design-expert version 71 what’s new in design-expert version 7 pat whitcomb september 13, 2005

Design-Expert version 7 1

What’s New inDesign-Expert version 7

Pat WhitcombSeptember 13, 2005

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 Evaluation

User specifies what order terms to ignore.

Can evaluate by design or response.

New options for more flexibility. User specifies ratios for power calculation. User specifies what to report. User specified options for standard error plots.

Annotation added to design evaluation report.

Design Evaluation Specify Order of Terms to Ignore

Focus attention on what is most

important.

Design Evaluation Evaluate by Design or Response

Useful when a response has missing data.

Design Evaluation New Options for More Flexibility

User specifies ratios for power calculation.

User specifies what to report.

User specified options for standard error plots.

Design EvaluationAnnotated Design Evaluation Report

Diagnostics

Diagnostics Tool has two sets of buttons: “Diagnostics” and “Influence”.

New names and limits. Internally studentized residual = studentized residual v6. Externally studentized residual = outlier t v6.

• The externally studentized residual has exact limits.

New – DFFITS

New – DFBETAS

Diagnostics Diagnostics Tool has Two Sets of Buttons

ei = residuali

DiagnosticsExact Limits

Design-Expert® SoftwareConversion

Color points by value ofConversion:

Run Number

Externally Studentized Residuals

1 4 7 10 13 16 19

t(/n, n-p'-1) p' is the number of model terms including the interceptn is the total number of runs

Diagnostics DFFITS

DFFITS measures the influence the ith observation has on the predicted value.(See Myers, Raymond: “Classical andModern Regression with Applications”,1986, Duxbury Press, page 284.) It isthe studentized difference between thepredicted value with observation i andthe predicted value without observation i. DFFITS is the externally studentized residual magnified by high leverage points and shrunk by low leverage points. It is a sensitive test for influence and points outside the limits are not necessarily bad just influential. These runs associated with points outside the limits should be investigated to for potential problems.

DFFITS is very sensitive and it is not surprising to have a point or two falling outside the limits, especially for small designs.

Run Number

DFFITS vs. Run

1 4 7 10 13 16 19

Diagnostics DFBETAS

DFBETAS measures the influence the ith observation has on each regression coefficient. (See Myers, Raymond: “Classical and Modern Regression with Applications”, 1986, Duxbury Press, page 284.) The DFBETAS j,i is the number of standard errors that the jth coefficient changes if the ith observation is removed.

Run Number

DFBETAS for Intercept vs. Run

1 4 7 10 13 16 19

Run Number

DFBETAS for A vs. Run

1 4 7 10 13 16 19

Updated Graphics

New color by option.

Full color contour and 3D plots.

Design points and their projection lines added to 3D plots.

Grid lines on contour plots.

Cross hairs read coordinates on plots.

Magnification on contour plots.

User specified detail on contour “Flags”.

Choice of “LSD Bars”, “Confidence Bands” or “None” on one factor and interaction plots.

New Color by Option

Design-Expert® SoftwareConversion

Color points by value ofConversion:

Internally Studentized Residuals

Normal Plot of Residuals

-1.39 -0.51 0.36 1.23 2.10

Full Color Contour and 3D Plots

A: Water5.000

B: Alcohol4.000

C: Urea4.000

2.000 2.000

Turbidity

517.398

626.122734.847

843.571

952.296

Design Points on 3D Plots

A (5.000)B (2.000)

C (4.000)

A (3.000)

B (4.000)

C (2.000)

Grid lines on contour plots

40.00 42.50 45.00 47.50 50.00

90.00Conversion

A: time

666666

A: Water

B: AlcoholC: Urea

Turbidity

Cross Hairs

Magnification on Contour Plots

A: TEA-LS28.000

B: Cocamide9.000

C: Lauramide9.000

1.000 1.000

20.000

Height

A: TEA-LS26.105

B: Cocamide6.283

C: Lauramide4.877

1.000 1.000

22.868

Specify Detail on Contour “Flags”

40.00 42.50 45.00 47.50 50.00

90.00Conversion

A: time

666666

Prediction 81.6Observed 81.095% CI Lo 77.895% CI Hig 85.495% PI Lo 71.695% PI Hig 91.6SE Mean 1.68337SE Pred 4.43914X1 45.00X2 85.00

“LSD Bars” & “Confidence Bands”

0.00 10.00 20.00 30.00 40.00

A: Departure

One Factor

0.00 10.00 20.00 30.00 40.00

A: Departure

One Factor

Better Help

Improved help

Screen tips

Movies (mini tutorials)

Miscellaneous Cool New Stuff

“Graph Columns” now has its own node.

