Transcript What’s New in Design-Expert version 7 - Stat-Ease

What’s New in
Design-Expert version 7
Factorial and RSM Design
Pat Whitcomb
November, 2006
Design-Expert version 7
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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
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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 for
6 to 31 factors.
 Minimum run equireplicated resolution IV designs for
5 to 50 factors.
Design-Expert version 7
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2k-p Factorial Designs
More Choices
Need to “check” box to see factor generators
Design-Expert version 7
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2k-p Factorial Designs
Specify Base Factors for Generators
Design-Expert version 7
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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
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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.
Design-Expert version 7
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MR5 Designs
Provide 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
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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 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
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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
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MR4 Designs
Provide 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
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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
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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)
Design-Expert version 7
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Two-Level Factorial Analysis
Colored Positive and Negative Effects
Design-Expert® Software
Filtration Rate
99
Half-Normal % Probability
Shapiro-Wilk test
W-value = 0.974
p-value = 0.927
A: Temperature
B: Pressure
C: Concentration
D: Stir Rate
Positive Effects
Negative Effects
Half-Normal Plot
A
95
90
AC
AD
80
D
70
C
50
30
20
10
0
0.00
5.41
10.81
16.22
21.63
|Standardized Effect|
Design-Expert version 7
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Two-Level Factorial Analysis
Select Model Terms by “Boxing” Them.
Half-Normal Plot
Warning! No terms are selected.
99
Half-Normal % Probability
Half-Normal % Probability
99
Half-Normal Plot
95
90
80
70
50
30
20
10
0
A
95
90
AC
AD
80
D
70
C
50
30
20
10
0
0.00
5.41
10.81
16.22
|Standardized Effect|
Design-Expert version 7
21.63
0.00
5.41
10.81
16.22
21.63
|Standardized Effect|
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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 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
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Two-Level Factorial Analysis
Pareto Chart to Select Effects
Pareto Chart
C
11.27
t-Value of |Effect|
8.45
AC
A
5.63
Bonferroni Limit 5.06751
2.82
t-Value Limit 2.77645
0.00
1
2
3
4
5
6
7
Rank
Design-Expert version 7
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Two-Level Factorial Analysis
Select Aliased terms via Right Click
Design-Expert version 7
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Two-Level Factorial Analysis
Better Effect Estimates in Irregular Factions
Design-Expert version 6
Half Normal plot
Shapiro-Wilk test
W-value = 0.876
p-value = 0.171
A: water temp
B: cycle time
C: soap
D: softener
Positive Effects
A
Negative Effects
p
99
97
Half Normal %probability
95
90
C
85
80
AC
70
B
60
40
Half-Normal Plot
99
Half-Normal % Probability
PERT Plot
Design-Expert version 7
Design-Expert® Software
clean
AC
95
90
C
80
A
70
50
30
20
10
0
20
0
0.00
14.83
29.67
|Effect|
Design-Expert version 7
44.50
59.33
0.00
17.81
35.62
53.44
71.25
|Standardized Effect|
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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]
Sum of
Mean
F
Source
Squares
DF
Square
Value
Model
38135.17
4
9533.79
130.22
A
10561.33
1
10561.33
144.25
B
8.17
1
8.17
0.11
C
11285.33
1
11285.33
154.14
AC
14701.50
1
14701.50
200.80
Residual
512.50
7
73.21
Cor Total
38647.67
11
Design-Expert version 7
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
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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.
Design-Expert version 7
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Two-Level Factorial Analysis
Better Effect Estimates in Irregular Fractions
 Irregular fractions – Use the “Recalculate” key when
selecting effects.
Design-Expert version 7
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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.
Design-Expert version 7
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General Factorial Design
D-optimal Categoric Balance
Design-Expert version 7
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General Factorial Design
Choice of Nominal or Ordinal Factor Coding
Design-Expert version 7
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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.
Design-Expert version 7
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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.
