Design of Experiments Guide Version 9 9.0.2

Design of Experiments Guide Version 9 9.0.2
Version 9
Design of Experiments
Guide
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9.0.2
The correct bibliographic citation for this manual is as follows: SAS Institute Inc. 2010. JMP® 9 Design
of Experiments Guide . Cary, NC: SAS Institute Inc.
JMP® 9 Design of Experiments Guide
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ISBN 978-1-60764-597-9
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Contents
Design of Experiments
1
Introduction to Designing Experiments
A Beginner’s Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
About Designing Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
My First Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 1: Design the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 2: Define Factor Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 3: Add Interaction Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 4: Determine the Number of Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 5: Check the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 6: Gather and Enter the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Step 7: Analyze the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
3
3
5
6
7
8
9
10
Examples Using the Custom Designer
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Creating Screening Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Creating a Main-Effects-Only Screening Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating a Screening Design to Fit All Two-Factor Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Compromise Design Between Main Effects Only and All Interactions . . . . . . . . . . . . . . . . . . . .
Creating ‘Super’ Screening Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Screening Designs with Flexible Block Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Checking for Curvature Using One Extra Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
19
21
23
27
31
Creating Response Surface Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Exploring the Prediction Variance Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introducing I-Optimal Designs for Response Surface Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Three-Factor Response Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Response Surface with a Blocking Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
37
38
40
Creating Mixture Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Mixtures Having Nonmixture Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Experiments that are Mixtures of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
ii
Special-Purpose Uses of the Custom Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Designing Experiments with Fixed Covariate Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Creating a Design with Two Hard-to-Change Factors: Split Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Technical Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3
Building Custom Designs
The Basic Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Creating a Custom Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Enter Responses and Factors into the Custom Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Describe the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specifying Alias Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Select the Number of Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Understanding Design Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specify Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Make the JMP Design Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
69
70
71
72
78
79
Creating Random Block Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Creating Split Plot Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Creating Split-Split Plot Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Creating Strip Plot Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Special Custom Design Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Save Responses and Save Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Load Responses and Load Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Save Constraints and Load Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Set Random Seed: Setting the Number Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Simulate Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Save X Matrix: Viewing the Number of Rows in the Moments Matrix and the Design Matrix (X) in the
Log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Optimality Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Number of Starts: Changing the Number of Random Starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Sphere Radius: Constraining a Design to a Hypersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Disallowed Combinations: Accounting for Factor Level Restrictions . . . . . . . . . . . . . . . . . . . . . . . 90
Advanced Options for the Custom Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Save Script to Script Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Assigning Column Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Define Low and High Values (DOE Coding) for Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Set Columns as Factors for Mixture Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Define Response Column Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
iii
Assign Columns a Design Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Identify Factor Changes Column Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
How Custom Designs Work: Behind the Scenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4
Screening Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Screening Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Using Two Continuous Factors and One Categorical Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Using Five Continuous Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Creating a Screening Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Enter Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Enter Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Choose a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display and Modify a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specify Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
View the Design Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
110
111
115
119
120
Create a Plackett-Burman design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Analysis of Screening Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Using the Screening Analysis Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Using the Fit Model Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5
Response Surface Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A Box-Behnken Design: The Tennis Ball Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
The Prediction Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A Response Surface Plot (Contour Profiler) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Geometry of a Box-Behnken Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Creating a Response Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Enter Responses and Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Choose a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specify Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
View the Design Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
137
138
139
140
Full Factorial Designs
.............................................................................
143
The Five-Factor Reactor Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Analyze the Reactor Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Creating a Factorial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
iv
Enter Responses and Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Select Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Make the Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7
Mixture Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Mixture Design Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
The Optimal Mixture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
The Simplex Centroid Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Creating the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Simplex Centroid Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
The Simplex Lattice Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
The Extreme Vertices Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Creating the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Extreme Vertices Example with Range Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Extreme Vertices Example with Linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Extreme Vertices Method: How It Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164
164
166
167
The ABCD Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Creating Ternary Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Fitting Mixture Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Whole Model Tests and Analysis of Variance Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Understanding Response Surface Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A Chemical Mixture Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Create the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analyze the Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Prediction Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Mixture Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Ternary Plot of the Mixture Response Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
171
174
175
176
177
Discrete Choice Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Create an Example Choice Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Analyze the Example Choice Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Design a Choice Experiment Using Prior Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Administer the Survey and Analyze Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
v
Initial Choice Platform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Find Unit Cost and Trade Off Costs with the Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9
Space-Filling Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Introduction to Space-Filling Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Sphere-Packing Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Creating a Sphere-Packing Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Visualizing the Sphere-Packing Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Latin Hypercube Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Creating a Latin Hypercube Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Visualizing the Latin Hypercube Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Uniform Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Comparing Sphere-Packing, Latin Hypercube, and Uniform Methods . . . . . . . . . . . . . . . . . . . . . . . . 206
Minimum Potential Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Maximum Entropy Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Gaussian Process IMSE Optimal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Borehole Model: A Sphere-Packing Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Create the Sphere-Packing Design for the Borehole Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Guidelines for the Analysis of Deterministic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Results of the Borehole Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10 Accelerated Life Test Designs
Designing Experiments for Accelerated Life Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Overview of Accelerated Life Test Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Using the ALT Design Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11 Nonlinear Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Examples of Nonlinear Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Using Nonlinear Fit to Find Prior Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Creating a Nonlinear Design with No Prior Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Creating a Nonlinear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Identify the Response and Factor Column with Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
vi
Set Up Factors and Parameters in the Nonlinear Design Dialog . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Enter the Number of Runs and Preview the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Make Table or Augment the Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Advanced Options for the Nonlinear Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
12 Taguchi Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
The Taguchi Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Taguchi Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Analyze the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Creating a Taguchi Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Detail the Response and Add Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Choose Inner and Outer Array Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display Coded Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Make the Design Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
254
255
256
257
13 Augmented Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
A D-Optimal Augmentation of the Reactor Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Analyze the Augmented Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Creating an Augmented Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Replicate a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Add Center Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating a Foldover Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adding Axial Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adding New Runs and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
272
275
276
277
278
Special Augment Design Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Save the Design (X) Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modify the Design Criterion (D- or I- Optimality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Select the Number of Random Starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specify the Sphere Radius Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Disallow Factor Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
282
282
283
283
284
14 Prospective Sample Size and Power
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Launching the Sample Size and Power Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
One-Sample and Two-Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Single-Sample Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
vii
Sample Size and Power Animation for One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Two-Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
k-Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
One Sample Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
One Sample Standard Deviation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
One-Sample and Two-Sample Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
One Sample Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Two Sample Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Counts per Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Counts per Unit Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Sigma Quality Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Sigma Quality Level Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Number of Defects Computation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Reliability Test Plan and Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Reliability Test Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Reliability Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Index
Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
317
viii
Credits and Acknowledgments
Origin
JMP was developed by SAS Institute Inc., Cary, NC. JMP is not a part of the SAS System, though portions
of JMP were adapted from routines in the SAS System, particularly for linear algebra and probability
calculations. Version 1 of JMP went into production in October 1989.
Credits
JMP was conceived and started by John Sall. Design and development were done by John Sall, Chung-Wei
Ng, Michael Hecht, Richard Potter, Brian Corcoran, Annie Dudley Zangi, Bradley Jones, Craige Hales,
Chris Gotwalt, Paul Nelson, Xan Gregg, Jianfeng Ding, Eric Hill, John Schroedl, Laura Lancaster, Scott
McQuiggan, Melinda Thielbar, Clay Barker, Peng Liu, Dave Barbour, Jeff Polzin, John Ponte, and Steve
Amerige.
In the SAS Institute Technical Support division, Duane Hayes, Wendy Murphrey, Rosemary Lucas, Win
LeDinh, Bobby Riggs, Glen Grimme, Sue Walsh, Mike Stockstill, Kathleen Kiernan, and Liz Edwards
provide technical support.
Nicole Jones, Kyoko Keener, Hui Di, Joseph Morgan, Wenjun Bao, Fang Chen, Susan Shao, Michael
Crotty, Jong-Seok Lee, Tonya Mauldin, Audrey Ventura, Ani Eloyan, Bo Meng, and Sequola McNeill
provide ongoing quality assurance. Additional testing and technical support are provided by Noriki Inoue,
Kyoko Takenaka, Yusuke Ono, Masakazu Okada, and Naohiro Masukawa from SAS Japan.
Bob Hickey and Jim Borek are the release engineers.
The JMP books were written by Ann Lehman, Lee Creighton, John Sall, Bradley Jones, Erin Vang, Melanie
Drake, Meredith Blackwelder, Diane Perhac, Jonathan Gatlin, Susan Conaghan, and Sheila Loring, with
contributions from Annie Dudley Zangi and Brian Corcoran. Creative services and production was done by
SAS Publications. Melanie Drake implemented the Help system.
Jon Weisz and Jeff Perkinson provided project management. Also thanks to Lou Valente, Ian Cox, Mark
Bailey, and Malcolm Moore for technical advice.
Thanks also to Georges Guirguis, Warren Sarle, Gordon Johnston, Duane Hayes, Russell Wolfinger,
Randall Tobias, Robert N. Rodriguez, Ying So, Warren Kuhfeld, George MacKensie, Bob Lucas, Warren
Kuhfeld, Mike Leonard, and Padraic Neville for statistical R&D support. Thanks are also due to Doug
Melzer, Bryan Wolfe, Vincent DelGobbo, Biff Beers, Russell Gonsalves, Mitchel Soltys, Dave Mackie, and
Stephanie Smith, who helped us get started with SAS Foundation Services from JMP.
Acknowledgments
We owe special gratitude to the people that encouraged us to start JMP, to the alpha and beta testers of JMP,
and to the reviewers of the documentation. In particular we thank Michael Benson, Howard Yetter (d),
x
Andy Mauromoustakos, Al Best, Stan Young, Robert Muenchen, Lenore Herzenberg, Ramon Leon, Tom
Lange, Homer Hegedus, Skip Weed, Michael Emptage, Pat Spagan, Paul Wenz, Mike Bowen, Lori Gates,
Georgia Morgan, David Tanaka, Zoe Jewell, Sky Alibhai, David Coleman, Linda Blazek, Michael Friendly,
Joe Hockman, Frank Shen, J.H. Goodman, David Iklé, Barry Hembree, Dan Obermiller, Jeff Sweeney,
Lynn Vanatta, and Kris Ghosh.
Also, we thank Dick DeVeaux, Gray McQuarrie, Robert Stine, George Fraction, Avigdor Cahaner, José
Ramirez, Gudmunder Axelsson, Al Fulmer, Cary Tuckfield, Ron Thisted, Nancy McDermott, Veronica
Czitrom, Tom Johnson, Cy Wegman, Paul Dwyer, DaRon Huffaker, Kevin Norwood, Mike Thompson,
Jack Reese, Francois Mainville, and John Wass.
We also thank the following individuals for expert advice in their statistical specialties: R. Hocking and P.
Spector for advice on effective hypotheses; Robert Mee for screening design generators; Roselinde Kessels
for advice on choice experiments; Greg Piepel, Peter Goos, J. Stuart Hunter, Dennis Lin, Doug
Montgomery, and Chris Nachtsheim for advice on design of experiments; Jason Hsu for advice on multiple
comparisons methods (not all of which we were able to incorporate in JMP); Ralph O’Brien for advice on
homogeneity of variance tests; Ralph O’Brien and S. Paul Wright for advice on statistical power; Keith
Muller for advice in multivariate methods, Harry Martz, Wayne Nelson, Ramon Leon, Dave Trindade, Paul
Tobias, and William Q. Meeker for advice on reliability plots; Lijian Yang and J.S. Marron for bivariate
smoothing design; George Milliken and Yurii Bulavski for development of mixed models; Will Potts and
Cathy Maahs-Fladung for data mining; Clay Thompson for advice on contour plotting algorithms; and
Tom Little, Damon Stoddard, Blanton Godfrey, Tim Clapp, and Joe Ficalora for advice in the area of Six
Sigma; and Josef Schmee and Alan Bowman for advice on simulation and tolerance design.
For sample data, thanks to Patrice Strahle for Pareto examples, the Texas air control board for the pollution
data, and David Coleman for the pollen (eureka) data.
Translations
Trish O'Grady coordinates localization. Special thanks to Noriki Inoue, Kyoko Takenaka, Masakazu Okada,
Naohiro Masukawa, and Yusuke Ono (SAS Japan); and Professor Toshiro Haga (retired, Tokyo University
of Science) and Professor Hirohiko Asano (Tokyo Metropolitan University) for reviewing our Japanese
translation; François Bergeret for reviewing the French translation; Bertram Schäfer and David Meintrup
(consultants, StatCon) for reviewing the German translation; Patrizia Omodei, Maria Scaccabarozzi, and
Letizia Bazzani (SAS Italy) for reviewing the Italian translation; RuiQi Qiao, Rula Li, and Molly Li for
reviewing Simplified Chinese translation (SAS R&D Beijing); Finally, thanks to all the members of our
outstanding translation and engineering teams.
