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### What’s New inDesign-Expert version 7Factorial and RSM DesignPat 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

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.

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 III – 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

* 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)

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

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.

C

11.27

8.45

AC

A

t-Value of |Effect|

5.63

Bonferroni Limit 5.06751

2.82

t-Value Limit 2.77645

0.00

1

2

3

4

5

6

7

Rank

Two-Level Factorial AnalysisPareto Chart to Select EffectsTwo-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions

- Design-Expert version 6 Design-Expert version 7

Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions

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

Sum ofMeanFSourceSquaresDFSquareValueProb > F

Model 38135.17 4 9533.79 130.22 < 0.0001A10561.33110561.33144.25< 0.0001B8.1718.170.110.7482C11285.33111285.33154.14< 0.0001AC14701.50114701.50200.80< 0.0001 Residual 512.50 7 73.21 Cor Total 38647.67 11

Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions

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

Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions

- 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.

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:

- Nominal – coefficients compare each factor level mean to the overall mean.
- Ordinal – uses orthogonal polynomials to give coefficients for linear, quadratic, cubic, …, contributions.

Battery LifeInterpreting the coefficients

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

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=10, a = 1.778)Properties closer to regular CCD

A-B slice

210-3 CCD MR-5 CCD Small CCD

158 runs 82 runs 71 runs

different y-axis scale

MR-5 CCDs(k=10, a = 1.778)Properties closer to regular CCD

A-C slice

210-3 CCD MR-5 CCD Small CCD

158 runs 82 runs 71 runs

all on the same y-axis scale

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

a = 1.000 a = 1.565 a = 2.449

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

Face Centered Practical Spherical

a = 1.000 a = 1.565 a = 2.449

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

Face Centered Practical Spherical

a = 1.000 a = 2.340 a = 5.477

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

Face Centered Practical Spherical

a = 1.000 a = 2.340 a = 5.477

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
- Start with a nonsingular set of model points.
- 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.)
- 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.

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