Transcript What’s New in Design-Expert version 7 - Stat-Ease
Design-Expert version 7
What’s New in Design-Expert version 7 Pat Whitcomb September 13, 2005
1
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 2
Design Evaluation
User specifies what order terms to ignore.
Can evaluate by design or response.
New options for more flexibility.
User specifies D/s 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 3
Design-Expert version 7
Design Evaluation
Specify Order of Terms to Ignore Focus attention on what is most important.
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Design-Expert version 7
Design Evaluation
Evaluate by Design or Response Useful when a response has missing data.
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Design Evaluation
New Options for More Flexibility User specifies D/s ratios for power calculation.
User specifies what to report.
User specified options for standard error plots.
Design-Expert version 7 6
Design-Expert version 7
Design Evaluation
Annotated Design Evaluation Report 7
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 8
Diagnostics
Diagnostics Tool has Two Sets of Buttons e i = residual i Design-Expert version 7 9
Diagnostics
Exact Limits t( a /n, n-p'-1) p' is the number of model terms including the intercept n is the total number of runs Design-Expert® Software Conversion Color points by value of Conversion: 97.0
51.0
4.33
Externally Studentized Residuals 2.17
0.00
-2.17
Design-Expert version 7 -4.33
1 4 7 10 13 Run Number 16 19 10
Diagnostics
DFFITS DFFITS measures the influence the i th observation has on the predicted value.
(See Myers, Raymond: “Classical and Modern Regression with Applications”, 1986, Duxbury Press, page 284.) It is the studentized difference between the predicted value with observation i and the 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.
2.00
0.96
-0.07
-1.11
-2.15
1 4 DFFITS vs. Run 7 10 13 Run Number 16 DFFITS is very sensitive and it is not surprising to have a point or two falling outside the limits, especially for small designs. 19 Design-Expert version 7 11
Diagnostics
DFBETAS DFBETAS measures the influence the i th 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 j th coefficient changes if the i th observation is removed.
DFBETAS for Intercept vs. Run DFBETAS for A vs. Run 2.00
2.00
1.00
0.00
-1.00
-2.00
1 4 7 10 13 Run Number 16 19 Design-Expert version 7 1.00
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-1.00
-2.00
1 4 7 10 13 Run Number 16 19 12
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.
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Design-Expert® Software Conversion Color points by value of Conversion: 97.0
51.0
Design-Expert version 7
New Color by Option
Normal Plot of Residuals 99 95 90 80 70 50 30 20 10 5 1 -1.39
-0.51
0.36
1.23
Internally Studentized Residuals 2.10
14
Full Color Contour and 3D Plots
A: Water 5.000
626.122
2.000
517.398
734.847
843.571
2.000
4.000
B: Al cohol Design-Expert version 7 952.296
3.000
Turbidity 4.000
C: Urea 15
Design-Expert version 7 1200 1100 1000 900 800 700 600 500 400 300 A (5.000)
Design Points on 3D Plots
B (2.000) C (4.000) C (2.000) A (3.000) B (4.000) 16
Grid lines on contour plots
90.00
87.50
85.00
82.50
80.00
40.00
80.0
Conversion 88.0
86.0
84.0
82.0
42.50
45.00
A: time 5.000
2.000
2 80.0
47.50
3.500
78.0
50.00
3.000
4.000
4.500
4.000
C: Urea 500 600 1000 700 800 900 2.500
3.000
B: Alcohol 3.500
3.000
A: Water Turbidity 2.500
3.500
4.000
2 2.000
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Cross Hairs
Design-Expert version 7 18
Magnification on Contour Plots
A: T EA-LS 28.000
160 160 2 1.000
160 150 175 170 140 9.000
B: Cocami de 20.000
Height 1.000
9.000
C: Laurami de 1.000
A: T EA-LS 26.105
160 1.000
170 170 175 160 6.283
B: Cocami de 22.868
Height C: Laurami de 4.877
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Specify Detail on Contour “Flags”
90.00
87.50
Conversion 88.0
Prediction 81.6
Observed 81.0
95% CI Lo 77.8
95% CI Hig 85.4
95% PI Lo 71.6
95% PI Hig 91.6
SE Mean 1.68337
SE Pred X1 4.43914
X2 85.00
86.0
85.00
80.0
80.0
82.50
80.00
40.00
Design-Expert version 7 42.50
45.00
A: time 47.50
78.0
50.00
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“LSD Bars” & “Confidence Bands”
One Factor 54 47.75
41.5
35.25
29 0.00
10.00
20.00
A: Departure 30.00
40.00
One Factor 54 47.75
41.5
35.25
29 0.00
10.00
20.00
A: Departure 30.00
40.00
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Improved help Screen tips Movies (mini tutorials)
Better Help
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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).