Past Support
Many people were important in the evolution of JMP. Special thanks to David DeLong, Mary Cole, Kristin
Nauta, Aaron Walker, Ike Walker, Eric Gjertsen, Dave Tilley, Ruth Lee, Annette Sanders, Tim Christensen,
Eric Wasserman, Charles Soper, Wenjie Bao, and Junji Kishimoto. Thanks to SAS Institute quality
assurance by Jeanne Martin, Fouad Younan, and Frank Lassiter. Additional testing for Versions 3 and 4 was
done by Li Yang, Brenda Sun, Katrina Hauser, and Andrea Ritter.
Also thanks to Jenny Kendall, John Hansen, Eddie Routten, David Schlotzhauer, and James Mulherin.
Thanks to Steve Shack, Greg Weier, and Maura Stokes for testing JMP Version 1.
xi
Thanks for support from Charles Shipp, Harold Gugel (d), Jim Winters, Matthew Lay, Tim Rey, Rubin
Gabriel, Brian Ruff, William Lisowski, David Morganstein, Tom Esposito, Susan West, Chris Fehily, Dan
Chilko, Jim Shook, Ken Bodner, Rick Blahunka, Dana C. Aultman, and William Fehlner.
Technology License Notices
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xii
Chapter 1
Introduction to Designing Experiments
A Beginner’s Tutorial
This tutorial chapter introduces you to the design of experiments (DOE) using JMP’s custom designer. It
gives a general understanding of how to design an experiment using JMP. Refer to subsequent chapters in
this book for more examples and procedures on how to design an experiment for your specific project.
Contents
About Designing Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
My First Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Step 1: Design the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Step 2: Define Factor Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Step 3: Add Interaction Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Step 4: Determine the Number of Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Step 5: Check the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Step 6: Gather and Enter the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Step 7: Analyze the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Chapter 1
Introduction to Designing Experiments
About Designing Experiments
3
About Designing Experiments
Increasing productivity and improving quality are important goals in any business. The methods for
determining how to increase productivity and improve quality are evolving. They have changed from costly
and time-consuming trial-and-error searches to the powerful, elegant, and cost-effective statistical methods
that JMP provides.
Designing experiments in JMP is centered around factors, responses, a model, and runs. JMP helps you
determine if and how a factor affects a response.
My First Experiment
If you have never used JMP to design an experiment, this section shows you how to design the experiment
and how to understand JMP’s output.
Tip: The recommended way to create an experiment is to use the custom designer. JMP also provides
classical designs for use in textbook situations.
The Situation
Your goal is to find the best way to microwave a bag of popcorn. Because you have some experience with
this, it is easy to decide on reasonable ranges for the important factors:
•
how long to cook the popcorn (between 3 and 5 minutes)
•
what level of power to use on the microwave oven (between settings 5 and 10)
•
which brand of popcorn to use (Top Secret or Wilbur)
When a bag of popcorn is popped, most of the kernels pop, but some remain unpopped. You prefer to have
all (or nearly all) of the kernels popped and no (or very few) unpopped kernels. Therefore, you define “the
best popped bag” based on the ratio of popped kernels to the total number of kernels.
A good way to improve any procedure is to conduct an experiment. For each experimental run, JMP’s
custom designer determines which brand to use, how long to cook each bag in the microwave and what
power setting to use. Each run involves popping one bag of corn. After popping a bag, enter the total
number of kernels and the number of popped kernels into the appropriate row of a JMP data table. After
doing all the experimental runs, use JMP’s model fitting capabilities to do the data analysis. Then, you can
use JMP’s profiling tools to determine the optimal settings of popping time, power level, and brand.
Step 1: Design the Experiment
The first step is to select DOE > Custom Design. Then, define the responses and factors.
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Chapter 1
Define the Responses: Popped Kernels and Total Kernels
There are two responses in this experiment:
•
the number of popped kernels
•
the total number of kernels in the bag. After popping the bag add the number of unpopped kernels to
the number of popped kernels to get the total number of kernels in the bag.
By default, the custom designer contains one response labeled Y (Figure 1.1).
Figure 1.1 Custom Design Responses Panel
You want to add a second response to the Responses panel and change the names to be more descriptive:
1. To rename the Y response, double-click the name and type “Number Popped.” Since you want to
increase the number of popped kernels, leave the goal at Maximize.
2. To add the second response (total number of kernels), click Add Response and choose None from the
menu that appears. JMP labels this response Y2 by default.
3. Double-click Y2 and type “Total Kernels” to rename it.
The completed Responses panel looks like Figure 1.2.
Figure 1.2 Renamed Responses with Specified Goals
Define the Factors: Time, Power, and Brand
In this experiment, the factors are:
Chapter 1
Introduction to Designing Experiments
My First Experiment
•
brand of popcorn (Top Secret or Wilbur)
•
cooking time for the popcorn (3 or 5 minutes)
•
microwave oven power level (setting 5 or 10)
5
In the Factors panel, add Brand as a two-level categorical factor:
1. Click Add Factor and select Categorical > 2 Level.
2. To change the name of the factor (currently named X1), type Brand.
3. To rename the default levels (L1 and L2), click the level names and type Top Secret and Wilbur.
Add Time as a two-level continuous factor:
4. Click Add Factor and select Continuous.
5. Change the default name of the factor (X2) by typing Time.
6. Likewise, to rename the default levels (–1 and 1) as 3 and 5, click the current level name and type in the
new value.
Add Power as a two-level continuous factor:
7. Click Add Factor and select Continuous.
8. Change the name of the factor (currently named X3) by typing Power.
9. Rename the default levels (currently named -1 and 1) as 5 and 10 by clicking the current name and
typing. The completed Factors panel looks like Figure 1.3.
Figure 1.3 Renamed Factors with Specified Values
10. Click Continue.
Step 2: Define Factor Constraints
The popping time for this experiment is either 3 or 5 minutes, and the power settings on the microwave are
5 and 10. From experience, you know that
•
popping corn for a long time on a high setting tends to scorch kernels.
•
not many kernels pop when the popping time is brief and the power setting is low.
You want to constrain the combined popping time and power settings to be less than or equal to 13, but
greater than or equal to 10. To define these limits:
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1. Open the Constraints panel by clicking the disclosure button beside the Define Factor Constraints title
bar (see Figure 1.4).
2. Click the Add Constraint button twice, once for each of the known constraints.
3. Complete the information, as shown to the right in Figure 1.4. These constraints tell the Custom
Designer to avoid combinations of Power and Time that sum to less than 10 and more than 13. Be sure
to change <= to >= in the second constraint.
The area inside the parallelogram, illustrated on the left in Figure 1.4, is the allowable region for the runs.
You can see that popping for 5 minutes at a power of 10 is not allowed and neither is popping for 3 minutes
at a power of 5.
Figure 1.4 Defining Factor Constraints
Step 3: Add Interaction Terms
You are interested in the possibility that the effect of any factor on the proportion of popped kernels may
depend on the value of some other factor. For example, the effect of a change in popping time for the
Wilbur popcorn brand could be larger than the same change in time for the Top Secret brand. This kind of
synergistic effect of factors acting in concert is called a two-factor interaction. You can examine all possible
two-factor interactions in your a priori model of the popcorn popping process.
1. Click Interactions in the Model panel and select 2nd. JMP adds two-factor interactions to the model as
shown to the left in Figure 1.5.
In addition, you suspect the graph of the relationship between any factor and any response might be curved.
You can see whether this kind of curvature exists with a quadratic model formed by adding the second order
powers of effects to the model, as follows.
2. Click Powers and select 2nd to add quadratic effects of the continuous factors, Power and Time.
The completed Model should look like the one to the right in Figure 1.5.
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My First Experiment
7
Figure 1.5 Add Interaction and Power Terms to the Model
Step 4: Determine the Number of Runs
The Design Generation panel in Figure 1.6 shows the minimum number of runs needed to perform the
experiment with the effects you’ve added to the model. You can use that minimum or the default number of
runs, or you can specify your own number of runs as long as that number is more than the minimum. JMP
has no restrictions on the number of runs you request. For this example, use the default number of runs, 16.
Click Make Design to continue.
Figure 1.6 Model and Design Generation Panels
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Step 5: Check the Design
When you click Make Design, JMP generates and displays a design, as shown on the left in Figure 1.7.
Note that because JMP uses a random seed to generate custom designs and there is no unique optimal
design for this problem, your table may be different than the one shown here. You can see in the table that
the custom design requires 8 runs using each brand of popcorn.
Scroll to the bottom of the Custom Design window and look at the Output Options area (shown to the
right in Figure 1.7. The Run Order option lets you designate the order you want the runs to appear in the
data table when it is created. Keep the selection at Randomize so the rows (runs) in the output table appear
in a random order.
Now click Make Table in the Output Options section.
Figure 1.7 Design and Output Options Section of Custom Designer
The resulting data table (Figure 1.8) shows the order in which you should do the experimental runs and
provides columns for you to enter the number of popped and total kernels. Note that the design matrix is
updated to match the order of runs, the Time and Power values, and the Number Popped and Total Kernels
columns are added.
You do not have fractional control over the power and time settings on a microwave oven, so you should
round the power and time settings, as shown in the data table. Although this altered design is slightly less
optimal than the one the custom designer suggested, the difference is negligible.
Tip: Note that optionally, before clicking Make Table in the Output Options, you could select Sort Left to
Right in the Run Order menu to have JMP present the results in the data table according to the brand. We
have conducted this experiment for you and placed the results, called Popcorn DOE Results.jmp, in the
Sample Data folder installed with JMP. These results have the columns sorted from left to right.
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Introduction to Designing Experiments
My First Experiment
9
Figure 1.8 JMP Data Table of Design Runs Generated by Custom Designer
Step 6: Gather and Enter the Data
Pop the popcorn according to the design JMP provided. Then, count the number of popped and unpopped
kernels left in each bag. Finally, enter the numbers shown below into the appropriate columns of the data
table.
We have conducted this experiment for you and placed the results in the Sample Data folder installed with
JMP. To see the results, open Popcorn DOE Results.jmp from the Design Experiment folder in the sample
data. The data table is shown in Figure 1.9.
Figure 1.9 Results of the Popcorn DOE Experiment
scripts to
analyze data
results from
experiment
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My First Experiment
Chapter 1
Step 7: Analyze the Results
After the experiment is finished and the number of popped kernels and total kernels have been entered into
the data table, it is time to analyze the data. The design data table has a script, labeled Model, that shows in
the top left panel of the table. When you created the design, a standard least squares analysis was stored in
the Model script with the data table.
1. Click the red triangle for Model and select Run Script.
The default fitting personality in the model dialog is Standard Least Squares. One assumption of
standard least squares is that your responses are normally distributed. But because you are modeling the
proportion of popped kernels it is more appropriate to assume that your responses come from a binomial
distribution. You can use this assumption by changing to a generalized linear model.
2. Change the Personality to Generalized Linear Model, Distribution to Binomial, and Link Function to
Logit, as shown in Figure 1.10.
Figure 1.10 Fitting the Model
3. Click Run.
4. Scroll down to view the Effect Tests table (Figure 1.11) and look in the column labeled Prob>Chisq.
This column lists p-values. A low p-value (a value less than 0.05) indicates that results are statistically
significant. There are asterisks that identify the low p-values. You can therefore conclude that, in this
experiment, all the model effects except for Time*Time are highly significant. You have confirmed that
there is a strong relationship between popping time (Time), microwave setting (Power), popcorn brand
(Brand), and the proportion of popped kernels.
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Introduction to Designing Experiments
My First Experiment
11
Figure 1.11 Investigating p-Values
p-values indicate significance.
Values with * beside them are
p-values that indicate the results
are statistically significant.
To further investigate, use the Prediction Profiler to see how changes in the factor settings affect the
numbers of popped and unpopped kernels:
1. Choose Profilers > Profiler from the red triangle menu on the Generalized Linear Model Fit title bar.
The Prediction Profiler is shown at the bottom of the report. Figure 1.12 shows the Prediction Profiler
for the popcorn experiment. Prediction traces are displayed for each factor.
Figure 1.12 The Prediction Profiler
Disclosure icon to
open or close the
Prediction Profiler
Prediction trace
for Brand
Prediction trace
for Time
Prediction trace
for Power
predicted value
of the response
95% confidence
interval on the mean
response
Factor values (here, time = 4)
2. Move the vertical red dotted lines to see the effect that changing a factor value has on the response. For
example, drag the red line in the Time graph to the right and left (Figure 1.13).
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Figure 1.13 Moving the Time Value from 4 to Near 5
As Time increases and decreases, the curved Brand and Power prediction traces shift their slope and
maximum/minimum values. The substantial slope shift tells you there is an interaction (synergistic effect)
between Time and Brand and Time and Power.
Furthermore, the steepness of a prediction trace reveals a factor’s importance. Because the prediction trace
for Time is steeper than that for Brand or Power for the values shown in Figure 1.13, you can predict that
cooking time is more important than the brand of popcorn or the microwave power setting.