Highlight points in the design layout or on a diagnostic graph for easy identification.

Right click and response cell and ignore it.

Improved design summary.

Numerical optimization results now carried over to graphical optimization and point prediction.

Export graph to enhanced metafile (*.emf).

Graph Columns Node

Highlight Points

Run Number

DFBETAS for C vs. Run

1 6 11 16 21 26 31

Ignore Response Cells

Improved Design Summary

New in version 7: Means and standard deviations for factors and responses. The ratio of maximum to minimum added for responses.

Numerical optimization results carried over to graphical optimization and point prediction.

What’s New

Two-Level Factorial Designs

2k-p factorials for up to 512 runs (256 in v6) and 21 factors (15 in v6). Design screen now shows resolution and updates with

blocking choices. Generators are hidden by default. User can specify base factors for generators. Block names are entered during build.

Minimum run equireplicated resolution V designs for6 to 31 factors.

Minimum run equireplicated resolution IV designs for 5 to 50 factors.

2k-p Factorial DesignsMore Choices

Need to “check” box to see factor generators

2k-p Factorial DesignsSpecify Base Factors for Generators

MR5 Designs Motivation

Regular fractions (2k-p fractional factorials) of 2k designs often contain considerably more runs than necessary to estimate the [1+k+k(k-1)/2] effects in the 2FI model.

For example, the smallest regular resolution V design for k=7 uses 64 runs (27-1) to estimate 29 coefficients.

Our balanced minimum run resolution V design for k=7 has 30 runs, a savings of 34 runs.

“Small, Efficient, Equireplicated Resolution V Fractions of 2k designs and their Application to Central Composite Designs”, Gary Oehlert and Pat Whitcomb, 46th Annual Fall Technical Conference, Friday, October 18, 2002.

Available as PDF at: http://www.statease.com/pubs/small5.pdf

MR5 DesignsConstruction

Designs have equireplication, so each column contains the same number of +1s and −1s.

Used the columnwise-pairwise of Li and Wu (1997) with the D-optimality criterion to find designs.

Overall our CP-type designs have better properties than the algebraically derived irregular fractions.

Efficiencies tend to be higher.

Correlations among the effects tend be lower.

MR5 DesignsProvide Considerable Savings

k 2k-p MR5 k 2k-p MR5

6 32 22 15 256 122

7 64 30 16 256 138

8 64 38 17 256 154

9 128 46 18 512 172

10 128 56 19 512 192

11 128 68 20 512 212

12 256 80 21 512 232

13 256 92 25 1024 326

14 256 106 30 1024 466

MR4 DesignsMitigate the use of Resolution III Designs

The minimum number of runs for resolution IV designs is only two times the number of factors (runs = 2k). This can offer quite a savings when compared to a regular resolution IV 2k-p fraction.

32 runs are required for 9 through 16 factors to obtain a resolution IV regular fraction.

The minimum-run resolution IV designs require 18 to 32 runs, depending on the number of factors.

• A savings of (32 – 18) 14 runs for 9 factors.

• No savings for 16 factors.

“Screening Process Factors In The Presence of Interactions”, Mark Anderson and Pat Whitcomb, presented at AQC 2004 Toronto. May 2004. Available as PDF at: http://www.statease.com/pubs/aqc2004.pdf.

MR4 DesignsSuggest using “MR4+2” Designs

Problems: If even 1 run lost, design becomes resolution IIIIII –

main effects become badly aliased.

Reduction in runs causes power loss – may miss significant effects.

Evaluate power before doing experiment.

Solution: To reduce chance of resolution loss and increase

power, consider adding some padding:

New Whitcomb & Oehlert “MR4+2” designs

MR4 DesignsProvide Considerable Savings

k 2k-p MR4+2 k 2k-p MR4+2

6 16 14 16 32 34*

7 16 16* 17 64 36

8 16 18* 18 64 38

9 32 20 19 64 40

10 32 22 20 64 42

11 32 24 21 64 44

12 32 26 22 64 46

13 32 28 23 64 48

14 32 30 24 64 50

15 32 32* 25 64 52* No savings

Two-Level Factorial Analysis

Effects Tool bar for model section tools.