Design-Expert version 7
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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] B[2]
15
-1
1
70
0
-2
125
1
1
Polynomial Contrasts
2
1
0
-1
B[1] = linear
-2
B[2] = quadratic
-3
15
Design-Expert version 7
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Temperature
125
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General Factorial Analysis
Backward Stepwise Model Reduction
Design-Expert version 7
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Select Factor Levels for Interaction Plot
Design-Expert version 7
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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
wood failure
96
81.5
67
52.5
38
LV-EPI-RT
EPI-RT
RF-RT
PRF-RT
B: Adhesive
PRF-ET
Design-Expert version 7
pine
maple
poplar
red oak
A:
chestnut
Wood
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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
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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
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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
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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 (210-3) = 158
• MR-5 = 82
• Small (Draper-Lin) = 71
Design-Expert version 7
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MR-5 CCDs (k = 6 to 30)
Number of runs closer to small CCD
600
CCD
500
n: # of runs
MR-5 CCD
400
SCCD
300
200
100
0
0
5
10
15
20
25
30
k: # of factors
Design-Expert version 7
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MR-5 CCDs (k=10, a = 1.778)
Regular, MR-5 and Small CCDs
210-3 CCD MR-5 CCD Small CCD
158 runs
82 runs
71 runs
Model
65
65
65
Residuals
92
16
5
Lack of Fit
83
11
1
Pure Error
9
5
4
157
81
70
Corr Total
Design-Expert version 7
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MR-5 CCDs (k=10, a = 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
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MR-5 CCDs (k=10, a = 1.778)
Properties closer to regular CCD
A-B slice
4
4
16
3
3
2
1
0
StdErr of Design
StdErr of Design
StdErr of Design
14
2
1
0
1.00
8
6
4
2
1.00
0.50
0.50
1.00
0.00
B: B
10
0
1.00
0.50
12
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
A: A
210-3 CCD
158 runs
Design-Expert version 7
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
MR-5 CCD
82 runs
A: A
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Small CCD
71 runs
different y-axis scale
40
MR-5 CCDs (k=10, a = 1.778)
Properties closer to regular CCD
4
4
3
3
3
2
1
0
StdErr of Design
4
StdErr of Design
StdErr of Design
A-C slice
2
1
1.00
1.00
0.50
0.50
0.50
1.00
0.00
C: C
1
0
0
1.00
2
0.50
C: C
0.00
-0.50
-0.50
-1.00 -1.00
A: A
210-3 CCD
158 runs
Design-Expert version 7
1.00
1.00
0.00
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
MR-5 CCD
82 runs
all on the same y-axis scale
0.00
C: C
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Small CCD
71 runs
41
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.
Design-Expert version 7
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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
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Standard Error Plots 26-1 CCD
1
1
0.8
0.8
0.8
0.6
0.4
0.2
0
StdErr of Design
1
StdErr of Design
StdErr of Design
Slice with the other four factors = 0
0.6
0.4
0.2
0
1.00
0.2
1.00
0.50
0.50
1.00
0.00
B: B
0.4
0
1.00
0.50
0.6
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Face Centered
a = 1.000
Design-Expert version 7
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
Practical
a = 1.565
A: A
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Spherical
a = 2.449
44
Standard Error Plots 26-1 CCD
1
1
0.8
0.8
0.8
0.6
0.4
0.2
0
StdErr of Design
1
StdErr of Design
StdErr of Design
Slice with two factors = +1 and two = 0
0.6
0.4
0.2
0
1.00
0.2
1.00
0.50
0.50
1.00
0.00
B: B
0.4
0
1.00
0.50
0.6
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Face Centered
a = 1.000
Design-Expert version 7
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
Practical
a = 1.565
A: A
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Spherical
a = 2.449
45
Standard Error Plots MR-5 CCD (k=30)
1
1
0.8
0.8
0.8
0.6
0.4
0.2
0
StdErr of Design
1
StdErr of Design
StdErr of Design
Slice with the other 28 factors = 0
0.6
0.4
0.2
0
1.00
0.2
1.00
0.50
0.50
1.00
0.00
B: B
0.4
0
1.00
0.50
0.6
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Face Centered
a = 1.000
Design-Expert version 7
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
Practical
a = 2.340
A: A
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Spherical
a = 5.477
46
Standard Error Plots MR-5 CCD (k=30)
2.4
2.4
2
2
2
1.6
1.2
0.8
0.4
0
StdErr of Design
2.4
StdErr of Design
StdErr of Design
Slice with 14 factors = +1 and 14 = 0
1.6
1.2
0.8
0.4
0
1.00
0.8
0.4
1.00
0.50
0.50
1.00
0.00
B: B
1.2
0
1.00
0.50
1.6
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Face Centered
a = 1.000
Design-Expert version 7
1.00
0.00
0.50
B: B
0.00
-0.50
-0.50
-1.00 -1.00
Practical
a = 2.340
A: A
0.50
0.00
-0.50
-0.50
-1.00 -1.00
A: A
Spherical
a = 5.477
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Choosing an alpha value for your CCD
k Practical Spherical
6
1.5651
2.4495
7
1.6266
2.6458
8
1.6818
2.8284
9
1.7321
3.0000
10
1.7783
3.1623
11
1.8212
3.3166
12
1.8612
3.4641
13
1.8988
3.6056
14
1.9343
3.7417
15
1.9680
3.8730
16
2.0000
4.0000
17
2.0305
4.1231
18
2.0598
4.2426
Design-Expert version 7
k Practical Spherical
19
2.0878
4.3589
20
2.1147
4.4721
21
2.1407
4.5826
22
2.1657
4.6904
23
2.1899
4.7958
24
2.2134
4.8990
25
2.2361
5.0000
26
2.2581
5.0990
27
2.2795
5.1962
28
2.3003
5.2915
29
2.3206
5.3852
30
2.3403
5.4772
48
D-optimal Design
Coordinate versus Point Exchange
There are two algorithms to select “optimal” points for
estimating model coefficients:
Point exchange
Coordinate exchange
Design-Expert version 7
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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.
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