Now for the final steps.
3. Click the red triangle icon in the Prediction Profiler title bar and select Desirability Functions.
4. Click the red triangle icon in the Prediction Profiler title bar and select Maximize Desirability. JMP
automatically adjusts the graph to display the optimal settings at which the most kernels will be popped
(Figure 1.14).
Our experiment found how to cook the bag of popcorn with the greatest proportion of popped kernels: use
Top Secret, cook for five minutes, and use a power level of 8. The experiment predicts that cooking at these
settings will yield greater than 96.5% popped kernels.
Chapter 1
Introduction to Designing Experiments
My First Experiment
Figure 1.14 The Most Desirable Settings
The best settings are the Top Secret brand, cooking time at 5, and power set at 8.
13
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Chapter 1
Chapter 2
Examples Using the Custom Designer
The use of statistical methods in industry is increasing. Arguably, the most cost-beneficial of these methods
for quality and productivity improvement is statistical design of experiments. A trial-and -error search for
the vital few factors that most affect quality is costly and time-consuming. The purpose of experimental
design is to characterize, predict, and then improve the behavior of any system or process. Designed
experiments are a cost-effective way to accomplish these goals.
JMP’s custom designer is the recommended way to describe your process and create a design that works for
your situation. To use the custom designer, you first enter the process variables and constraints, then JMP
tailors a design to suit your unique case. This approach is more general and requires less experience and
expertise than previous tools supporting the statistical design of experiments.
Custom designs accommodate any number of factors of any type. You can also control the number of
experimental runs. This makes custom design more flexible and more cost effective than alternative
approaches.
This chapter presents several examples showing the use of custom designs. It shows how to drive its interface
to build a design using this easy step-by-step approach:
Figure 2.1 Approach to Experimental Design
Key engineering steps: process knowledge and
engineering judgement are important.
Describe
Identify factors
and responses.
Design
Compute design
for maximum
information from
runs.
Collect
Use design to set
factors: measure
response for each
run.
Fit
Compute best fit
of mathematical
model to data
from test runs.
Key mathematical steps: appropriate
computer-based tools are empowering.
Predict
Use model to find
best factor settings
for on-target
responses and
minimum variability.
Contents
Creating Screening Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Creating a Main-Effects-Only Screening Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Creating a Screening Design to Fit All Two-Factor Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
A Compromise Design Between Main Effects Only and All Interactions. . . . . . . . . . . . . . . . . . . . . . . 21
Creating ‘Super’ Screening Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
Screening Designs with Flexible Block Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Checking for Curvature Using One Extra Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Creating Response Surface Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Exploring the Prediction Variance Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Introducing I-Optimal Designs for Response Surface Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
A Three-Factor Response Surface Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
Response Surface with a Blocking Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Creating Mixture Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Mixtures Having Nonmixture Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Experiments that are Mixtures of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Special-Purpose Uses of the Custom Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Designing Experiments with Fixed Covariate Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Creating a Design with Two Hard-to-Change Factors: Split Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Technical Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 2
Examples Using the Custom Designer
Creating Screening Experiments
17
Creating Screening Experiments
You can use the screening designer in JMP to create screening designs, but the custom designer is more
flexible and general. The straightforward screening examples described below show that ‘custom’ does not
mean ‘exotic.’ The custom designer is a general purpose design environment that can create screening
designs.
Creating a Main-Effects-Only Screening Design
To create a main-effects-only screening design using the custom designer:
1. Select DOE > Custom Design.
2. Enter six continuous factors into the Factors panel (see “Step 1: Design the Experiment,” p. 3, for
details). Figure 2.2 shows the six factors.
3. Click Continue. The default model contains only the main effects.
4. Using the default of eight runs, click Make Design. Click the disclosure button (
on the Macintosh) to open the Design Evaluation outline node.
on Windows and
Note to DOE experts: The result is a resolution-three screening design. All the main effects are
estimable, but they are confounded with two factor interactions.
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Creating Screening Experiments
Chapter 2
Figure 2.2 A Main-Effects-Only Screening Design
5. Click the disclosure buttons beside Design Evaluation and then beside Alias Matrix (
on Windows
and
on the Macintosh) to open the Alias Matrix. Figure 2.3 shows the Alias Matrix, which is a
table of zeros, ones, and negative ones.
The Alias Matrix shows how the coefficients of the constant and main effect terms in the model are biased
by any active two-factor interaction effects not already added to the model. The column labels identify
interactions. For example, the columns labeled X2*X6 and X3*X4 in the table have a 1 and -1 in the row
for X1. This means that the expected value of the main effect of X1 is actually the sum of the main effect of
X1 and X2*X6, minus the effect of X3*X4. You are assuming that these interactions are negligible in size
compared to the effect of X1.
Figure 2.3 The Alias Matrix
Chapter 2
Examples Using the Custom Designer
Creating Screening Experiments
19
Note to DOE experts: The Alias matrix is a generalization of the confounding pattern in fractional
factorial designs.
Creating a Screening Design to Fit All Two-Factor Interactions
There is risk involved in designs for main effects only. The risk is that two-factor interactions, if they are
strong, can confuse the results of such experiments. To avoid this risk, you can create experiments resolving
all the two-factor interactions.
Note to DOE experts: The result in this example is a resolution-five screening design. Two-factor
interactions are estimable but are confounded with three-factor interactions.
1. Select DOE > Custom Design.
2. Enter five continuous factors into the Factors panel (see “Step 1: Design the Experiment,” p. 3 in the
“Introduction to Designing Experiments” chapter for details).
3. Click Continue.
4. In the Model panel, select Interactions > 2nd.
5. In the Design Generation Panel choose Minimum for Number of Runs and click Make Design.
Figure 2.4 shows the runs of the two-factor design with all interactions. The sample size, 16 (a power of
two) is large enough to fit all the terms in the model. The values in your table may be different from those
shown below.
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Examples Using the Custom Designer
Creating Screening Experiments
Chapter 2
Figure 2.4 All Two-Factor Interactions
Open
outline
nodes
6. Click the disclosure button (
on Windows and
on the Macintosh) and to open the Design
Evaluation outlines, then open Alias Matrix. Figure 2.5 shows the alias matrix table of zeros and ones.
The columns labels identify an interaction. For example, the column labelled X1*X2 refers to the
interaction of the first and second effect, the column labelled X2*X3 refers to the interaction between the
second and third effect, and so forth.
Look at the column labelled X1*X2. There is only one value of 1 in that column. All others are 0. The 1
occurs in the row labelled X1*X2. All the other rows and columns are similar. This means that the
expected value of the two-factor interaction X1*X2 is not biased by any other terms. All the rows above
the row labelled X1*X2 contain only zeros, which means that the Intercept and main effect terms are not
biased by any two-factor interactions.
Chapter 2
Examples Using the Custom Designer
Creating Screening Experiments
21
Figure 2.5 Alias Matrix Showing all Two-Factor Interactions Clear of all Main Effects
A Compromise Design Between Main Effects Only and All Interactions
In a screening situation, suppose there are six continuous factors and resources for n = 16 runs. The first
example in this section showed an eight-run design that fit all the main effects. With six factors, there are 15
possible two-factor interactions. The minimum number of runs that could fit the constant, six main effects
and 15 two-factor interactions is 22. This is more than the resource budget of 16 runs. It would be good to
find a compromise between the main-effects only design and a design capable of fitting all the two-factor
interactions.
This example shows how to obtain such a design compromise using the custom designer.
1. Select DOE > Custom Design.
2. Define six continuous factors (X1 - X6).
3. Click Continue. The model includes the main effect terms by default. The default estimability of these
terms is Necessary.
4. Click the Interactions button and choose 2nd to add all the two-factor interactions.
5. Select all the interaction terms and click the current estimability (Necessary) to reveal a menu. Change
Necessary to If Possible, as shown in Figure 2.6.
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Creating Screening Experiments
Chapter 2
Figure 2.6 Model for Six-Variable Design with Two-Factor Interactions Designated If Possible
6. Type 16 in the User Specified edit box in the Number of Runs section, as shown. Although the desired
number of runs (16) is less than the total number of model terms, the custom designer builds a design to
estimate as many two-factor interactions as possible.
7. Click Make Design.
After the custom designer creates the design, click the disclosure button beside Design Evaluation to open
the Alias Matrix (Figure 2.7). The values in your table may be different from those shown below, but with a
similar pattern.
Chapter 2
Examples Using the Custom Designer
Creating Screening Experiments
23
Figure 2.7 Alias Matrix
All the rows above the row labelled X1*X2 contain only zeros, which means that the Intercept and main
effect terms are not biased by any two-factor interactions. The row labelled X1*X2 has the value 0.333 in the
X1*X2 column and the same value in the X3*X5 column. That means the expected value of the estimate for
X1*X2 is actually the sum of X1*X2 and any real effect due to X3*X5.
Note to DOE experts: The result in this particular example is a resolution-four screening design.
Two-factor interactions are estimable but are aliased with other two-factor interactions.
Creating ‘Super’ Screening Designs
This section shows how to use the technique shown in the previous example to create ‘super’
(supersaturated) screening designs. Supersaturated designs have fewer runs than factors, which makes them
attractive for factor screening when there are many factors and experimental runs are expensive.
In a saturated design, the number of runs equals the number of model terms. In a supersaturated design, as
the name suggests, the number of model terms exceeds the number of runs (Lin, 1993). A supersaturated
design can examine dozens of factors using fewer than half as many runs as factors.
The Need for Supersaturated Designs
The 27–4 and the 215–11 fractional factorial designs available using the screening designer are both saturated
with respect to a main effects model. In the analysis of a saturated design, you can (barely) fit the model, but
there are no degrees of freedom for error or for lack of fit. Until recently, saturated designs represented the
limit of efficiency in designs for screening.
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Creating Screening Experiments
Chapter 2
Factor screening relies on the sparsity principle. The experimenter expects that only a few of the factors in a
screening experiment are active. The problem is not knowing which are the vital few factors and which are
the trivial many. It is common for brainstorming sessions to turn up dozens of factors. Yet, in practice,
screening experiments rarely involve more than ten factors. What happens to winnow the list from dozens to
ten or so?
If the experimenter is limited to designs that have more runs than factors, then dozens of factors translate
into dozens of runs. Often, this is not economically feasible. The result is that the factor list is reduced
without the benefit of data. In a supersaturated design, the number of model terms exceeds the number of
runs, and you can examine dozens of factors using less than half as many runs.
There are drawbacks:
•
If the number of active factors approaches the number of runs in the experiment, then it is likely that
these factors will be impossible to identify. A rule of thumb is that the number of runs should be at least
four times larger than the number of active factors. If you expect that there might be as many as five
active factors, you should have at least 20 runs.
•
Analysis of supersaturated designs cannot yet be reduced to an automatic procedure. However, using
forward stepwise regression is reasonable and the Screening platform (Analyze > Modeling >
Screening) offers a more streamlined analysis.
Example: Twelve Factors in Eight Runs
As an example, consider a supersaturated design with twelve factors. Using model terms designated If
Possible provides the software machinery for creating a supersaturated design.
In the last example, two-factor interaction terms were specified If Possible. In a supersaturated design, all
terms—including main effects—are If Possible. Note in Figure 2.8, the only primary term is the intercept.
To see an example of a supersaturated design with twelve factors in eight runs:
1. Select DOE > Custom Design.
2. Add 12 continuous factors and click Continue.
3. Highlight all terms except the Intercept and click the current estimability (Necessary) to reveal the
menu. Change Necessary to If Possible, as shown in Figure 2.8.
Chapter 2
Examples Using the Custom Designer
Creating Screening Experiments
25
Figure 2.8 Changing the Estimability
4. The desired number of runs is eight so type 8 in the User Specified edit box in the Number of Runs
section.
5. Click the red triangle on the Custom Design title bar and select Simulate Responses.
6. Click Make Design, then click Make Table. A window named Simulate Responses and a design table
appear, similar to the one in Figure 2.9. The Y column values are controlled by the coefficients of the
model in the Simulate Responses window. The values in your table may be different from those shown
below.
Figure 2.9 Simulated Responses and Design Table
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Examples Using the Custom Designer
Creating Screening Experiments
Chapter 2
7. Change the default settings of the coefficients in the Simulate Responses dialog to match those in
Figure 2.10 and click Apply. The numbers in the Y column change. Because you have set X2 and X10 as
active factors in the simulation, the analysis should be able to identify the same two factors.
Note that random noise is added to the Y column formula, so the numbers you see might not necessarily
match those in the figure. The values in your table may be different from those shown below.
Figure 2.10 Give Values to Two Main Effects and Specify the Standard Error as 0.5
To identify active factors using stepwise regression:
1. To run the Model script in the design table, click the red triangle beside Model and select Run Script.
2. Change the Personality in the Model Specification window from Standard Least Squares to
Stepwise.