Colored positive and negative effects and Shapiro-Wilk test statistic add to probability plots.

Select model terms by “boxing” them.

Pareto chart of t-effects.

Select aliased terms for model with right click.

Better initial estimates of effects in irregular factions by using “Design Model”. Recalculate and clear buttons.

Two-Level Factorial AnalysisEffects Tool Bar

New – Effects Tool on the factorial effects screen makes all the options obvious.

New – Pareto Chart

New – Clear Selection button

New – Recalculate button (discuss later in respect to irregular fractions)

Design-Expert® SoftwareFiltration Rate

Shapiro-Wilk testW-value = 0.974p-value = 0.927A: TemperatureB: PressureC: ConcentrationD: Stir Rate

Positive Effects Negative Effects

Half-Normal Plot

|Standardized Effect|

0.00 5.41 10.81 16.22 21.63

0102030

Two-Level Factorial AnalysisColored Positive and Negative Effects

Two-Level Factorial AnalysisSelect Model Terms by “Boxing” Them.

Half-Normal Plot

0.00 5.41 10.81 16.22 21.63

0102030

Half-Normal Plot

0.00 5.41 10.81 16.22 21.63

0102030

99Warning! No terms are selected.

Two-Level Factorial AnalysisPareto Chart to Select Effects

The Pareto chart is useful for showing the relative size of effects, especially to non-statisticians.

Problem: If the 2k-p factorial design is not orthogonal and balanced the effects have differing standard errors, so the size of an effect may not reflect its statistical significance.

Solution: Plotting the t-values of the effects addresses the standard error problems for non-orthogonal and/or unbalanced designs.

Problem: The largest effects always look large, but what is statistically significant?

Solution: Put the t-value and the Bonferroni corrected t-value on the Pareto chart as guidelines.

Two-Level Factorial AnalysisPareto Chart to Select Effects

Pareto Chartt-

Bonferroni Limit 5.06751

t-Value Limit 2.77645

1 2 3 4 5 6 7

Two-Level Factorial AnalysisSelect Aliased terms via Right Click

DESIGN-EXPERT Plotclean

A: water temp B: cy cle timeC: soapD: sof tener

Half Normal plot

bability

|Effect|

0.00 14.83 29.67 44.50 59.33

Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions

Design-Expert version 6 Design-Expert version 7Design-Expert® Softwareclean

Shapiro-Wilk testW-value = 0.876p-value = 0.171A: water temp B: cycle timeC: soapD: softener

Positive Effects Negative Effects

Half-Normal Plot

0.00 17.81 35.62 53.44 71.25

0102030

ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > F

Model 38135.17 4 9533.79 130.22 < 0.0001A 10561.33 1 10561.33 144.25 < 0.0001B 8.17 1 8.17 0.11 0.7482C 11285.33 1 11285.33 154.14 < 0.0001

AC 14701.50 1 14701.50 200.80 < 0.0001Residual 512.50 7 73.21Cor Total 38647.67 11

Main effects only model: [Intercept] = Intercept - 0.333*CD - 0.333*ABC - 0.333*ABD [A] = A - 0.333*BC - 0.333*BD - 0.333*ACD [B] = B - 0.333*AC - 0.333*AD - 0.333*BCD [C] = C - 0.5*AB [D] = D - 0.5*AB

Main effects & 2fi model: [Intercept] = Intercept - 0.5*ABC - 0.5*ABD [A] = A - ACD [B] = B - BCD [C] = C [D] = D [AB] = AB [AC] = AC - BCD [AD] = AD - BCD [BC] = BC - ACD [BD] = BD - ACD [CD] = CD - 0.5*ABC - 0.5*ABD

Design-Expert version 6 calculates the initial effects using sequential SS via hierarchy.

Design-Expert version 7 calculates the initial effects using partial SS for the “Base model for the design”.

The recalculate button (next slide) calculates the chosen (model) effects using partial SS and then remaining effects using sequential SS via hierarchy.

Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Fractions

Irregular fractions – Use the “Recalculate” key when selecting effects.

