Design-Expert version 71 What’s New in Design-Expert version 7 Mixture and Combined Design Pat Whitcomb March 25, 2006.

Design-Expert version 7 1

What’s New inDesign-Expert version 7

Mixture and Combined Design

Pat WhitcombMarch 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

Mixture design and analysis

Combined design and analysis


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


Inverted Simplex

When component proportions are restricted by upper bounds it can lead to an inverted simplex.

For example:

x1 ≤ 0.4

x2 ≤ 0.6

x3 ≤ 0.3

90

50

70

30

10

X1

X2 X3


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


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.


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”


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


Inverted SimplexUpper Bounded Pseudo Values

The upper value becomes 0 and the lower value 1!

U_Pseudo values:

Real Pseudo

Li Ui Li Ui

x1 0.1 0.4 1 0

x2 0.3 0.6 1 0

x3 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


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

0.000

22

22

22

22

22

22


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

0.000

22

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


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


A (1.000)B (0.000)

C (1.000)

4

6

8

10

12

14

R

1

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

R

1

A (0.000)

B (1.000)

C (0.000)


Mixture Design“Historical Data”


D-optimal DesignCoordinate versus Point Exchange

There are two algorithms to select “optimal” points for estimating model coefficients:

Point exchange

Coordinate exchange


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.


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)


Mixture Analysis Cox Model

Cox model option for mixtures: May be more informative for formulators when they are interested in a particular composition.


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


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.)


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

R

1

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


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


Orthogonal Component EffectsRange Adjusted versus Constraint Bounded

Bounded AdjustedComponent Effect Effect

A-X1 0.60 1.80

B-X2 0.00 0.00

C-X3 -0.30 -0.30

In constrained mixtures the “Adjusted”

effect is almost never realized.


Orthogonal Component GradientsConstraint Bounded

GradientComponent at Base Pt.

A-X1 3.00

B-X2 0.00

C-X3 -3.00

A has a positive slope

B has no slope

C has a negative slope

Trace (Orthogonal)


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


Screening DesignsCox Direction Component Effect

Trace (Cox)


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


Cox Component EffectsConstraint Bounded


A-X1 2.50

B-X2 -0.91

C-X3 -2.94

ComponentComponent Effect

A-X1 1.00

B-X2 -0.33

C-X3 -0.29

Trace (Cox)


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


Screening DesignsPiepel Direction Component Effect

Trace (Piepel)


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


Piepel Component EffectsConstraint Bounded


A-X1 2.25

B-X2 -1.43

C-X3 -2.92

ComponentComponent Effect

A-X1 1.35

B-X2 -1.00

C-X3 -0.29

Trace (Piepel)


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


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.


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


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


Combined Design


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 Predicted

Order Order R-Squared R-Squared

M M

M L < 0.0001 0.0027 0.3929 0.3393

M 2FI 0.9550 0.0024 0.3630 0.2678

M Q * * 0.0024 0.3630 0.2678 Aliased

M C * * 0.6965 0.0023 0.3528 0.2418 Aliased

M ML M < 0.0001 0.0032 0.4350 0.3825

L L < 0.0001 < 0.0001 0.1534 0.9042 0.8715

L 2FI < 0.0001 0.5856 0.1415 0.9013 0.8142

L Q * < 0.0001 * 0.1415 0.9013 0.8142 Aliased

L C * < 0.0001 * 0.7605 0.1280 0.8966 0.7536 Aliased


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:

ove

n t

em

p

1.50

1.752.00

2.25

2.50


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

A

ve T

ext

ure

A: mullet D: oven temp B: sheepshead

Design-Expert version 71 What’s New in Design-Expert version 7 Mixture and Combined Design Pat Whitcomb March 25, 2006.

Documents

designexpert version

new design

pseudo slide

design space

previous design info

pseudo page

following inverted simplex

upper bounded pseudo