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Graph Columns Node
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DFBETAS for C vs. Run 4.93
3.20
1.46
-0.27
-2.00
1 6 11 16 Run Number 21 26 31 Design-Expert version 7
Highlight Points
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Ignore Response Cells
Design-Expert version 7 26
Improved Design Summary
New in version 7: Means and standard deviations for factors and responses.
The ratio of maximum to minimum added for responses.
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Numerical optimization results carried over to graphical optimization and point prediction.
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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 29
Two-Level Factorial Designs
2 k-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 for 6 to 31 factors.
Minimum run equireplicated resolution IV designs for 5 to 50 factors.
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2
k-p
Factorial Designs
More Choices Need to “check” box to see factor generators Design-Expert version 7 31
2
k-p
Factorial Designs
Specify Base Factors for Generators Design-Expert version 7 32
MR5 Designs
Motivation Regular fractions (2 k-p fractional factorials) of 2 k 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 (2 7-1 ) 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 2 k designs and their Application to Central Composite Design
s”, 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
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MR5 Designs
Construction 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.
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k 6 7 8 9 10 11 12 13 14 2 k-p 32 64 64 128 128 128 256 256 256 MR5 22 30 38 46 56 68 80 92 106
MR5 Designs
Provide Considerable Savings k 15 16 17 18 19 20 21 25 30 2 k-p 256 256 256 512 512 512 512 1024 1024 MR5 122 138 154 172 192 212 232 326 466 Design-Expert version 7 35
MR4 Designs
Mitigate 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 2 k-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.
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MR4 Designs
Suggest 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 Design-Expert version 7 37
MR4 Designs
Provide Considerable Savings k 6 7 8 9 10 11 12 13 14 15 2 k-p 16 16 16 32 32 32 32 32 32 32 MR4+2 14 16* 18* 20 22 24 26 28 30 32*
* No savings
k 16 17 18 19 20 21 22 23 24 25 Design-Expert version 7 2 k-p 32 64 64 64 64 64 64 64 64 64 MR4+2 34* 36 38 40 42 44 46 48 50 52 38
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.
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Two-Level Factorial Analysis
Effects 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)
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Design-Expert® Software Filtration Rate Shapiro-Wilk test W-value = 0.974
p-value = 0.927
A: T emperature B: Pressure C: Concentration D: Stir Rate Positive Effects Negative Effects
Two-Level Factorial Analysis
Colored Positive and Negative Effects Half-Normal Plot 99 A 95 90 80 70 50 30 20 10 0 C D AD AC 0.00
5.41
10.81
|Standardized Effect| 16.22
21.63
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95 90 80 70 50 30 20 10 0 99 Half-Normal Plot Warning! No terms are selected.
Two-Level Factorial Analysis
Select Model Terms by “Boxing” Them.
Half-Normal Plot 99 95 90 80 70 50 30 20 10 0 C D AD AC A 0.00
5.41
10.81
|Standardized Effect| 16.22
21.63
0.00
5.41
10.81
|Standardized Effect| 16.22
21.63
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Two-Level Factorial Analysis
Pareto Chart to Select Effects The Pareto chart is useful for showing the relative size of effects, especially to non-statisticians.
Problem : If the 2 k-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.
