what’s new in design-expert version 7 mixture and combined design pat whitcomb march 25, 2006
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What’s New in Design-Expert version 7 Mixture and Combined Design Pat Whitcomb March 25, 2006. What’s New. General improvements Design evaluation Diagnostics Updated graphics Better help Miscellaneous Cool New Stuff Factorial design and analysis Response surface design - PowerPoint PPT PresentationTRANSCRIPT
Design-Expert version 7 1
What’s New inDesign-Expert version 7
Mixture and Combined Design
Pat WhitcombMarch 25, 2006
Design-Expert version 7 2
What’s New
General improvements Design evaluation Diagnostics Updated graphics Better help Miscellaneous Cool New Stuff
Factorial design and analysis
Response surface design
Mixture design and analysis Combined design and analysis
Design-Expert version 7 3
Mixture Design
More components Simplex lattice 2 to 30 components (v6 2 to 24) Screening 6 to 40 components (v6 6 to 24)
Detect inverted simplex Upper bounded pseudo values: U_Pseudo and
L_Pseudo
New mixture design “Historical Data”
Coordinate exchange
Design-Expert version 7 4
Inverted Simplex
When component proportions are restricted by upper bounds it can lead to an inverted simplex.
For example:x1 ≤ 0.4x2 ≤ 0.6x3 ≤ 0.3
90
50
70
30
10
X1
X2 X3
Design-Expert version 7 5
A: x11.000
B: x21.000
C: x31.000
0.000 0.000
0.000
22 22
22
22
22 22
Inverted Simplex3 component L_Pseudo
Using lower bounded L_Pseudo values leads to the following inverted simplex.
Open “I-simplex L_P.dx7” andmodel the response. 0.50 in L_Pseudo
Design-Expert version 7 6
Inverted Simplex3 component U_Pseudo (page 1 of 2)
1. Build a new design and say “Yes” to “Use previous design info”.
2. Change “User-Defined” to “Simplex Centroid”.
3. When asked say “Yes” to switch to upper bounded pseudo values “U_Pseudo.
Design-Expert version 7 7
Inverted Simplex3 component U_Pseudo (page 1 of 3)
4. Change the replicates from 4 to 6 and
5. Right click on the “Block”column header and“Display Point Type”
Design-Expert version 7 8
Inverted SimplexUpper Bounded Pseudo Values
The high value becomes 0 and the low value becomes 1!A: x11.000
B: x21.000
C: x31.000
0.000 0.000
0.000
22 22
22
22
22 22
0 in U_Pseudo1 in U_Pseudo
Design-Expert version 7 9
Inverted SimplexUpper Bounded Pseudo Values
The upper value becomes 0 and the lower value 1!
U_Pseudo values:
Real PseudoLi Ui Li Ui
x1 0.1 0.4 1 0x2 0.3 0.6 1 0x3 0.0 0.3 1 0
i
i
i ii
11
22
33
U Real U_Pseudo
U 1
U Xu '
1.3 1
0.4 Xu '
0.3
0.6 Xu '
0.3
0.3 Xu '
0.3
Design-Expert version 7 10
Inverted Simplex3 component U_Pseudo
Go to the “Evaluation” and view the design space:A: x11.000
B: x21.000
C: x31.000
0.000 0.000
0.00022
22
22
22
22
22
Design-Expert version 7 11
Inverted SimplexNote the Improved Values
Coding is U_Pseudo. Term StdErr** VIF Ri-Sq
A 0.69 1.74 0.4255 B 0.69 1.74 0.4255 C 0.69 1.74 0.4255
AB 3.45 1.94 0.4844 AC 3.45 1.94 0.4844 BC 3.45 1.94 0.4844
ABC 27.03 1.75 0.4300**Basis Std. Dev. = 1.0
A: x11.000
B: x21.000
C: x31.000
0.000 0.000
0.00022
22
22
22
22
22
Coding is L_Pseudo. Term StdErr**VIF Ri-Sq
A 26.33 1550.78 0.9994B 26.33 1550.78 0.9994C 26.33 1550.78 0.9994
AB 104.19 2686.10 0.9996AC 104.19 2686.10 0.9996BC 104.19 2686.10 0.9996
ABC 216.27 455.72 0.9978**Basis Std. Dev. = 1.0
A: x11.000
B: x21.000
C: x31.000
0.000 0.000
0.000
22 22
22
22
22 22
Design-Expert version 7 12
Inverted Simplex 3 component U_Pseudo
1. Simulate the response using “I-simplex U_P.sim”
2. Model the response.A: x10.100
B: x20.300
C: x30.000
0.300 0.600
0.400
R1
5.0
6.0
7.0
8.0
8.0
9.0
9.0
10.011.0 12.0
22
22 22
22 22
22
Design-Expert version 7 13
A (1.000)B (0.000)
C (1.000)
4
6
8
10
12
14
R1
A (0.000)
B (1.000)
C (0.000)
Inverted Simplex Upper vs Lower Bounded Pseudo Values
Low becomes high and high becomes low:U_Pseudo L_Psuedo
A (1.000)B (0.000)
C (1.000)
4
6
8
10
12
14
R1
A (0.000)
B (1.000)
C (0.000)
Design-Expert version 7 14
Mixture Design“Historical Data”
Design-Expert version 7 15
D-optimal DesignCoordinate versus Point Exchange
There are two algorithms to select “optimal” points for estimating model coefficients:
Point exchange Coordinate exchange
Design-Expert version 7 16
D-optimal Coordinate Exchange*
Cyclic Coordinate Exchange Algorithm
1. Start with a nonsingular set of model points.
2. Step through the coordinates of each design point determining if replacing the current value increases the optimality criterion. If the criterion is improved, the new coordinate replaces the old. (The default number of steps is twelve. Therefore 13 levels are tested between the low and high factor constraints; usually ±1.)
3. The exchanges continue and cycle through the model points until there is no further improvement in the optimality criterion.
* R.K. Meyer, C.J. Nachtsheim (1995), “The Coordinate-Exchange Algorithm for Constructing Exact Optimal Experimental Designs”, Technometrics, 37, 60-69.
Design-Expert version 7 17
Mixture Analysis
Cox Model; a new mixture parameterization
New screening tools for linear models: Constraint Region Bounded Component Effects for
Piepel Direction Constraint Region Bounded Component Effects for
Cox Direction Constraint Region Bounded Component Effects for
Orthogonal Direction Range Adjusted Component Effects for Orthogonal
Direction (this is the only measure in v6)
Design-Expert version 7 18
Mixture Analysis Cox Model
Cox model option for mixtures: May be more informative for formulators when they are interested in a particular composition.
Design-Expert version 7 19
Screening DesignsComponent Effects Concepts
Basis for screening is a linear model:
In a mixture it is impossible to change one component while holding the others fixed.
Changes in the component of interest must be offset by changes in the other components (so the components still sum to their total).
Choosing a direction through the mixture space to vary to component of interest defines how the offsetting changes are made.
1 1 2 2 3 3 q qx x x x
Design-Expert version 7 20
Screening DesignsComponent Effect Directions
Three directions in which component effects are assessed:1. Orthogonal2. Cox3. Piepel
The most meaningful direction (or directions) to use for computing effects for a particular mixture DOE depends on the shape of the mixture region.
In an unconstrained simplex theCox and Piepel directions are the same.
In a constrained simplex they’re not!(Remember the ABS Pipe example.)
Design-Expert version 7 21
Screening DesignsComponent Effect Directions
Example (equation in actuals):
A (0.800)B (0.100)
C (0.800)
7.50
8.00
8.50
9.00
9.50
10.00
R1
A (0.100)
B (0.800)
C (0.100)
A: X11.000
B: X21.000
C: X31.000
0.000 0.000
0.000
R1
8.00
8.50
9.00
9.50
1 2 3y 10x 8x 6x
Design-Expert version 7 22
Screening DesignsOrthogonal Direction Component Effect
Trace (Orthogonal)
Deviation from Reference Blend (L_Pseudo Units)
R1
-0.143 -0.071 0.000 0.071 0.143
7.50
8.00
8.50
9.00
9.50
10.00
A
A
B B
C
C
1
2
X
X X 3
Design-Expert version 7 23
Orthogonal Component EffectsRange Adjusted versus Constraint Bounded
Bounded AdjustedComponent Effect Effect
A-X1 0.60 1.80B-X2 0.00 0.00C-X3 -0.30 -0.30
In constrained mixtures the “Adjusted”effect is almost never realized.
Design-Expert version 7 24
Orthogonal Component GradientsConstraint Bounded
GradientComponent at Base Pt.
A-X1 3.00B-X2 0.00C-X3 -3.00
A has a positive slopeB has no slopeC has a negative slope
Trace (Orthogonal)
Deviation from Reference Blend (L_Pseudo Units)
R1
-0.143 -0.071 0.000 0.071 0.143
7.50
8.00
8.50
9.00
9.50
10.00
A
A
B B
C
C
Slope = 3.0
Design-Expert version 7 25
Screening DesignsCox Direction Component Effect
Trace (Cox)
Deviation from Reference Blend (L_Pseudo Units)
R1
-0.286 -0.143 0.000 0.143 0.286
7.50
8.00
8.50
9.00
9.50
10.00
A
A
B
B
C
C
1
2
X
X X 3
Design-Expert version 7 26
Cox Component EffectsConstraint Bounded
GradientComponent at Base Pt.
