Download - 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
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
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-Expert version 7 4
Design Evaluation Specify Order of Terms to Ignore
Focus attention on what is most
important.
Design-Expert version 7 5
Design Evaluation Evaluate by Design or Response
Useful when a response has missing data.
Design-Expert version 7 6
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-Expert version 7 7
Design EvaluationAnnotated Design Evaluation Report
Design-Expert version 7 8
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
Design-Expert version 7 9
Diagnostics Diagnostics Tool has Two Sets of Buttons
ei = residuali
Design-Expert version 7 10
DiagnosticsExact Limits
Design-Expert® SoftwareConversion
Color points by value ofConversion:
97.0
51.0
Run Number
Ext
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Stu
de
ntiz
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ua
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Externally Studentized Residuals
-4.33
-2.17
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t(/n, n-p'-1) p' is the number of model terms including the interceptn is the total number of runs
Design-Expert version 7 11
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
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S
DFFITS vs. Run
-2.15
-1.11
-0.07
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Design-Expert version 7 12
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
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or
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DFBETAS for Intercept vs. Run
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Design-Expert version 7 13
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.
Design-Expert version 7 14
New Color by Option
Design-Expert® SoftwareConversion
Color points by value ofConversion:
97.0
51.0
Internally Studentized Residuals
No
rma
l % P
rob
ab
ility
Normal Plot of Residuals
-1.39 -0.51 0.36 1.23 2.10
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Design-Expert version 7 15
Full Color Contour and 3D Plots
A: Water5.000
B: Alcohol4.000
C: Urea4.000
2.000 2.000
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Turbidity
517.398
626.122734.847
843.571
952.296
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Design-Expert version 7 16
Design Points on 3D Plots
A (5.000)B (2.000)
C (4.000)
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urbi
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A (3.000)
B (4.000)
C (2.000)
Design-Expert version 7 17
Grid lines on contour plots
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B:
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666666
A: Water
B: AlcoholC: Urea
Turbidity
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Design-Expert version 7 18
Cross Hairs
Design-Expert version 7 19
Magnification on Contour Plots
A: TEA-LS28.000
B: Cocamide9.000
C: Lauramide9.000
1.000 1.000
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Height
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A: TEA-LS26.105
B: Cocamide6.283
C: Lauramide4.877
1.000 1.000
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Design-Expert version 7 20
Specify Detail on Contour “Flags”
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90.00Conversion
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B:
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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
Design-Expert version 7 21
“LSD Bars” & “Confidence Bands”
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Design-Expert version 7 22
Better Help
Improved help
Screen tips
Movies (mini tutorials)
Design-Expert version 7 23
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).
Design-Expert version 7 24
Graph Columns Node
Design-Expert version 7 25
Highlight Points
Run Number
DF
BE
TA
S f
or
C
DFBETAS for C vs. Run
-2.00
-0.27
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3.20
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Design-Expert version 7 26
Ignore Response Cells
Design-Expert version 7 27
Improved Design Summary
New in version 7: Means and standard deviations for factors and responses. The ratio of maximum to minimum added for responses.
Design-Expert version 7 28
Numerical optimization results carried over to graphical optimization and point prediction.
Design-Expert version 7 29
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 30
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.
Design-Expert version 7 31
2k-p Factorial DesignsMore Choices
Need to “check” box to see factor generators
Design-Expert version 7 32
2k-p Factorial DesignsSpecify Base Factors for Generators
Design-Expert version 7 33
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
Design-Expert version 7 34
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.
Design-Expert version 7 35
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
Design-Expert version 7 36
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.
Design-Expert version 7 37
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
Design-Expert version 7 38
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
Design-Expert version 7 39
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.
Design-Expert version 7 40
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 version 7 41
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
Ha
lf-N
orm
al %
Pro
ba
bili
ty
|Standardized Effect|
0.00 5.41 10.81 16.22 21.63
0102030
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AD
Two-Level Factorial AnalysisColored Positive and Negative Effects
Design-Expert version 7 42
Two-Level Factorial AnalysisSelect Model Terms by “Boxing” Them.
Half-Normal Plot
Ha
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orm
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|Standardized Effect|
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Half-Normal Plot
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0.00 5.41 10.81 16.22 21.63
0102030
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99Warning! No terms are selected.
Design-Expert version 7 43
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.