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Two-Level Factorial Analysis
Pareto Chart to Select Effects Pareto Chart 11.27
C 8.45
AC 5.63
A 2.82
Bonferroni Limit 5.06751
t-Value Limit 2.77645
0.00
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Design-Expert version 7
Two-Level Factorial Analysis
Select Aliased terms via Right Click 45
Two-Level Factorial Analysis
Better Effect Estimates in Irregular Factions DESIGN-EXPERT Plot clean A: water temp B: cy cle time C: soap D: sof tener Design-Expert version 6 Half Normal plot Design-Expert® Software clean 99 97 95 Shapiro-Wilk test W-value = 0.876
p-value = 0.171
A: water temp B: cycle time C: soap D: softener A Negative Effects 90 85 80 70 60 B C AC 40 20 0 0.00
14.83
29.67
|Effect| 44.50
59.33
95 90 80 70 50 30 20 10 0 Design-Expert version 7 Half-Normal Plot 99 AC A C 0.00
17.81
35.62
53.44
|Standardized Effect| 71.25
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Two-Level Factorial Analysis
Better Effect Estimates in Irregular Factions
ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Source Sum of Squares DF Mean Square F Value
Model
A
B
C AC
Residual Cor Total 38135.17
10561.33
8.17
11285.33
14701.50
512.50
38647.67
1
7 11 4
1
1
1
9533.79
10561.33
8.17
11285.33
14701.50
73.21
130.22
144.25
0.11
154.14
200.80
Prob > F
< 0.0001
< 0.0001
0.7482
< 0.0001
< 0.0001
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Two-Level Factorial Analysis
Better 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 Design-Expert version 7 48
Two-Level Factorial Analysis
Better 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.
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Two-Level Factorial Analysis
Better Effect Estimates in Irregular Fractions Irregular fractions – Use the “Recalculate” key when selecting effects.
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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.
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General Factorial Design
D-optimal Categoric Balance Design-Expert version 7 52
General Factorial Design
Choice of Nominal or Ordinal Factor Coding Design-Expert version 7 53
Categoric Factors
Nominal 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.
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Battery Life
Interpreting the coefficients Nominal contrasts – coefficients compare each factor level mean to the overall mean.
Name A1 A2 A3 A[1] 1 0 -1 A[2] 0 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.
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Battery Life
Interpreting the coefficients Ordinal contrasts – using orthogonal polynomials the first coefficient gives the linear contribution and the second the quadratic: Name B[1] 15 -1 70 125 0 1 B[2] 1 -2 1 2 1 0 Polynomial Contrasts B[1] = linear B[2] = quadratic -1 -2 -3 15 70 Temperature 125 Design-Expert version 7 56
General Factorial Analysis
Backward Stepwise Model Reduction Design-Expert version 7 57
Select Factor Levels for Interaction Plot
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General Factorial Analysis
3D Response Plot Design-Expert® Software wood failure X1 = A: Wood X2 = B: Adhesive Actual Factors C: Applicator = brush D: Clamp = pneumatic E: Pressure = firm 96 81.5 67 52.5 38 LV-EPI-RT EPI-RT RF-RT PRF-RT B: Adhesive PRF-ET pine poplar maple red oak chestnut A: Wood Design-Expert version 7 59
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.
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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 61
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 62
MR-5 CCDs
Response 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 (2 10-3 ) = 158 • • MR-5 = 82 Small (Draper-Lin) = 71 Design-Expert version 7 63
MR-5 CCDs
(k = 6 to 30) Number of runs closer to small CCD 600 500 400 300 200 100 0 0 CCD MR-5 CCD SCCD Design-Expert version 7 5 10 15 k: # of factors 20 25 30 64
MR-5 CCDs
(k=10,
a
= 1.778)
Regular, MR-5 and Small CCDs Model Residuals Lack of Fit Pure Error 2 10-3 CCD 158 runs MR-5 CCD 82 runs Small CCD 71 runs 65 65 65 92 83 9 16 11 5 5 1 4 Corr Total 157 81 70 Design-Expert version 7 65
MR-5 CCDs
(k=10,
a
= 1.778)
Properties of Regular, MR-5 and Small CCDs Max coefficient SE Max VIF Max leverage Ave leverage Scaled D-optimality 2 10-3 CCD 158 runs 0.214
1.543
0.498
0.418
1.568
MR-5 CCD 82 runs 0.227
Small CCD 71 runs 16.514
2.892
0.991
12,529 1.000
0.805
2.076
0.930
3.824
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MR-5 CCDs
(k=10,
a
= 1.778)
Properties closer to regular CCD A-B slice 4 3 2 1 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
4 3 2 1 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
16 14 12 10 8 6 4 2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
2 10-3 CCD 158 runs Design-Expert version 7 MR-5 CCD 82 runs Small CCD 71 runs different y-axis scale 67
MR-5 CCDs
(k=10,
a
= 1.778)
Properties closer to regular CCD A-C slice 4 3 2 1 0 1.00 0.50 C: C 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
4 3 2 1 0 1.00 0.50 C: C 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
4 3 2 1 0 1.00 0.50 C: C 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
2 10-3 CCD 158 runs MR-5 CCD 82 runs all on the same y-axis scale Small CCD 71 runs Design-Expert version 7 68
MR-5 CCDs
Conclusion 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.