3. Click Run on the Fit Model dialog.
4. In the resulting display click the Step button two times. JMP enters the factors with the largest effects.
From the report that appears, you should identify two active factors, X2 and X10, as shown in
Figure 2.11. The step history appears in the bottom part of the report. Because random noise is added,
your estimates will be slightly different from those shown below.
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Creating Screening Experiments
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Figure 2.11 Stepwise Regression Identifies Active Factors
Note: This example defines two large main effects and sets the rest to zero. In real-world situations, it may
be less likely to have such clearly differentiated effects.
Screening Designs with Flexible Block Sizes
When you create a design using the Screening designer (DOE > Screening), the available block sizes for the
listed designs are a power of two. However, custom designs in JMP can have blocks of any size. The
blocking example shown in this section is flexible because it is using three runs per block, instead of a power
of two.
After you select DOE > Custom Design and enter factors, the blocking factor shows only one level in the
Values section of the Factors panel because the sample size is unknown at this point. After you complete the
design, JMP shows the appropriate number of blocks, which is calculated as the sample size divided by the
number of runs per block.
For example, Figure 2.12 shows that when you enter three continuous factors and one blocking factor with
three runs per block, only one block appears in the Factors panel.
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Figure 2.12 One Block Appears in the Factors Panel
The default sample size of nine requires three blocks. After you click Continue, there are three blocks in the
Factors panel (Figure 2.13). This is because the default sample size is nine, which requires three blocks with
three runs each.
Figure 2.13 Three Blocks in the Factors Panel
If you enter 24 runs in the User Specified box of the Number of Runs section, the Factors panel changes
and now contains 8 blocks (Figure 2.14).
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Creating Screening Experiments
Figure 2.14 Number of Runs is 24 Gives Eight Blocks
If you add all the two-factor interactions and change the number of runs to 15, three runs per block
produces five blocks (as shown in Figure 2.15), so the Factors panel displays five blocks in the Values
section.
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Figure 2.15 Changing the Runs to 15
Click Make Design, then click the disclosure button (
on Windows and
on the Macintosh) to
open the Design Evaluation outline node. Then, click the disclosure button to open the Relative Variance of
Coefficients report. Figure 2.16 shows the variance of each coefficient in the model relative to the unknown
error variance.
The values in your table may be slightly different from those shown below. Notice that the variance of each
coefficient is about one-tenth the error variance and that all the variances are roughly the same size. The
error variance is assumed to be 1.
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Creating Screening Experiments
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Figure 2.16 Table of Relative Variance of the Model Coefficients
The main question here is whether the relative size of the coefficient variance is acceptably small. If not,
adding more runs (18 or more) will lower the variance of each coefficient.
For more details, see “The Relative Variance of Coefficients and Power Table,” p. 75.
Note to DOE experts: There are four rows associated with X4 (the block factor). That is because X4 has
5 blocks and, therefore, 4 degrees of freedom. Each degree of freedom is associated with one unknown
coefficient in the model.
Checking for Curvature Using One Extra Run
In screening designs, experimenters often add center points and other check points to a design to help
determine whether the assumed model is adequate. Although this is good practice, it is also ad hoc. The
custom designer provides a way to improve on this ad hoc practice while supplying a theoretical foundation
and an easy-to-use interface for choosing a design robust to the modeling assumptions.
The purpose of check points in a design is to provide a detection mechanism for higher-order effects that are
contained in the assumed model. These higher-order terms are called potential terms. (Let q denote the
potential terms, designated If Possible in JMP.) The assumed model consists of the primary terms. (Let p
denote the primary terms designated Necessary in JMP.)
To take advantage of the benefits of the approach using If Possible model terms, the sample size should be
larger than the number of Necessary (primary) terms but smaller than the sum of the Necessary and If
Possible (potential) terms. That is, p < n < p+q. The formal name of the approach using If Possible model
terms is Bayesian D-Optimal design. This type of design allows the precise estimation of all of the Necessary
terms while providing omnibus detectability (and some estimability) for the If Possible terms.
For a two-factor design having a model with an intercept, two main effects, and an interaction, there are
p = 4 primary terms. When you enter this model in the custom designer, the default minimum runs value is
a four-run design with the factor settings shown in Figure 2.17.
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Figure 2.17 Two Continuous Factors with Interaction
Now suppose you can afford an extra run (n = 5). You would like to use this point as a check point for
curvature. If you leave the model the same and increase the sample size, the custom designer replicates one
of the four vertices. Replicating any run is the optimal choice for improving the estimates of the terms in the
model, but it provides no way to check for lack of fit.
Adding the two quadratic terms to the model makes a total of six terms. This is a way to model curvature
directly. However, to do this the custom designer requires two additional runs (at a minimum), which
exceeds your budget of five runs.
The Bayesian D-Optimal design provides a way to check for curvature while adding only one extra run. To
create this design:
1. Select DOE > Custom Design.
2. Define two continuous factors (X1 and X2).
3. Click Continue.
4. Choose 2nd from the Interactions menu in the Model panel. The results appear as shown in
Figure 2.18.
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Creating Screening Experiments
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Figure 2.18 Second-Level Interactions
5. Choose 2nd from the Powers button in the Model panel. This adds two quadratic terms.
6. Select the two quadratic terms (X1*X1 and X2*X2) and click the current estimability (Necessary) to see
the menu and change Necessary to If Possible, as shown in Figure 2.19.
Figure 2.19 Changing the Estimability
Now, the p = 4 primary terms (the intercept, two main effects, and the interaction) are designated as
Necessary while the q = 2 potential terms (the two quadratic terms) are designated as If Possible. The
desired number of runs, five, is between p = 4 and p + q = 6.
7. Enter 5 into the User Specified edit box in the Number of Runs section of the Design Generation panel.
8. Click Make Design. The resulting factor settings appear in Figure 2.20. The values in your design may
be different from those shown below.
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Figure 2.20 Five-Run Bayesian D-Optimal Design
9. Click Make Table to create a JMP data table of the runs.
10. Create the overlay plot in Figure 2.21 with Graph > Overlay Plot, and assign X1 as Y and X2 as X. The
overlay plot illustrates how the design incorporates the single extra run. In this example the design places
the factor settings at the center of the design instead of at one of the corners.
Figure 2.21 Overlay Plot of Five-run Bayesian D-Optimal Design
Creating Response Surface Experiments
Response surface experiments traditionally involve a small number (generally 2 to 8) of continuous factors.
The a priori model for a response surface experiment is usually quadratic.
In contrast to screening experiments, researchers use response surface experiments when they already know
which factors are important. The main goal of response surface experiments is to create a predictive model
of the relationship between the factors and the response. Using this predictive model allows the
experimenter to find better operating settings for the process.
In screening experiments one measure of the quality of the design is the size of the relative variance of the
coefficients. In response surface experiments, the prediction variance over the range of the factors is more
important than the variance of the coefficients. One way to visualize the prediction variance is JMP’s
prediction variance profile plot. This plot is a powerful diagnostic tool for evaluating and comparing
response surface designs.
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Creating Response Surface Experiments
35
Exploring the Prediction Variance Surface
The purpose of the example below is to generate and interpret a simple Prediction Variance Profile Plot.
Follow the steps below to create a design for a quadratic model with a single continuous factor.
1. Select DOE > Custom Design.
2. Add one continuous factor by selecting Add Factor > Continuous (Figure 2.22), and click Continue.
3. In the Model panel, select Powers > 2nd to create a quadratic term (Figure 2.22).
Figure 2.22 Adding a Factor and a Quadratic Term
4. In the Design Generation panel, use the default number of runs (six) and click Make Design
(Figure 2.23). The number of runs is inversely proportional to the size of variance of the predicted
response. As the number of runs increases, the prediction variances decrease.
Figure 2.23 Using the Default Number of Runs
5. Click the disclosure button (
on Windows and
on the Macintosh) to open the Design
Evaluation outline node, and then the Prediction Variance Profile, as shown in Figure 2.24.
For continuous factors, the initial setting is at the mid-range of the factor values. For categorical factors, the
initial setting is the first level. If the design model is quadratic, then the prediction variance function is
quartic. The y-axis is the relative variance of prediction of the expected value of the response.
In this design, the three design points are –1, 0, and 1. The prediction variance profile shows that the
variance is a maximum at each of these points on the interval –1 to 1.
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Figure 2.24 Prediction Profile for Single Factor Quadratic Model
The prediction variance is relative to the error variance. When the relative prediction variance is one, the
absolute variance is equal to the error variance of the regression model. More detail on the Prediction
Variance Profiler is in “Understanding Design Evaluation,” p. 72.
6. To compare profile plots, click the Back button and choose Minimum in the Design Generation panel,
which gives a sample size of three.
7. Click Make Design and then open the Prediction Variance Profile again.
Now you see a curve that has the same shape as the previous plot, but the maxima are at one instead of 0.5.
Figure 2.25 compares plots for a sample size of six and sample size of three for this quadratic model. You can
see the prediction variance increase as the sample size decreases. Since the prediction variance is inversely
proportional to the sample size, doubling the number of runs halves the prediction variance. These profiles
show settings for the maximum variance and minimum variance, for sample sizes six (top charts) and sample
size three (bottom charts). The axes on the bottom plots are adjusted to match the axes on the top plot.
Figure 2.25 Comparison of Prediction Variance Profiles
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Creating Response Surface Experiments
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Tip: Hold the CTRL key and click (hold the COMMAND key and click on Macintosh) on the factor to set
a factor level precisely.
8. To create an unbalanced design, click the Back button and enter a sample size of 7 in the User Specified
text edit box in the Design Generation panel, then click Make Design. The results are shown in
Figure 2.26.
You can see that the variance of prediction at –1 is lower than the other sample points (its value is 0.33
instead of 0.5). The symmetry of the plot is related to the balance of the factor settings. When the design is
balanced, the plot is symmetric, as shown in Figure 2.25. When the design is unbalanced, the prediction
plot might not be symmetric, as shown in Figure 2.26.
Figure 2.26 Sample Size of Seven for the One-Factor Quadratic Model
Prediction variance at X1= 0
Prediction variance at X1= –1
Introducing I-Optimal Designs for Response Surface Modeling
The custom designer generates designs using a mathematical optimality criterion. All the designs in this
chapter so far have been D-Optimal designs. D-Optimal designs are most appropriate for screening
experiments because the optimality criterion focuses on precise estimates of the coefficients. If an
experimenter has precise estimates of the factor effects, then it is easy to tell which factors’ effects are
important and which are negligible. However, D-Optimal designs are not as appropriate for designing
experiments where the primary goal is prediction.
I-Optimal designs minimize the average prediction variance inside the region of the factors. This makes
I-Optimal designs more appropriate for prediction. As a result I-Optimality is the recommended criterion
for JMP response surface designs.
An I-Optimal design tends to place fewer runs at the extremes of the design space than does a D-Optimal
design. As an example, consider a one-factor design for a quadratic model using n = 12 experimental runs.
The D-Optimal design for this model puts four runs at each end of the range of interest and four runs in the
middle. The I-Optimal design puts three runs at each end point and six runs in the middle. In this case, the
D-Optimal design places two-thirds of its runs at the extremes versus one-half for the I-Optimal design.
Figure 2.27 compares prediction variance profiles of the one-factor I- and D-Optimal designs for a
quadratic model with (n = 12) runs. The variance function for the I-Optimal design is less than the
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corresponding function for the D-Optimal design in the center of the design space; the converse is true at
the edges.
Figure 2.27 Prediction Variance Profiles for 12-Run I-Optimal (left) and D-Optimal (right) Designs
At the center of the design space, the average variance (relative to the error variance) for the I-Optimal
design is 0.1667 compared to the D-Optimal design, which is 0.25. This means that confidence intervals
for prediction will be nearly 10% shorter on average for the I-Optimal design.
To compare the two design criteria, create a one-factor design with a quadratic model that uses the
I-Optimality criterion, and another one that uses D-Optimality:
1. Select DOE > Custom Design.
2. Add one continuous factor: X1.
3. Click Continue.
4. Click the RSM button in the Model panel to make the design I-Optimal.
5. Change the number of runs to 12.
6. Click Make Design.
7. Click the disclosure button (
Evaluation outline node.
on Windows and
on the Macintosh) to open the Design
8. Click the disclosure button (
on Windows and
on the Macintosh) to open the Prediction
Variance Profile. (The Prediction Variance Profile is shown on the left in Figure 2.27.)
9. Repeat the same steps to create a D-Optimal design, but select Optimality Criterion > Make D-Optimal
Design from the red triangle menu on the custom design title bar. The results in the Prediction Variance
Profile should look the same as those on the right in Figure 2.27.
A Three-Factor Response Surface Design
In higher dimensions, the I-Optimal design continues to place more emphasis on the center of the region of
the factors. The D-Optimal and I-Optimal designs for fitting a full quadratic model in three factors using
16 runs are shown in Figure 2.28.
