Design-Expert version 71 What’s New in Design-Expert version 7 Factorial and RSM Design Pat...

Design-Expert version 7 1

What’s New inDesign-Expert version 7

Factorial and RSM Design

Pat WhitcombNovember, 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


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

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Two-Level Factorial AnalysisColored Positive and Negative Effects


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

Half-Normal Plot

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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

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t-Value Limit 2.77645

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Two-Level Factorial AnalysisSelect Aliased terms via Right Click


DESIGN-EXPERT Plotclean

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

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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

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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

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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

maple

pine

PRF-ET

PRF-RT

RF-RT

EPI-RT

LV-EPI-RT 38

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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

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


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

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SCCD


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

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

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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

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


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


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


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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 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.

Design-Expert version 71 What’s New in Design-Expert version 7 Factorial and RSM Design Pat...

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