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Practical alpha
Choosing 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 70
Standard Error Plots
2 6-1 CCD Slice with the other four factors = 0 1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
Face Centered a = 1.000
Practical a = 1.565
Spherical a = 2.449
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Standard Error Plots
2 6-1 CCD Slice with two factors = +1 and two = 0 1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
Face Centered a = 1.000
Practical a = 1.565
Spherical a = 2.449
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Standard Error Plots MR-5 CCD (k=30)
Slice with the other 28 factors = 0 1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
1 0.8 0.6 0.4 0.2 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
Face Centered a = 1.000
Practical a = 2.340
Spherical a = 5.477
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Standard Error Plots MR-5 CCD (k=30)
Slice with 14 factors = +1 and 14 = 0 2.4 2 1.6 1.2 0.8 0.4 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
2.4 2 1.6 1.2 0.8 0.4 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
2.4 2 1.6 1.2 0.8 0.4 0 1.00 0.50 B: B 0.00 -0.50 -1.00 -1.00
-0.50
0.50
0.00
A: A 1.00
Face Centered a = 1.000
Practical a = 2.340
Spherical a = 5.477
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Choosing an alpha value for your CCD
11 12 13 14 15 16 17 18 8 9 10
k Practical Spherical 6
7
1.5651
1.6266
2.4495
2.6458
1.6818
1.7321
1.7783
2.8284
3.0000
3.1623
1.8212
1.8612
1.8988
1.9343
1.9680
2.0000
2.0305
2.0598
3.3166
3.4641
3.6056
3.7417
3.8730
4.0000
4.1231
4.2426
Design-Expert version 7 24 25 26 27 28 29
30 k Practical Spherical
19 20 2.0878
2.1147
4.3589
4.4721
21 22 23 2.1407
2.1657
2.1899
4.5826
4.6904
4.7958
2.2134
2.2361
2.2581
2.2795
2.3003
2.3206
2.3403
4.8990
5.0000
5.0990
5.1962
5.2915
5.3852
5.4772
75
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 76
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 77
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 78
Inverted Simplex
When component proportions are restricted by upper bounds it can lead to an inverted simplex.
X 1 For example: x1 ≤ 0.4
90
x2 ≤ 0.6
x3 ≤ 0.3
70 10 50 50 30 30 30 50 1 0 70 7 0 10 90 9 0
Design-Expert version 7 X 2 X 3 79
Inverted Simplex
3 component L_Pseudo Using lower bounded L_Pseudo values leads to the following inverted simplex.
A: x1 1.000
Open “
I-simplex L_P.dx7
” and model the response. 0.50 in L_Pseudo 0.000
0.000
Design-Expert version 7 1.000
B: x2 0.000
1.000
C: x3 80
Inverted Simplex
3 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 81
Inverted Simplex
3 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 82
Inverted Simplex
Upper Bounded Pseudo Values The high value becomes 0 and the low value becomes 1!
A: x1 1.000
1 in U_Pseudo 0 in U_Pseudo 0.000
0.000
Design-Expert version 7 1.000
B: x2 0.000
1.000
C: x3 83
Inverted Simplex
Upper Bounded Pseudo Values The upper value becomes 0 and the lower value 1!