Design-Expert version 7 44
Two-Level Factorial AnalysisPareto Chart to Select Effects
Pareto Chartt-
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of
|Eff
ect
|
Rank
0.00
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5.63
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Bonferroni Limit 5.06751
t-Value Limit 2.77645
1 2 3 4 5 6 7
C
AC
A
Design-Expert version 7 45
Two-Level Factorial AnalysisSelect Aliased terms via Right Click
Design-Expert version 7 46
DESIGN-EXPERT Plotclean
A: water temp B: cy cle timeC: soapD: sof tener
Half Normal plot
Half
Norm
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pro
bability
|Effect|
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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
Half-Normal Plot
Ha
lf-N
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Pro
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|Standardized Effect|
0.00 17.81 35.62 53.44 71.25
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Design-Expert version 7 47
Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions
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
Design-Expert version 7 48
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 7 49
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.
Design-Expert version 7 50
Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Fractions
Irregular fractions – Use the “Recalculate” key when selecting effects.
Design-Expert version 7 51
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.
Design-Expert version 7 52
General Factorial DesignD-optimal Categoric Balance
Design-Expert version 7 53
General Factorial DesignChoice of Nominal or Ordinal Factor Coding
Design-Expert version 7 54
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.
Design-Expert version 7 55
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
Design-Expert version 7 56
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
-3
-2
-1
0
1
2
15 70 125Temperature
Design-Expert version 7 57
General Factorial AnalysisBackward Stepwise Model Reduction
Design-Expert version 7 58
Select Factor Levels for Interaction Plot
Design-Expert version 7 59
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
52.5
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ood
failu
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A: Wood B: Adhesive
Design-Expert version 7 60
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.
Design-Expert version 7 61
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 62
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
Design-Expert version 7 63
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
Design-Expert version 7 64
MR-5 CCDs (k = 6 to 30)Number of runs closer to small CCD
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of r
uns
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SCCD
Design-Expert version 7 65
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
Design-Expert version 7 66
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
Design-Expert version 7 67
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
all on the same y-axis scale
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MR-5 CCDs (k=10, = 1.778)
Properties closer to regular CCD
Design-Expert version 7 69
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.
Design-Expert version 7 70
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
Design-Expert version 7 71
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
Face Centered Practical Spherical = 1.000 = 1.565 = 2.449
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tdE
rr o
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esig
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B: B
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0.00 0.50
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B: B
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B: B
Design-Expert version 7 73
Standard Error Plots MR-5 CCD (k=30) Slice with the other 28 factors = 0
Face Centered Practical Spherical = 1.000 = 2.340 = 5.477
-1.00 -0.50
0.00 0.50
1.00
-1.00
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0.00
0.50
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0
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B: B
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B: B
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0
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A: A
B: B
Design-Expert version 7 74
Standard Error Plots MR-5 CCD (k=30) Slice with 14 factors = +1 and 14 = 0
Face Centered Practical Spherical = 1.000 = 2.340 = 5.477
-1.00 -0.50
0.00 0.50
1.00
-1.00
-0.50
0.00
0.50
1.00
0
0.4
0.8
1.2
1.6
2
2.4
S
tdE
rr o
f D
esig
n
A: A
B: B
-1.00 -0.50
0.00 0.50
1.00
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1.00
0
0.4
0.8
1.2
1.6
2
2.4
S
tdE
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esig
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A: A
B: B
-1.00 -0.50
0.00 0.50
1.00
-1.00
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0
0.4
0.8
1.2
1.6
2
2.4
S
tdE
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A: A
B: B
Design-Expert version 7 75
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
Design-Expert version 7 76
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 77
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 78
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 79
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
Design-Expert version 7 80
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 81
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 82
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 83
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 84
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
Design-Expert version 7 85
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
Design-Expert version 7 86
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
Design-Expert version 7 87
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 88
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)
Design-Expert version 7 89
Mixture Design“Historical Data”
Design-Expert version 7 90
D-optimal DesignCoordinate versus Point Exchange
There are two algorithms to select “optimal” points for estimating model coefficients:
Coordinate exchange
Point exchange
Design-Expert version 7 91
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 92
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 93
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 94
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 95
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 96
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
Design-Expert version 7 97
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 98
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.
Design-Expert version 7 99
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)
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 7 100
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 7 101
Cox Component EffectsConstraint Bounded
GradientComponent at Base Pt.
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)
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 7 102
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 7 103
Piepel Component EffectsConstraint Bounded
GradientComponent at Base Pt.
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)
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 7 104
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 7 105
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 7 106
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 7 107
Combined Design
Design-Expert version 7 108
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
Design-Expert version 7 109
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
Design-Expert version 7 110
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