General Factorials

Design:

Bigger designs than possible in v6.

D-optimal now can force categoric balance (or impose a balance penalty).

Choice of nominal or ordinal factor coding.

Analysis:

Backward stepwise model reduction.

Select factor levels for interaction plot.

3D response plot.

General Factorial DesignD-optimal Categoric Balance

General Factorial DesignChoice of Nominal or Ordinal Factor Coding

Categoric FactorsNominal versus Ordinal

The choice of nominal or ordinal for coding categoric factors has no effect on the ANOVA or the model graphs. It only affects the coefficients and their interpretation:

1. Nominal – coefficients compare each factor level mean to the overall mean.

2. Ordinal – uses orthogonal polynomials to give coefficients for linear, quadratic, cubic, …, contributions.

Nominal contrasts – coefficients compare each factor level mean to the overall mean.

Name A[1] A[2] A1 1 0 A2 0 1 A3 -1 -1

The first coefficient is the difference between the overall mean and the mean for the first level of the treatment.

The second coefficient is the difference between the overall mean and the mean for the second level of the treatment.

The negative sum of all the coefficients is the difference between the overall mean and the mean for the last level of the treatment.

Battery LifeInterpreting the coefficients

Ordinal contrasts – using orthogonal polynomials the first coefficient gives the linear contribution and the second the quadratic:

Name B[1] B[2] 15 -1 1 70 0 -2

125 1 1

B[1] = linear

B[2] = quadratic

Battery LifeInterpreting the coefficients

Polynomial Contrasts

15 70 125Temperature

General Factorial AnalysisBackward Stepwise Model Reduction

Select Factor Levels for Interaction Plot

General Factorial Analysis3D Response Plot

Design-Expert® Software

wood failure

X1 = A: WoodX2 = B: Adhesive

Actual FactorsC: Applicator = brushD: Clamp = pneumaticE: Pressure = firm

chestnut

red oak

poplar

PRF-ET

PRF-RT

EPI-RT

LV-EPI-RT 38

A: Wood B: Adhesive

Factorial Design Augmentation

Semifold: Use to augment 2k-p resolution IV; usually as many additional two-factor interactions can be estimated with half the runs as required for a full foldover.

Add Center Points.

Replicate Design.

Add Blocks.

What’s New

Response Surface Designs

More “canned” designs; more factors and choices. CCDs for ≤ 30 factors (v6 ≤ 10 factors)

• New CCD designs based on MR5 factorials.

• New choices for alpha “practical”, “orthogonal quadratic” and “spherical”.

Box-Behnken for 3–30 factors (v6 3, 4, 5, 6, 7, 9 & 10)

“Odd” designs moved to “Miscellaneous”.

Improved D-optimal design. for ≤ 30 factors (v6 ≤ 10 factors)

Coordinate exchange

MR-5 CCDsResponse Surface Design

Minimum run resolution V (MR-5) CCDs:

Add six center points to the MR-5 factorial design.

Add 2(k) axial points.

For k=10 the quadratic model has 66 coefficients. The number of runs for various CCDs:

• Regular (210-3) = 158

• MR-5 = 82

• Small (Draper-Lin) = 71

MR-5 CCDs (k = 6 to 30)Number of runs closer to small CCD

0 5 10 15 20 25 30

k: # of factors

MR-5 CCD

MR-5 CCDs (k=10, = 1.778)

Regular, MR-5 and Small CCDs

210-3 CCD

158 runs

MR-5 CCD

82 runs

Small CCD

71 runs

Model 65 65 65

Residuals 92 16 5

Lack of Fit 83 11 1

Pure Error 9 5 4

Corr Total 157 81 70

MR-5 CCDs (k=10, = 1.778)

Properties of Regular, MR-5 and Small CCDs

210-3 CCD

158 runs

MR-5 CCD

82 runs

Small CCD

71 runs

Max coefficient SE 0.214 0.227 16.514

Max VIF 1.543 2.892 12,529

Max leverage 0.498 0.991 1.000

Ave leverage 0.418 0.805 0.930

Scaled D-optimality 1.568 2.076 3.824

MR-5 CCDs (k=10, = 1.778)

Properties closer to regular CCD

-1.00 -0.50

0.00 0.50

210-3 CCD MR-5 CCD Small CCD158 runs 82 runs 71 runs

different y-axis scale

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

A-B slice

210-3 CCD MR-5 CCD Small CCD158 runs 82 runs 71 runs

all on the same y-axis scale

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

A-C slice

MR-5 CCDs (k=10, = 1.778)

Properties closer to regular CCD

MR-5 CCDsConclusion

Best of both worlds:

The number of runs are closer to the number in the small than in the regular CCDs.