U_Pseudo values: U_Pseudo U i Re al U i 1 Real Pseudo U i X i x1 x2 x3 L i 0.1
0.3
0.0
U i 0.4
0.6
0.3
L i 1 1 1 U i 0 0 0 0.4
0.3
X 1 0.6
0.3
X 2 0.3
0.3
X 3 Design-Expert version 7 84
Inverted Simplex
3 component U_Pseudo Go to the “ Evaluation ” and view the design space: A: x1 1.000
0.000
0.000
Design-Expert version 7 1.000
B: x2 0.000
1.000
C: x3 85
A: x1 1.000
0.000
0.000
0.000
1.000
B: x2
Coding is U_Pseudo. Term StdErr**
A B C AB AC BC ABC 0.69
0.69
0.69
3.45
3.45
3.45
27.03
**Basis Std. Dev. = 1.0
VIF
1.74
1.74
1.74
1.94
1.94
1.94
1.75
1.000
C: x3
Ri-Sq
0.4255
0.4255
0.4255
0.4844
0.4844
0.4844
0.4300
Inverted Simplex
Note the Improved Values A: x1 1.000
0.000
0.000
0.000
1.000
B: x2
Coding is L_Pseudo. Term StdErr**
A B C AB AC BC ABC 26.33
26.33
26.33
104.19
104.19
104.19
216.27
**Basis Std. Dev. = 1.0
VIF
1550.78
1550.78
1550.78
2686.10
2686.10
2686.10
455.72
1.000
C: x3
Ri-Sq
0.9994
0.9994
0.9994
0.9996
0.9996
0.9996
0.9978
Design-Expert version 7 86
Inverted Simplex
3 component U_Pseudo 1.
Simulate the response using “
I-simplex U_P.sim
” 2.
Model the response.
A: x1 0.100
2 5.0
6.0
7.0
8.0
9.0
0.300
10.0 11.0 12.0
0.600
0.300
B: x2 Design-Expert version 7 8.0
9.0
0.400
R1 0.000
C: x3 87
Inverted Simplex
Upper vs Lower Bounded Pseudo Values Low becomes high and high becomes low: U_Pseudo L_Psuedo 14 12 10 8 6 4 A (1.000) B (0.000) 14 12 10 8 6 4 A (1.000) B (0.000) C (1.000) C (0.000) C (0.000) A (0.000) A (0.000) C (1.000) B (1.000) B (1.000) Design-Expert version 7 88
Design-Expert version 7
Mixture Design
“Historical Data” 89
D-optimal Design
Coordinate versus Point Exchange There are two algorithms to select “optimal” points for estimating model coefficients: Coordinate exchange Point exchange Design-Expert version 7 90
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 91
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 92
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 93
Screening Designs
Component Effects Concepts Basis for screening is a linear model:
x 1 1
2 x 2
3 x 3
q x q 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.
Design-Expert version 7 94
Screening Designs
Component Effect Directions Three directions in which component effects are assessed: 1.
Orthogonal 2.
Cox 3.
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 the Cox and Piepel directions are the same.
In a constrained simplex they’re not!
(Remember the ABS Pipe example.)
Design-Expert version 7 95
Screening Designs
Component Effect Directions Example (equation in actuals) : 1 8x 2 6x 3 A: X1 1.000
9.50
0.000
8.50
9.00
8.00
1.000
B: X2 0.000
R1 Design-Expert version 7 0.000
1.000
C: X3 10.00 9.50 9.00 8.50 8.00 7.50 A (0.800) C (0.100) B (0.800) B (0.100) A (0.100) C (0.800) 96
X 2 Design-Expert version 7
Screening Designs
Orthogonal Direction Component Effect X 1 Trace (Orthogonal) 10.00
9.50
9.00
8.50
B A 8.00
C C A B 7.50
-0.143
-0.071
0.000
0.071
0.143
Deviation from Reference Blend (L_Pseudo Units) X 3 97
Orthogonal Component Effects
Range Adjusted versus Constraint Bounded
Component
A-X1 B-X2 C-X3
Bounded Effect 0.60
0.00
-0.30
Adjusted Effect 1.80
0.00
-0.30
In constrained mixtures the “Adjusted” effect is almost never realized.
Design-Expert version 7 98
Orthogonal Component Gradients
Constraint Bounded
Component
A-X1 B-X2 C-X3
Gradient at Base Pt.