A-X1 2.50B-X2 -0.91C-X3 -2.94
ComponentComponent Effect
A-X1 1.00B-X2 -0.33C-X3 -0.29
Trace (Cox)
Deviation from Reference Blend (L_Pseudo Units)
R1
-0.286 -0.143 0.000 0.143 0.286
7.50
8.00
8.50
9.00
9.50
10.00
A
A
B
B
C
C
Slope = 2.5
Design-Expert version 7 27
Screening DesignsPiepel Direction Component Effect
Trace (Piepel)
Deviation from Reference Blend (L_Pseudo Units)
R1
-0.500 -0.250 0.000 0.250 0.500
7.50
8.00
8.50
9.00
9.50
10.00
A
A
B
B
C
C
1
2
X
X X 3
Design-Expert version 7 28
Piepel Component EffectsConstraint Bounded
GradientComponent at Base Pt.
A-X1 2.25B-X2 -1.43C-X3 -2.92
ComponentComponent Effect
A-X1 1.35B-X2 -1.00C-X3 -0.29
Trace (Piepel)
Deviation from Reference Blend (L_Pseudo Units)
R1
-0.500 -0.250 0.000 0.250 0.500
7.50
8.00
8.50
9.00
9.50
10.00
A
A
B
B
C
C
Slope = 2.25
Design-Expert version 7 29
SummaryComponent Effect Directions
1. Orthogonal: The direction for the ith component along a line that is orthogonal to space spanned by the other q-1 components. Appropriate only for simplex regions.
2. Cox: The direction for the ith component along a line joining the reference blend to the ith vertex (in real values). The line is also extended in the opposite direction to its end point. Appropriate for all regions.
3. Piepel: The same as the Cox direction after applying the pseudo component transformation. Appropriate for all regions.
Design-Expert version 7 30
What’s New
General improvements Design evaluation Diagnostics Updated graphics Better help Miscellaneous Cool New Stuff
Factorial design and analysis
Response surface design
Mixture design and analysis
Combined design and analysis
Design-Expert version 7 31
Combined Design
Design: Big new feature: combine two mixture designs!
Analysis: New fit summary layout. New model graphs:
• Mix-Process contour plot• Mix-Process 3D plot
Design-Expert version 7 32
Combined Design
Design-Expert version 7 33
Combined Design: Analysis New Fit Summary Layout
Order Abbreviations in Fit Summary Table M = Mean L = Linear Q = Quadratic SC = Special Cubic C = Cubic
Combined Model Mixture Process Fit Summary Table Sequential p-value Summary Statistics
Mix Process Mix Process Lack of Fit Adjusted PredictedOrder Order R-Squared R-Squared
M MM L < 0.0001 0.0027 0.3929 0.3393M 2FI 0.9550 0.0024 0.3630 0.2678M Q * * 0.0024 0.3630 0.2678 AliasedM C * * 0.6965 0.0023 0.3528 0.2418 AliasedM ML M < 0.0001 0.0032 0.4350 0.3825L L < 0.0001 < 0.0001 0.1534 0.9042 0.8715L 2FI < 0.0001 0.5856 0.1415 0.9013 0.8142L Q * < 0.0001 * 0.1415 0.9013 0.8142 AliasedL C * < 0.0001 * 0.7605 0.1280 0.8966 0.7536 Aliased
Design-Expert version 7 34
Combined Design: Analysis Mix-Process Contour Plot
Design-Expert® Software
Ave Texture4.13
0.58
X1 = A: mulletX2 = B: sheepsheadX3 = D: oven temp
Actual ComponentC: croaker = 33.333
Actual FactorsE: oven time = 32.50F: deep fry = 32.50
0.00
66.67
16.67
50.00
33.33
33.33
50.00
16.67
66.67
0.00
Actual mullet
Actual sheepshead
375.00
387.50
400.00
412.50
425.00Ave Texture
D: o
ven
tem
p
1.50
1.752.00
2.25
2.50
Design-Expert version 7 35
Combined Design: Analysis Mix-Process 3D Plot
Design-Expert® Software
Ave Texture4.13
0.58
X1 = A: mulletX2 = B: sheepsheadX3 = D: oven temp
Actual ComponentC: croaker = 33.333
Actual FactorsE: oven time = 32.50F: deep fry = 32.50
0.00 66.67
16.67 50.00
33.33 33.33
50.00 16.67
66.67 0.00
375.00
387.50
400.00
412.50
425.00
1.30
1.65
2.00
2.35
2.70
Ave
Tex
ture
A: mullet D: oven temp B: sheepshead