Properties of the MR-5 designs are closer to those of the regular than the small CCDs.

• The standard errors of prediction are higher than regular CCDs, but not extremely so.

• Blocking options are limited to 1 or 2 blocks.

Practical alphaChoosing an alpha value for your CCD

Problems arise as the number of factors increase: The standard error of prediction for the face centered

CCD (alpha = 1) increases rapidly. We feel that an alpha > 1 should be used when k > 5.

The rotatable and spherical alpha values become too large to be practical.

Solution: Use an in between value for alpha, i.e. use a practical

alpha value.

practical alpha = (k)¼

Standard Error Plots 26-1 CCDSlice with the other four factors = 0

Face Centered Practical Spherical = 1.000 = 1.565 = 2.449

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

Standard Error Plots 26-1 CCDSlice with two factors = +1 and two = 0

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

Standard Error Plots MR-5 CCD (k=30) Slice with the other 28 factors = 0

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

Standard Error Plots MR-5 CCD (k=30) Slice with 14 factors = +1 and 14 = 0

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

-1.00 -0.50

0.00 0.50

Choosing an alpha value for your CCD

k Practical Spherical k Practical Spherical6 1.5651 2.4495 19 2.0878 4.35897 1.6266 2.6458 20 2.1147 4.47218 1.6818 2.8284 21 2.1407 4.58269 1.7321 3.0000 22 2.1657 4.6904

10 1.7783 3.1623 23 2.1899 4.795811 1.8212 3.3166 24 2.2134 4.899012 1.8612 3.4641 25 2.2361 5.000013 1.8988 3.6056 26 2.2581 5.099014 1.9343 3.7417 27 2.2795 5.196215 1.9680 3.8730 28 2.3003 5.291516 2.0000 4.0000 29 2.3206 5.385217 2.0305 4.1231 30 2.3403 5.477218 2.0598 4.2426

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.

What’s New

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

A: x11.000

B: x21.000

C: x31.000

0.000 0.000

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 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

U Real U_Pseudo

U Xu '

0.4 Xu '

0.6 Xu '

0.3 Xu '

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

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

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

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

10.011.0 12.0

A (1.000)B (0.000)

C (1.000)

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)

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:

Coordinate exchange

Point 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)

A (0.100)

B (0.800)

C (0.100)

A: X11.000

B: X21.000

C: X31.000

0.000 0.000

1 2 3y 10x 8x 6x

Screening DesignsOrthogonal Direction Component Effect

Trace (Orthogonal)

Deviation from Reference Blend (L_Pseudo Units)

-0.143 -0.071 0.000 0.071 0.143

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)

-0.143 -0.071 0.000 0.071 0.143

Slope = 3.0

Screening DesignsCox Direction Component Effect

Trace (Cox)

-0.286 -0.143 0.000 0.143 0.286

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)

-0.286 -0.143 0.000 0.143 0.286

Slope = 2.5

Screening DesignsPiepel Direction Component Effect

Trace (Piepel)

-0.500 -0.250 0.000 0.250 0.500

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)

-0.500 -0.250 0.000 0.250 0.500

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

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 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

Ave Texture4.13

X1 = A: mulletX2 = B: sheepsheadX3 = D: oven temp

Actual ComponentC: croaker = 33.333

Actual FactorsE: oven time = 32.50F: deep fry = 32.50

Actual mullet

Actual sheepshead

375.00

387.50

400.00

412.50

425.00Ave Texture

1.752.00

Combined Design: Analysis Mix-Process 3D Plot

Ave Texture4.13

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

A: mullet D: oven temp B: sheepshead

design-expert version 71 what’s new in design-expert version 7 pat whitcomb september 13, 2005

design evaluation new

analysiscombined design

contour plotsspecify

new names

interaction plots

optionfull color contour

standard error plots

studentized residual

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