3.00
0.00
-3.00
A has a positive slope B has no slope C has a negative slope Trace (Orthogonal) 10.00
9.50
9.00
B 8.50
A 8.00
C Slope = 3.0
C A B 7.50
-0.143
-0.071
0.000
0.071
0.143
Deviation from Reference Blend (L_Pseudo Units) Design-Expert version 7 99
X 2 Design-Expert version 7 X 1
Screening Designs
Cox Direction Component Effect Trace (Cox) 10.00
9.50
9.00
B 8.50
8.00
A C C B A 7.50
-0.286
-0.143
0.000
0.143
0.286
Deviation from Reference Blend (L_Pseudo Units) X 3 100
Component
A-X1 B-X2 C-X3
Gradient at Base Pt.
2.50
-0.91
-2.94
Component
A-X1 B-X2 C-X3
Component Effect
1.00
-0.33
-0.29
Cox Component Effects
Constraint Bounded Trace (Cox) 10.00
9.50
9.00
B A 8.50
8.00
A C C Slope = 2.5
B 7.50
-0.286
-0.143
0.000
0.143
0.286
Deviation from Reference Blend (L_Pseudo Units) Design-Expert version 7 101
X 2 Design-Expert version 7
Screening Designs
Piepel Direction Component Effect X 1 Trace (Piepel) 10.00
A 9.50
B 9.00
C C 8.50
B 8.00
A 7.50
-0.500
-0.250
0.000
0.250
0.500
Deviation from Reference Blend (L_Pseudo Units) X 3 102
Component
A-X1 B-X2 C-X3
Gradient at Base Pt.
2.25
-1.43
-2.92
Component
A-X1 B-X2 C-X3
Component Effect
1.35
-1.00
-0.29
Piepel Component Effects
Constraint Bounded Trace (Piepel) 10.00
A 9.50
B 9.00
8.50
8.00
C C Slope = 2.25
B A 7.50
-0.500
-0.250
0.000
0.250
0.500
Deviation from Reference Blend (L_Pseudo Units) Design-Expert version 7 103
Summary
Component Effect Directions 1. Orthogonal : The direction for the i th 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 i th component along a line joining the reference blend to the i th 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 7 104
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 7 105
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 7 106
Combined Design
Design-Expert version 7 107
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 Mix Process Mix Process Lack of Fit Summary Statistics Adjusted Predicted Order
M
Order
M
R-Squared R-Squared
M M M M L 2FI Q C * * < 0.0001
0.9550
* * 0.6965
0.0027
0.0024
0.0024
0.0023
0.3929
0.3630
0.3630
0.3528
0.3393
0.2678
0.2678
0.2418
M L L L L L M M L 2FI Q C < 0.0001
< 0.0001
< 0.0001
* < 0.0001
* < 0.0001
< 0.0001
0.5856
* * 0.7605
0.0032
0.1534
0.1415
0.1415
0.1280
0.4350
0.9042
0.9013
0.9013
0.8966
0.3825
0.8715
0.8142
0.8142
0.7536
Aliased Aliased Aliased Aliased Design-Expert version 7 108
Design-Expert® Software Ave T exture 4.13
0.58
X1 = A: mullet X2 = B: sheepshead X3 = D: oven temp Actual Component C: croaker = 33.333
Actual Factors E: oven time = 32.50
F: deep fry = 32.50
425.00
Combined Design: Analysis
Mix-Process Contour Plot Ave Texture 2.50
412.50
2.25
1.75
400.00
2.00
1.50
387.50
375.00
Actual mullet 0.00
Actual sheepshead 66.67
16.67
50.00
33.33
33.33
50.00
16.67
66.67
0.00
Design-Expert version 7 109
Combined Design: Analysis
Mix-Process 3D Plot Design-Expert® Software Ave T exture 4.13
0.58
X1 = A: mullet X2 = B: sheepshead X3 = D: oven temp Actual Component C: croaker = 33.333
Actual Factors E: oven time = 32.50
F: deep fry = 32.50
Design-Expert version 7 2.70 2.35 2.00 1.65 1.30 425.00 412.50 400.00 D: oven temp 387.50 375.00 0.00
66.67
66.67
0.00
50.00
16.67
33.33
33.33
16.67
50.00
A: mullet B: sheepshead 110