To compare the two design criteria, create a three-factor design that uses the I-Optimality criterion, and
another one that uses D-Optimality:
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Creating Response Surface Experiments
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1. Select DOE > Custom Design.
2. Add three continuous factors: X1, X2, and X3.
3. Click Continue.
4. Click the RSM button in the Model panel to add interaction and quadratic terms to the model and to
change the default optimality criterion to I-Optimal.
5. Use the default of 16 runs.
6. Click Make Design.
The design is shown in the Design panel (the left in Figure 2.28).
7. If you want to create a D-Optimal design for comparison, repeat the same steps but select Optimality
Criterion > Make D-Optimal Design from the red triangle menu on the custom design title bar. The
design should look similar to those on the right in Figure 2.28. The values in your design may be
different from those shown below.
Figure 2.28 16-run I-Optimal and D-Optimal designs for RSM Model
Profile plots of the variance function are displayed in Figure 2.29. These plots show slices of the variance
function as a function of each factor, with all other factors fixed at zero. The I-Optimal design has the lowest
prediction variance at the center. Note that there are two center points in this design.
The D-Optimal design has no center points and its prediction variance at the center of the factor space is
almost three times the variance of the I-Optimal design. The variance at the vertices of the D-Optimal
design is not shown. However, note that the D-Optimal design predicts better than the I-Optimal design
near the vertices.
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Creating Response Surface Experiments
Chapter 2
Figure 2.29 Variance Profile Plots for 16 run I-Optimal and D-Optimal RSM Designs
I-Optimal RSM Design
with 16 runs
D-Optimal RSM Design
with 16 runs
Response Surface with a Blocking Factor
It is not unusual for a process to depend on both qualitative and quantitative factors. For example, in the
chemical industry, the yield of a process might depend not only on the quantitative factors temperature and
pressure, but also on such qualitative factors as the batch of raw material and the type of reactor. Likewise,
an antibiotic might be given orally or by injection, a qualitative factor with two levels. The composition and
dosage of the antibiotic could be quantitative factors (Atkinson and Donev, 1992).
The response surface designer (described in “Response Surface Designs,” p. 127) only deals with
quantitative factors. You could use the response surface designer to produce a Response Surface Model
(RSM) design with a qualitative factor by replicating the design over each level of the factor. But, this is
unnecessarily time-consuming and expensive. Using custom designer is simpler and more cost-effective
because fewer runs are required. The following steps show how to accommodate a blocking factor in a
response surface design using the custom designer:
1. First, define two continuous factors (X1 and X2).
2. Now, click Add Factor and select Blocking > 4 runs per block to create a blocking factor(X3). The
blocking factor appears with one level, as shown in Figure 2.30, but the number of levels adjusts later to
accommodate the number of runs specified for the design.
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Creating Response Surface Experiments
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Figure 2.30 Add Two Continuous Factors and a Blocking Factor
3. Click Continue, and then click RSM in the Model panel to add the quadratic terms to the model
(Figure 2.31). This automatically changes the recommended optimality criterion from D-Optimal to
I-Optimal. Note that when you click RSM, a message reminds you that nominal factors (such as the
blocking factor) cannot have quadratic effects.
Figure 2.31 Add Response Surface Terms
4. Enter 12 in the User Specified text edit box in the Design Generation panel, and note that the Factors
panel now shows the Blocking factor, X3, with three levels (Figure 2.32). Twelve runs defines three
blocks with four runs per block.
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Figure 2.32 Blocking Factor Now Shows Three Levels
5. Click Make Design.
6. In the Output Options, select Sort Right to Left from the Run Order list.
7. Click Make Table to see an I-Optimal table similar to the one on the left in Figure 2.33.
Figure 2.33 compares the results of a 12-run I-Optimal design and a 12-run D-Optimal Design.
To see the D-Optimal design:
1. Click the Back button.
2. Click the red triangle icon on the Custom Design title bar and select Optimality Criterion > Make
D-Optimal Design.
3. Click Make Design, then click Make Table.
Figure 2.33 JMP Design Tables for 12-Run I-Optimal and D-Optimal Designs
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Creating Response Surface Experiments
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Figure 2.34 gives a graphical view of the designs generated by this example. These plots were generated for
the runs in each JMP table by choosing Graph > Overlay Plot from the main menu andusing the blocking
factor (X3) as the Grouping variable.
Note that there is a center point in each block of the I-Optimal design. The D-Optimal design has only one
center point. The values in your graph may be different from those shown in Figure 2.34.
Figure 2.34 Plots of I-Optimal (left) and D-Optimal (right) Design Points by Block.
Either of the designs in Figure 2.34 supports fitting the specified model. The D-Optimal design does a
slightly better job of estimating the model coefficients. The diagnostics (Figure 2.35) for the designs show
beneath the design tables. In this example, the D-efficiency of the I-Optimal design is about 51%, and is
55% for the D-Optimal design.
The I-Optimal design is preferable for predicting the response inside the design region. Using the formulas
given in “Technical Discussion,” p. 60, you can compute the relative average variance for these designs. The
average variance (relative to the error variance) for the I-Optimal design is 0.5 compared to 0.59 for the
D-Optimal design (See Figure 2.35). This means confidence intervals for prediction will be almost 20%
longer on average for D-Optimal designs.
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Creating Mixture Experiments
Chapter 2
Figure 2.35 Design Diagnostics for I-Optimal and D-Optimal Designs
Creating Mixture Experiments
If you have factors that are ingredients in a mixture, you can use either the custom designer or the
specialized mixture designer. However, the mixture designer is limited because it requires all factors to be
mixture components and you might want to vary the process settings along with the percentages of the
mixture ingredients. The optimal formulation could change depending on the operating environment. The
custom designer can handle mixture ingredients and process variables in the same study. You are not forced
to modify your problem to conform to the restrictions of a special-purpose design approach.
Mixtures Having Nonmixture Factors
The following example from Atkinson and Donev (1992) shows how to create designs for experiments with
mixtures where one or more factors are not ingredients in the mixture. In this example:
•
The response is the electromagnetic damping of an acrylonitrile powder.
•
The three mixture ingredients are copper sulphate, sodium thiosulphate, and glyoxal.
•
The nonmixture environmental factor of interest is the wavelength of light.
Though wavelength is a continuous variable, the researchers were only interested in predictions at three
discrete wavelengths. As a result, they treated it as a categorical factor with three levels. To create this custom
design:
1. Select DOE > Custom Design.
2. Create Damping as the response. The authors do not mention how much damping is desirable, so
right-click the goal and create Damping’s response goal to be None.
3. In the Factors panel, add the three mixture ingredients and the categorical factor, Wavelength. The
mixture ingredients have range constraints that arise from the mechanism of the chemical reaction.
Rather than entering them by hand, load them from the Sample Data folder that was installed with
JMP: click the red triangle icon on the Custom Design title bar and select Load Factors. Open Donev
Mixture Factors.jmp, from the Design Experiment sample data folder. The custom design panels should
now look like those shown in Figure 2.36.
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Creating Mixture Experiments
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Figure 2.36 Mixture Experiment Response Panel and Factors Panel
The model, shown in Figure 2.37 is a response surface model in the mixture ingredients along with the
additive effect of the wavelength. To create this model:
1. Click Interactions, and choose 2nd. A warning dialog appears telling you that JMP removes the main
effect terms for non-mixture factors that interact with all the mixture factors. Click OK.
2. In the Design Generation panel, type 18 in the User Specified text edit box (Figure 2.37), which results
in six runs each for the three levels of the wavelength factor.
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Figure 2.37 Mixture Experiment Design Generation Panel
3. Click Make Design, and then click Make Table.
The resulting data table is shown in Figure 2.38. The values in your table may be different from those
shown below.
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Creating Mixture Experiments
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Figure 2.38 Mixture Experiment Design Table
Atkinson and Donev also discuss the design where the number of runs is limited to 10. In that case, it is not
possible to run a complete mixture response surface design for every wavelength.
To view this:
1. Click the Back button.
2. Remove all the effects by highlighting them and clicking Remove Term.
3. Add the main effects by clicking the Main Effects button.
4. In the Design Generation panel, change the number of runs to 10 (Figure 2.39) and click Make
Design. The Design table to the right in Figure 2.39 shows the factor settings for 10 runs.
Figure 2.39 Ten-Run Mixture Response Surface Design
Note that there are necessarily unequal numbers of runs for each wavelength. Because of this lack of balance
it is a good idea to look at the prediction variance plot (top plot in Figure 2.40).
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5. Open the Design Evaluation outline node, then open the Prediction Variance Profile.
The prediction variance is almost constant across the three wavelengths, which is a good indication that the
lack of balance is not a problem.
The values of the first three ingredients sum to one because they are mixture ingredients. If you vary one of
the values, the others adjust to keep the sum constant.
6. Select Maximize Desirability from red triangle menu on the Prediction Variance Profile title bar, as
shown in the bottom profiler in Figure 2.40.
The most desirable wavelength is L3, with the CuSO4 percentage decreasing from about 0.4 to 0.2, Glyoxal
percentage is zero, and Na2S2O3 is 0.8, which maintains the mixture.
Figure 2.40 Prediction Variance Plots for Ten-Run Design
Experiments that are Mixtures of Mixtures
As a way to illustrate the idea of a ‘mixture of mixtures’ situation, imagine the ingredients that go into
baking a cake and assume the following:
•
dry ingredients composed of flour, sugar, and cocoa
•
wet (or non-dry) ingredients consisting of milk, melted butter, and eggs.
These two components (wet and dry) of the cake are two mixtures that are first mixed separately and then
blended together.
The dessert chef knows that the dry component (the mixture of flour, sugar, and cocoa) contributes 45% of
the combined mixture and the wet component (butter, milk, and eggs) contributes 55%.
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Creating Mixture Experiments
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The objective of such an experiment might be to identify proportions within the two components that
maximize some measure of taste or consistency.
This is a main effects model except that you must leave out one of the factors in order to avoid singularity.
The choice of which factor to leave out of the model is arbitrary.
For now, consider these upper and lower levels of the various factors:
Within the dry mixture:
•
cocoa must be greater than 10% but less than 20%
•
sugar must be greater than 0% but less than 15%
•
flour must be greater than 20% but less than 30%
Within the wet mixture:
•
melted butter must be greater than 10% but less than 20%
•
milk must be greater than 25% and less than 35%
•
eggs constitute more than 5% but less than 20%
You want to bake cakes and measure taste on a scale from 1 to 10
Use the Custom Designer to set up this example, as follows:
1. In the Response Panel, enter one response and call it Taste.
2. Give Taste a Lower Limit of 1 and an Upper Limit of 10. (You are assuming a taste test where the
respondents reply on a scale of 1 to 10.)
3. In the Factors Panel, enter the six cake factors described above.
4. Enter the given percentage values of the factors as proportions in the Values section of the Factors panel.
The completed Response and Factors panels should look like those shown in Figure 2.41.
Figure 2.41 Completed Responses and Factors Panel for the Cake Example
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5. Next, click Continue.
6. Open the Define Factor Constraints pane and click Add Constraint twice.
7. Enter the constraints as shown in Figure 2.42. For the second constraint setting, click on the less than or
equal to button and select the greater than or equal to direction.
By confining the dry factors to exactly 45% in this way, the mixture role of all the factors ensures that the
wet factors constitute the remaining 55%.
8. Open the Model dialog and note that it lists all 6 effects. Because these are mixture factors, including all
effects would render the model singular. Highlight any one of the terms in the model and click Remove
Term, as shown.
Figure 2.42 Constraints to Define the Double Mixture Experiment
9. To see a completed example, choose Simulate Responses from the menu on the Custom Design title
bar.
10. In the Design Generation panel, enter 10 as the number of runs for the example. That is, you would
bake cakes with 10 different sets of ingredient proportions.
11. Click Make Design in the Design Generation panel, and then click Make Table.
The table inFigure 2.43 shows that the two sets of cake ingredients (dry and wet) adhere to the proportions
45% and 55% as defined by the entered constraints. In addition, the amount of each ingredient in each
cake recipe (run) conforms to the upper and lower limits given in the factors dialog.
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Special-Purpose Uses of the Custom Designer
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Figure 2.43 Cake Experiment Conforming to a Mixture of Mixture Design
Each run sums to 0.45 (45%)
Each run sums to 0.55 (55%)
Note: As a word of caution, keep in mind that it is easy to define constraints in such a way that it is
impossible to construct a design that fits the model. In such a case, you will get a message saying “Could not
find a valid starting design. Please check your constraints for consistency.”
Special-Purpose Uses of the Custom Designer
While some of the designs discussed in previous sections can be created using other designers in JMP or by
looking them up in a textbook containing tables of designs, the designs presented in this section cannot be
created without using the custom designer.
