture (RSM) Designs
Thank You for AttendingThank You for Attendinggg
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