Designing Experiments with Fixed Covariate Factors
Pre-tabulated designs rely on the assumption that the experimenter controls all the factors. Sometimes you
have quantitative measurements (a covariate) on the experimental units before the experiment begins. If this
variable affects the experimental response, the covariate should be a design factor. The pre-defined design
that allows only a few discrete values is too restrictive. The custom designer supplies a reasonable design
option.
For this example, suppose there are a group of students participating in a study. A physical education
researcher has proposed an experiment where you vary the number of hours of sleep and the calories for
breakfast and ask each student to run 1/4 mile. The weight of the student is known and it seems important
to include this information in the experimental design.
To follow along with this example that shows column properties, open Big Class.jmp from the Sample Data
folder that was installed when you installed JMP.
Build the custom design as follows:
1. Select DOE > Custom Design.
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Chapter 2
2. Add two continuous variables to the models by entering 2 beside Add N Factors, clicking Add Factor
and selecting Continuous, naming them calories and sleep.
3. Click Add Factor and select Covariate, as shown in Figure 2.44. The Covariate selection displays a list
of the variables in the current data table.
Figure 2.44 Add a Covariate Factor
4. Select weight from the variable list (Figure 2.45) and click OK.
Figure 2.45 Design with Fixed Covariate
5. Click Continue.
6. Add the interaction to the model by selecting calories in the Factors panel, selecting sleep in the Model
panel, and then clicking the Cross button (Figure 2.46).
Chapter 2
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Special-Purpose Uses of the Custom Designer
53
Figure 2.46 Design With Fixed Covariate Factor
7. Click Make Design, then click Make Table. The data table in Figure 2.47 shows the design table. Your
runs might not look the same because the order of the runs has been randomized.
Figure 2.47 Design Table for Covariate Example
Note: Covariate factors cannot have missing values.
Remember that weight is the covariate factor, measured for each student, but it is not controlled. The
custom designer has calculated settings for calories and sleep for each student. It would be desirable if the
correlations between calories, sleep and weight were as small as possible. You can see how well the custom
designer did by fitting a model of weight as a function of calories and sleep. If that fit has a small model
sum of squares, that means the custom designer has successfully separated the effect of weight from the
effects of calories and sleep.
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8. Click the red triangle icon beside Model in the data table and select Run Script, as shown on the left in
Figure 2.48.
Figure 2.48 Model Script
9. Rearrange the dialog so weight is Y and calories, sleep, and calories*sleep are the model effects, as
shown to the right in Figure 2.48. Click Run.
The leverage plots are nearly horizontal, and the analysis of variance table shows that the model sum of
squares is near zero compared to the residuals (Figure 2.49). Therefore, weight is independent of calories
and sleep. The values in your analysis may be a little different from those shown below.
Figure 2.49 Analysis to Check That Weight is Independent of Calories and Sleep
Chapter 2
Examples Using the Custom Designer
Special-Purpose Uses of the Custom Designer
55
Creating a Design with Two Hard-to-Change Factors: Split Plot
While there is substantial research literature covering the analysis of split plot designs, it has only been
possible in the last few years to create optimal split plot designs (Goos 2002). The split plot design
capability accessible in the JMP custom designer is the first commercially available tool for generating
optimal split plot designs.
The split plot design originated in agriculture, but is commonplace in manufacturing and engineering
studies. In split plot experiments, hard-to-change factors only change between one whole plot and the next.
The whole plot is divided into subplots, and the levels of the easy-to-change factors are randomly assigned
to each subplot.
The example in this section is adapted from Kowalski, Cornell, and Vining (2002). The experiment studies
the effect of five factors on the thickness of vinyl used to make automobile seat covers. The response and
factors in the experiment are described below:
•
Three of the factors are ingredients in a mixture. They are plasticizers whose proportions, m1, m2, and
m3, sum to one. Additionally, the mixture components are the subplot factors of the experiment.
•
Two of the factors are process variables. They are the rate of extrusion (extrusion rate) and the
temperature (temperature) of drying. These process variables are the whole plot factors of the
experiment. They are hard to change.
•
The response in the experiment is the thickness of the vinyl used for automobile seat covers. The
response of interest (thickness) depends both on the proportions of the mixtures and on the effects of
the process variables.
To create this design in JMP:
1. Select DOE > Custom Design.
2. By default, there is one response, Y, showing. Double-click the name and change it to thickness. Use the
default goal, Maximize (Figure 2.50).
3. Enter the lower limit of 10.
4. To add three mixture factors, type 3 in the box beside Add N Factors, and click Add Factor > Mixture.
5. Name the three mixture factors m1, m2, and m3. Use the default levels 0 and 1 for those three factors.
6. Add two continuous factors by typing 2 in the box beside Add N Factors, and click Add Factor >
Continuous. Name these factors extrusion rate and temperature.
7. Ensure that you are using the default levels, –1 and 1, in the Values area corresponding to these two
factors.
8. To make extrusion rate a whole plot factor, click Easy and select Hard.
9. To make temperature a whole plot factor, click Easy and select Hard. Your dialog should look like the
one in Figure 2.50.
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Chapter 2
Figure 2.50 Entering Responses and Factors
10. Click Continue.
11. Next, add interaction terms to the model by selecting Interactions > 2nd (Figure 2.51). This causes a
warning that JMP removes the main effect terms for non-mixture factors that interact with all the
mixture factors. Click OK.
Figure 2.51 Adding Interaction Terms
12. In the Design Generation panel, type 7 in the Number of Whole Plots text edit box.
13. For Number of Runs, type 28 in the User Specified text edit box (Figure 2.52).
Chapter 2
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Special-Purpose Uses of the Custom Designer
57
Figure 2.52 Assigning the Number of Whole Plots and Number of Runs
Note: If you enter a missing value in the Number of Whole Plots edit box, then JMP considers many
different numbers of whole plots and chooses the number that maximizes the information about the
–1
coefficients in the model. It maximizes the determinant of X ′ V X where V -1 is the inverse of the
variance matrix of the responses. The matrix, V, is a function of how many whole plots there are, so
changing the number of whole plots changes V, which can make a difference in the amount of information
a design contains.
14. Click Make Design. The result is shown in Figure 2.53.
Figure 2.53 Partial Listing of the Final Design Structure
15. Click Make Table.
16. From the Sample Data folder that was installed with JMP, open Vinyl Data.jmp from the Design
Experiment folder, which contains 28 runs as well as response values. The values in the table you
generated with the custom designer may be different from those from the Sample Data folder, shown in
Figure 2.54.
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Chapter 2
Figure 2.54 The Vinyl Data Design Table
17. Click the red triangle icon next to the Model script and select Run Script. The dialog in Figure 2.55
appears.
The models for split plots have a random effect associated with the whole plots’ effect.
As shown in the dialog in Figure 2.55, JMP designates the error term by appending &Random to the name
of the effect. REML will be used for the analysis, as indicated in the menu beside Method in Figure 2.55.
For more information about REML models, see Modeling and Multivariate Methods.
Chapter 2
Examples Using the Custom Designer
Special-Purpose Uses of the Custom Designer
Figure 2.55 Define the Model in the Fit Model Dialog
18. Click Run to run the analysis. The results are shown in Figure 2.56.
59
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Technical Discussion
Chapter 2
Figure 2.56 Split Plot Analysis Results
Technical Discussion
This section provides information about I-, D-, Bayesian I-, Bayesian D-, and Alias-Optimal designs.
D-Optimality:
•
is the default design type produced by the custom designer except when the RSM button has been
clicked to create a full quadratic model.
•
minimizes the variance of the model coefficient estimates. This is appropriate for first-order models and
in screening situations, because the experimental goal in such situations is often to identify the active
factors; parameter estimation is key.
•
is dependent on a pre-stated model. This is a limitation because in most real situations, the form of the
pre-stated model is not known in advance.
•
has runs whose purpose is to lower the variability of the coefficients of this pre-stated model. By focusing
on minimizing the standard errors of coefficients, a D-Optimal design may not allow for checking that
the model is correct. It will not include center points when investigating a first-order model. In the
extreme, a D-Optimal design may have just p distinct runs with no degrees of freedom for lack of fit.
Chapter 2
•
Examples Using the Custom Designer
Technical Discussion
61
maximizes D when
D = det [ X ′ X]
D-optimal split plot designs maximize D when
D = det [ X ′ V
–1
X]
where V -1is the block diagonal variance matrix of the responses (Goos 2002).
Bayesian D-Optimality:
•
is a modification of the D-Optimality criterion that effectively estimates the coefficients in a model, and
at the same time has the ability to detect and estimate some higher-order terms. If there are interactions
or curvature, the Bayesian D-Optimality criterion is advantageous.
•
works best when the sample size is larger than the number of Necessary terms but smaller than the sum
of the Necessary and If Possible terms. That is, p + q > n > p. The Bayesian D-Optimal design is an
approach that allows the precise estimation of all of the Necessary terms while providing omnibus
detectability (and some estimability) for the If Possible terms.
•
uses the If Possible terms to force in runs that allow for detecting any inadequacy in the model
containing only the Necessary terms. Let K be the (p + q) by (p + q) diagonal matrix whose first p
diagonal elements are equal to 0 and whose last q diagonal elements are the constant, k. If there are
2-factor interactions then k = 4. Otherwise k = 1. The Bayesian D-Optimal design maximizes the
determinant of (X'X + K). The difference between the criterion for D-Optimality and Bayesian
D-Optimality is this constant added to the diagonal elements corresponding to the If Possible terms in
the X'X matrix.
I-Optimality:
•
minimizes the average variance of prediction over the region of the data.
•
is more appropriate than D-Optimality if your goal is to predict the response rather than the
coefficients, such as in response surface design problems. Using the I-Optimality criterion is more
appropriate because you can predict the response anywhere inside the region of data and therefore find
the factor settings that produce the most desirable response value. It is more appropriate when your
objective is to determine optimum operating conditions, and also is appropriate to determine regions in
the design space where the response falls within an acceptable range. Precise estimation of the response
therefore takes precedence over precise estimation of the parameters.
•
minimizes this criterion: If f '(x) denotes a row of the X matrix corresponding to factor combinations x,
then
I = ∫ f ′ ( x ) ( X′X )
–1
R
where
M = ∫ f ( x ) f ( x )′dx
R
–1
f ( x )dx = Trace [ ( X ′ X ) M ]
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Technical Discussion
Chapter 2
is a moment matrix that is independent of the design and can be computed in advance.
Bayesian I-Optimality:
Bayesian I-Optimality has a different objective function to optimize than the Bayesian D-optimal design, so
the designs that result are different. The variance matrix of the coefficients for Bayesian I-optimality is X'X
+ K where K is a matrix having zeros for the Necessary model terms and some constant value for the
If Possible model terms.
The variance of the predicted value at a point x0 is:
ˆ
–1
var ⎛Y x 0⎞ = ⎛ x 0′ ⎛ X ′ X + K⎞ ⎞ x 0
⎝
⎠
⎝ ⎝
⎠ ⎠
The Bayesian I-Optimal design minimizes the average prediction variance over the design region:
–1
I B = Trace [ ( X ′ X + K ) M ]
where M is defined as before.
Alias Optimality:
•
seeks to minimize the aliasing between model effects and alias effects.
Specifically, let X1 be the design matrix corresponding to the model effects, and let X2 be the matrix of alias
effects, and let
A = ( X 1 ′X 1 ) – 1 X 1 ′X 2
be the alias matrix.
Then, alias optimality seeks to minimize the tr ( AA' ) , subject to the D-Efficiency being greater than some
lower bound. In other words, it seeks to minimize the sum of the squared diagonal elements of A.
Chapter 3
Building Custom Designs
The Basic Steps
JMP can build a custom design that both matches the description of your engineering problem and remains
within your budget for time and material. Custom designs are general, flexible, and good for routine factor
screening or response optimization. To create these tailor-made designs, use the Custom Design command
found on the DOE menu or the Custom Design button found on the DOE panel of the JMP Starter.
This chapter introduces you to the steps you need to complete to build a custom design.
Contents
Creating a Custom Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
Enter Responses and Factors into the Custom Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
Describe the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Specifying Alias Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Select the Number of Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
Understanding Design Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Specify Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Make the JMP Design Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Creating Random Block Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Creating Split Plot Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Creating Split-Split Plot Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Creating Strip Plot Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82
Special Custom Design Commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83
Save Responses and Save Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Load Responses and Load Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Save Constraints and Load Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Set Random Seed: Setting the Number Generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Simulate Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Save X Matrix: Viewing the Number of Rows in the Moments Matrix and the Design Matrix (X) in the
Log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Optimality Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88
Number of Starts: Changing the Number of Random Starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88
Sphere Radius: Constraining a Design to a Hypersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Disallowed Combinations: Accounting for Factor Level Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . 90
Advanced Options for the Custom Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
Assigning Column Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
Define Low and High Values (DOE Coding) for Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Set Columns as Factors for Mixture Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95
Define Response Column Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Assign Columns a Design Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Identify Factor Changes Column Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
How Custom Designs Work: Behind the Scenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 3
Building Custom Designs
Creating a Custom Design
65
Creating a Custom Design
To begin, select DOE > Custom Design, or click the Custom Design button on the JMP Starter DOE page.
Then, follow the steps below.
•
Enter responses and factors into the custom designer.
•
Describe the model.
•
Select the number of runs.
•
Check the design diagnostics, if desired.
•
Specify output options.
•
Make the JMP design table.
The following sections describe each of these steps.
Enter Responses and Factors into the Custom Designer
How to Enter Responses
To enter responses, follow the steps in Figure 3.1.
1. To enter one response at a time, click Add Response, and then select a goal type. Possible goal types are
Maximize, Match Target, Minimize, or None.
2. (Optional) Double-click to edit the response name.
3. (Optional) Click to change the response goal.
4. Click to enter lower and upper limits and importance weights.
Figure 3.1 Entering Responses
4
1
2
3
Tip: To quickly enter multiple responses, click Number of Responses and enter the number of responses
you want.
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Specifying Response Goal Types and Lower and Upper Limits
When entering responses, you can tell JMP that your goal is to obtain the maximum or minimum value
possible, to match a specific value, or that there is no response goal.
The following description explains the relationship between the goal type (step 3 in Figure 3.1) and the
lower and upper limits (step 4 in Figure 3.1):
•
For responses such as strength or yield, the best value is usually the largest possible. A goal of Maximize
supports this objective.
•
The Minimize goal supports an objective of having the smallest value, such as when the response is
impurity or defects.
•
The Match Target goal supports the objective when the best value for a response is a specific target
value, such as a dimension for a manufactured part. The default target value is assumed to be midway
between the given lower and upper limits.
Note: If your target response is not equidistant from the lower and upper acceptable bounds, you can alter
the default target after you make a table from the design. In the data table, open the Column Info dialog for
the response column (Cols > Column Info) and enter the desired target value.
Understanding Response Importance Weights
To compute and maximize overall desirability, JMP uses the value you enter as the importance weight (step
4 in Figure 3.1) of each response. If there is only one response, then importance weight is unnecessary. With
two responses you can give greater weight to one response by assigning it a higher importance value.
Adding Simulated Responses, If Desired
If you do not have values for specific responses, you might want to add simulated responses to see a
prospective analysis in advance of real data collection.
1. Create the design.
2. Before you click Make Table, click the red triangle icon in the title bar and select Simulate Responses.
3. Click Make Table to create the design table. The Y column contains values for simulated responses.
4. For custom and augment designs, a window (Figure 3.2) appears along with the design data table. In
this window, enter values you want to apply to the Y column in the data table and click Apply. The
numbers you enter represent the coefficients in an equation. An example of such an equation, as shown
in Figure 3.2, would be, y = 28 + 4X1 + 5X2 + random noise, where the random noise is distributed with
mean zero and standard deviation one.
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Creating a Custom Design
67
Figure 3.2 In Custom and Augment Designs, Specify Values for Simulated Responses
How to Enter Factors
To enter factors, follow the steps in Figure 3.3.
1. To add one factor, click Add Factor and select a factor type. Possible factor types are Continuous,
Categorical, Blocking, Covariate, Mixture, Constant, or Uncontrolled. See “Types of Factors,” p. 67.
2. Click a factor and select Add Level to increase the number of levels.
3. Double-click a factor to edit the factor name.
4. Click to indicate that changing a factor’s setting from run to run is Easy, Hard, or Very Hard. Changing
to Hard or Very Hard will cause the resulting design to be a split plot or split-split plot design.
5. Click to enter or change factor values. To remove a level, click it, press the delete key on the keyboard,
then press the Return or Enter key on the keyboard.
6. To add multiple factors, type the number of factors in the Add N Factors box, click the Add Factor
button, and select the factor type.
Figure 3.3 Entering Factors in a Custom Design
6
1
2
3
4
5
Types of Factors
When adding factors, click the Add Factor button and choose the type of factor.
Continuous Continuous factors are numeric data types only. In theory, you can set a continuous
factor to any value between the lower and upper limits you supply.
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Either numeric or character data types. Categorical data types have no implied order. If
the values are numbers, the order is the numeric magnitude. If the values are character, the order is
the sorting sequence. The settings of a categorical factor are discrete and have no intrinsic order.
Examples of categorical factors are machine, operator, and gender.
Categorical
Blocking Either numeric or character data types. Blocking factors are a special kind of categorical
factor. Blocking factors differ from other categorical factors in that there is a limit to the number of
runs that you can perform within one level of a blocking factor.
Covariate Either numeric or character data types. Covariate factors are not controllable, but their
values are known in advance of an experiment.
Mixture Mixture factors are continuous factors that are ingredients in a mixture. Factor settings for a
run are the proportion of that factor in a mixture and vary between zero and one.
Either numeric or character data types. Constant factors are factors whose values are fixed
during an experiment.
Constant
Uncontrolled Either numeric or character data types. Uncontrolled factors have values that cannot be
controlled during an experiment, but they are factors you want to include in the model.
Factors that are Easy, Hard, or Very Hard, to Change: Creating Optimal Split-Plot and
Split-Split-Plot Designs
Split plot experiments are performed in groups of runs where one or more factors are held constant within a
group but vary between groups. In industrial experimentation this structure is desirable because certain
factors may be difficult and expensive to change from one run to the next. It is convenient to make several
runs while keeping such factors constant. Until now, commercial software has not supplied a general
capability for the design and analysis of these experiments.
To indicate the difficulty level of changing a factor’s setting, click in Changes column of the Factors panel
for a given factor and select Easy, Hard, or Very Hard from the menu that appears. Changing to Hard
results in a split-plot design and Very Hard results in a split-split-plot design.
See “Creating Split Plot Designs,” p. 80, for more details.
Defining Factor Constraints, If Necessary
Sometimes it is impossible to vary factors simultaneously over their entire experimental range. For example,
if you are studying the affect of cooking time and microwave power level on the number of kernels popped
in a microwave popcorn bag, the study cannot simultaneously set high power and long time without
burning all the kernels. Therefore, you have factors whose levels are constrained.
To define the constraints:
1. After you add factors and click Continue, click the disclosure button (
the Macintosh) to open the Define Factor Constraints panel.
on Windows and
on
2. Click the Add Constraint button. Note that this feature is disabled if you have already controlled the
design region by entering disallowed combinations or chosen a sphere radius.
Chapter 3
Building Custom Designs
Creating a Custom Design
69
Figure 3.4 Add Constraint
3. Specify the coefficients and their limiting value in the boxes provided, as shown to the right. When you
need to change the direction of the constraint, click on the default less than or equal button and select
the greater than or equal to direction.
4. To add another constraint, click the Add Constraint button again and repeat the above steps.
Describe the Model
Initially, the Model panel lists only the main effects corresponding to the factors you entered, as shown in
Figure 3.5. However, you can add factor interactions or powers of continuous factors to the model. For
example, to add all the two-factor interactions and quadratic effects at once, click the RSM button.
Figure 3.5 Add Terms
Table 3.1 summarizes the ways to add specific factor types to the model.
Table 3.1 How to Add Terms to a Model
Action
Instructions
Add interaction terms involving selected
factors. If none are selected, JMP adds
all of the interactions to the specified
order.
Click the Interactions button and select 2nd, 3rd, 4th,
or 5th. For example, if the factors are X1 and X2 and
you click Interactions > 2nd, X1*X2 is added to the list
of model terms.
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Table 3.1 How to Add Terms to a Model (Continued)
Action
Instructions
Add all second-order effects, including
two-factor interactions and quadratic
effects
Click the RSM button. The design now uses
I-Optimality criterion rather than D-Optimality
criterion.
Add selected cross product terms
1. Highlight the factor names.
2. Highlight term(s) in the model list.
3. Click the Cross button.
Add powers of continuous factors to the
model effects
Click the Powers button and select 2nd, 3rd, 4th, or
5th.
Specifying Alias Terms
You can investigate the aliasing between the model terms and terms you specify in the Alias Terms panel.
For example, suppose you specify a design with three main effects in six runs, and you want to see how those
main effects are aliased by the two-way interactions and the three-way interaction. In the Alias Terms panel,
specify the interactions as shown in Figure 3.6. Also, specify six runs in the Design Generation panel.
Figure 3.6 Alias Terms
After you click the Make Design button at the bottom of the Custom Design panel, open the Alias Matrix
panel in the Design Evaluation panel to see the alias matrix. See Figure 3.7.
Figure 3.7 Aliasing
Chapter 3
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Creating a Custom Design
71
In this example, all the main effects are partially aliased with two of the interactions. Also see “The Alias
Matrix (Confounding Pattern),” p. 76.
Select the Number of Runs
The Design Generation panel (Figure 3.8) shows the minimum number of runs needed to perform the
experiment based on the effects you’ve added to the model (two main effects in the example above). It also
shows alternate (default) numbers of runs, or lets you choose your own number of runs. Balancing the cost
of each run with the information gained by extra runs you add is a judgment call that you control.
Figure 3.8 Options for Selecting the Number of Runs
The Design Generation panel has these options for selecting the number of runs you want:
is the smallest number of terms that can create a design. When you use Minimum, the
resulting design is saturated (no degrees of freedom for error). This is an extreme and risky choice,
and is appropriate only when the cost of extra runs is prohibitive.
Minimum
is a custom design suggestion for the number of runs. This value is based on heuristics for
creating balanced designs with a few additional runs above the minimum.
Default
User Specified is a value that specifies the number of runs you want. Enter that value into the
Number of Runs text box.
Note: In general, the custom design suggests a number of runs that is the smallest number that can be
evenly divided by the number of levels of each of the factors and is larger than the minimum possible sample
size. For designs with factors at two levels only, the default sample size is the smallest power of two larger
than the minimum sample size.
When the Design Generation panel shows the number of runs you want, click Make Design.
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Understanding Design Evaluation
After making the design, you can preview the design and investigate details by looking at various plots and
tables that serve as design diagnostic tools.
Although different tools are available depending on the model you specify, most designs display
•
the Prediction Variance Profile Plot
•
the Fraction of Design Space Plot
•
the Prediction Variance Surface Plot
•
the Relative Variance of Coefficients and Power Table
•
the Alias Matrix
•
Design Diagnostic Table
These diagnostic tools are outline nodes beneath the Design Evaluation panel, as shown in Figure 3.9. JMP
always provides the Prediction Variance Profile, but the Prediction Surface Plot only appears if there are two
or more variables.
Figure 3.9 Custom Design Evaluation and Diagnostic Tools
The Prediction Variance Profile
The example in Figure 3.10 shows the prediction variance profile for a response surface model (RSM) with
2 variables and 8 runs. To see a response surface design similar to this:
1. Chose DOE > Custom Design.
2. In the Factors panel, add 2 continuous factors.
3. Click Continue.
4. In the Model panel, click RSM.
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5. Click Make Design.
6. Open the Prediction Variance Profile.
Figure 3.10 A Factor Design Layout For a Response Surface Design with 2 Variables
The prediction variance for any factor setting is the product of the error variance and a quantity that
depends on the design and the factor setting. Before you collect the data the error variance is unknown, so
the prediction variance is also unknown. However, the ratio of the prediction variance to the error variance
is not a function of the error variance. This ratio, called the relative variance of prediction, depends only on
the design and the factor setting and can be calculated before acquiring the data. The prediction variance
profile plots the relative variance of prediction as a function of each factor at fixed values of the other factors
After you run the experiment, collect the data, and fit the model, you can estimate the actual variance of
prediction at any setting by multiplying the relative variance of prediction by the mean squared error (MSE)
of the least squares fit.
It is ideal for the prediction variance to be small throughout the allowable regions of the factors. Generally,
the error variance drops as the sample size increases. Comparing the prediction variance profilers for two
designs side-by-side, is one way to compare two designs. A design that has lower prediction variance on the
average is preferred.
In the profiler, drag the vertical lines in the plot to change the factor settings to different points. Dragging
the lines reveals any points that have prediction variances that are larger than you would like.
Another way to evaluate a design, or to compare designs, is to try and minimize the maximum variance. You
can use the Maximize Desirability command on the Prediction Variance Profile title bar to identify the
maximum prediction variance for a model. Consider the Prediction Variance profile for the two-factor RSM
model shown in Figure 3.11. The plots on the left are the default plots. The plots on the right identify the
factor values where the maximum variance (or worst-case scenario) occur, which helps you evaluate the
acceptability of the model.
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Figure 3.11 Find Maximum Prediction Variance
The Fraction of Design Space Plot
The Fraction of Design Space plot is a way to see how much of the model prediction variance lies above (or
below) a given value. As a simple example, consider the Prediction Variance plot for a single factor quadratic
model, shown on the left in Figure 3.12. The Prediction Variance plot shows that 100% of the values are
smaller than 0.5. You can move the vertical trace and also see that all the values are above 0.332. The
Fraction of Design Space plot displays the same information. The X axis is the proportion of prediction
variance values, ranging from 0 to 100%, and the Y axis is the range of prediction variance values. In this
simple example, the Fraction of Design plot verifies that 100% of the values are below 0.5 and 0% of the
values are below approximately 0.3. You can use the crosshair tool and find the percentage of values for any
value of the prediction variance. The example to the right in Figure 3.12 shows that 75% of the prediction
variance values are below approximately 0.46.
The Fraction of Design space is most useful when there are multiple factors. It summarizes the prediction
variance, showing the fractional design space for all the factors taken together.
Figure 3.12 Variance Profile and Fraction of Design Space
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The Prediction Variance Surface
When there are two or more factors, the Prediction Variance Surface plots the surface of the prediction
variance for any two variables. This feature uses the Graph > Surface Plot platform in JMP, and has all its
functionality. Drag on the plot to rotate and change the perspective. Figure 3.13 shows the Prediction
Variance Surface plot for a two-factor RSM model. The factors are on the x and y axes, and the prediction
variance is on the z axis. You can clearly see that there are high and low variance areas for both factors.
Compare this plot to the Prediction Variance Profile shown in Figure 3.11.
Figure 3.13 Prediction Variance Surface Plot for Two-Factor RSM Model
You can find complete documentation for the Surface Plot platform in the Modeling and Multivariate
Methods book.
The Relative Variance of Coefficients and Power Table
Before clicking Make Table in the custom designer, click the disclosure button (
on Windows and
on the Macintosh) to open Design Evaluation and then again to open the Relative Variance of
Coefficients table.
The Relative Variance of Coefficients table (Figure 3.14) shows the relative variance of all the coefficients for
the example RSM custom design (see Figure 3.10). The variances are relative to the error variance, which is
unknown before the experiment, and is assumed to be one. Once you complete the experiment and have an
estimate for the error variance, you can multiply it by the relative variance to get the estimated variance of
the coefficient. The square root of this value should match the standard error of prediction for the
coefficient when you fit a model using Analyze > Fit Model.
The Power column shows the power of the design as specified to detect effects of a certain size. In the text
edit boxes, you can change the alpha level of the test and the magnitude of the effects compared to the error
standard deviation. The alpha level edit box is called Significance Level. The magnitude of the effects edit
box is called Signal to Noise Ratio. This is the ratio of the absolute value of the regression parameter to
sigma (the square root of the error variance).
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If you enter a smaller alpha (requiring a more significant test), then the power falls. If you increase the
magnitude of the effect you want to detect, the power rises.
The power reported is the probability of finding a significant model parameter if the true effect is Signal to
Noise Ratio times sigma. The Relative Variance of Coefficients table on the left in Figure 3.14 shows the
results for the two-factor RSM model.
As another example, suppose you have a 3-factor 8-run experiment with a linear model and you want to
detect any regression coefficient that is twice as large as the error standard deviation, with an alpha level of
0.05. The Relative Variance of Coefficients table on the right in Figure 3.14 shows that the resulting power
is 0.984 for all the parameters.
Figure 3.14 Table of Relative Variance of Coefficients
The Alias Matrix (Confounding Pattern)
Click the Alias Matrix disclosure button (
matrix (Figure 3.15).
on Windows and
on the Macintosh) to open the alias
The alias matrix shows the aliasing between the model terms and the terms you specify in the Alias Terms
panel (see “Specifying Alias Terms,” p. 70). It allows you to see the confounding patterns.
Figure 3.15 Alias Matrix
Color Map on Correlations
The Color Map On Correlations panel (see Figure 3.16) shows the correlations between all model terms
and alias terms you specify in the Alias Terms panel (see “Specifying Alias Terms,” p. 70). The colors
correspond to the absolute value of the correlations.
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Figure 3.16 Color Map of Correlations
The Design Diagnostics Table
Open the Design Diagnostics outline node to display a table with relative D-, G-, and A-efficiencies, average
variance of prediction, and length of time to create the design. The design efficiencies are computed as
follows:
11 / p⎞
D-efficiency = 100 ⎛ -----⎝ N D X′X
⎠
⎛
⎞
p
A-efficiency = 100 ⎜ -----------------------------------------------⎟
⎝ trace ( N ( X′X ) – 1 )⎠
D
⎛ p⎞
⎜ ------⎟
G-efficiency = 100 ⎜ ND⎟
⎜ ---------σ ⎟
⎝ M⎠
where
•
ND is the number of points in the design
•
p is the number of effects in the model including the intercept
•
σM is the maximum standard error for prediction over the design points.
These efficiency measures are single numbers attempting to quantify one mathematical design characteristic.
While the maximum efficiency is 100 for any criterion, an efficiency of 100% is impossible for many design
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problems. It is best to use these design measures to compare two competitive designs with the same model
and number of runs rather than as some absolute measure of design quality.
Figure 3.17 Custom Design Showing Diagnostics
Specify Output Options
Use the Output Options panel to specify how you want the output data table to appear.
Figure 3.18 Output Options Panel
Run Order lets you designate the order you want the runs to appear in the data table when it is created.
Choices are:
Keep the Same
the rows (runs) in the output table will appear as they do in the Design panel.
Sort Left to Right the rows (runs) in the output table will appear sorted from left to right.
Randomize the rows (runs) in the output table will appear in a random order.
Sort Right to Left the rows (runs) in the output table will appear sorted from right to left.
Randomize within Blocks
the rows (runs) in the output table will appear in random order within the
blocks you set up.
Add additional points using options from Make JMP Table from design plus:
Number of Center Points: Specifies additional runs placed at the center of each continuous factor’s
range.
Number of Replicates: Specify the number of times to replicate the entire design, including
centerpoints. Type the number of times you want to replicate the design in the associated text box.
One replicate doubles the number of runs.
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Make the JMP Design Table
When the Design panel shows the layout you want, click Make Table. Parts of the table contain
information you might need to continue working with the table in JMP. The upper-left of the design table
can have one or more of the following scripts:
•
a Screening script runs the Analyze > Modeling > Screening platform when appropriate for the
generated design.
•
a Model script runs the Analyze > Fit Model platform with the model appropriate for the design.
•
a constraint script that shows any model constraints you entered in the Define Factor Constraints panel
of the Custom Design dialog.
•
a DOE Dialog script that recreates the dialog used to generate the design table, and regenerates the
design table.
Figure 3.19 Example Design Table
1
2
3
1. This area identifies the design type that generated the table. Click Custom Design to edit.
2. Model is a script. Click the red triangle icon and select Run Script to open the Fit Model dialog, which
is used to generate the analysis appropriate to the design.
3. DOE Dialog is a script. Click the red triangle icon and select Run Script to recreate the DOE Custom
Dialog and generate a new design table.
Creating Random Block Designs
It is often necessary to group the runs of an experiment into blocks. Runs within a block of runs are more
homogeneous than runs in different blocks. For example, the experiment described in Goos (2002),
describes a pastry dough mixing experiment that took several days to run. It is likely that random day-to-day
differences in environmental variables have some effect on all the runs performed on a given day. Random
block designs are useful in situations like this, where there is a non-reproducible shock to the system
between each block of runs. In Goos (2002), the purpose of the experiment was to understand how certain
properties of the dough depend on three factors: feed flow rate, initial moisture content, and rotational
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screw speed. It was only possible to conduct four runs a day. Because day-to-day variation was likely, it was
important to group the runs so that this variation would not compromise the information about the three
factors. Thus, blocking the runs into groups of four was necessary. Each day's experimentation was one
block. The factor, Day, is an example of a random block factor.
To create a random block, use the custom design and enter responses and factors, and define your model as
usual. In the Design Generation panel, check the Group runs into random blocks of size check box and
enter the number of runs you want in each block. When you select or enter the sample size, the number of
runs specified are assigned to the blocks.
Figure 3.20 Assigning Runs to Blocks
In this example, the Design Generation Panel shown here designates four runs per block, and the number of
runs (16) indicates there will be four days (blocks) of 4 runs. If the number of runs is not an even multiple
of the random block size, some blocks will have a fewer runs than others.
Creating Split Plot Designs
Split plot experiments happen when it is convenient to run an experiment in groups of runs (called whole
plots) where one or more factors stay constant within each group. Usually this is because these factors are
difficult or expensive to change from run to run. JMP calls these factors Hard to change because this is
usually how split plotting arises in industrial practice.
In a completely randomized design, any factor can change its setting from one run to the next. When
certain factors are hard to change, the completely randomized design may require more changes in the
settings of hard-to-change factors than desired.
If you know that a factor or two are difficult to change, then you can set the Changes setting of a factor from
the default of Easy to Hard. Before making the design, you can set the number of whole plots you are
willing to run.
For an example of creating a split plot design, see “Creating a Design with Two Hard-to-Change Factors:
Split Plot,” p. 55.
To create a split plot design using the custom designer:
1. In the factors table there is a column called Changes. By default, changes are Easy for all factors. If,
however, you click in the changes area for a factor, you can choose to make the factor Hard to change.
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2. Once you finish defining the factors and click continue, you see an edit box for supplying the number of
whole plots. You can supply any value as long as it is above the minimum necessary to fit all the model
parameters. You can also leave this field empty. In this case, JMP chooses a number of whole plots to
minimize the omnibus uncertainty of the fixed parameters.
Note: If you enter a missing value in the Number of Whole Plots edit box, then JMP considers many
different numbers of whole plots and chooses the number that maximizes the information about the
–1
coefficients in the model. It maximizes the determinant of X ′V X where V -1 is the inverse of the variance
matrix of the responses. The matrix, V, is a function of how many whole plots there are, so changing the
number of whole plots changes V, which can make a difference in the amount of information a design
contains.
To create a split plot design every time you use a certain factor, save steps by setting up that factor to be
“hard” in all experiments. See “Identify Factor Changes Column Property,” p. 98, for details.
Creating Split-Split Plot Designs
Split-split plot designs are a three stratum extension of split plot designs. Now there are factors that are
Very-Hard-to-change, Hard-to-change, and Easy-to-change. Here, in the top stratum, the Very-Hard-tochange factors stay fixed within each whole plot. In the middle stratum the Hard-to-change factors stay
fixed within each subplot. Finally, the Easy-to-change factors may vary (and should be reset) between runs
within a subplot. This structure is natural when an experiment covers three processing steps. The factors in
the first step are Very-Hard-to-change in the sense that once the material passes through the first processing
stage, these factor settings are fixed. Now the material passes to the second stage where the factors are all
Hard-to-change. In the third stage, the factors are Easy-to-change.
Schoen (1999) provides an example of three-stage processing involving the production of cheese that leads
to a split-split plot design. The first processing step is milk storage. Typically milk from one storage facility
provides the raw material for several curds processing units—the second processing stage. Then the curds are
further processed to yield individual cheeses.
In a split-split plot design the material from one processing stage passes to the next stage in such a way that
nests the subplots within a whole plot. In the example above, milk from a storage facility becomes divided
into two curds processing units. Each milk storage tank provided milk to a different set of curds processors.
So, the curds processors were nested within the milk storage unit.
Figure 3.21 shows an example of how factors might be defined for the cheese processing example.
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Figure 3.21 Example of Split-Split Response and Factors in Custom Designer Dialog
Creating Strip Plot Designs
In a strip plot design it is possible to reorder material between processing stages. Suppose units are labelled
and go through the first stage in a particular order. If it is possible to collect all the units at the end of the
first stage and reorder them for the second stage process, then the second stage variables are not nested
within the blocks of the first stage variables. For example, in semiconductor manufacturing a boat of wafers
may go through the first processing step together. However, after this step, the wafers in a given boat may be
divided among many boats for the second stage.
To set up a strip plot design, enter responses and factors as usual, designating factors as Very Hard, Hard, or
Easy to change. Then, in the Design Generation panel, check the box that says Hard to change factors
can vary independently of Very Hard to change factors, as shown in Figure 3.22. Note that the Design
Generation panel specified 6 whole plots, 12 subplots, and 24 runs.
When you click Make Design, the design table on the right in Figure 3.22 lists the run with subplots that
are not nested in the whole plots.
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Special Custom Design Commands
Figure 3.22 Example of Strip Split Factors and Design Generation panel in Custom Designer Dialog
Special Custom Design Commands
After you select DOE > Custom Design, click the red triangle icon on the title bar to see the list of
commands available to the Custom designer (Figure 3.23). The commands found on this menu vary,
depending on which DOE command you select. However, the commands to save and load responses and
factors, the command to set the random seed, and the command to simulate responses are available to all
designers. You should examine the red triangle menu for each designer you use to determine which
commands are available. If a designer has additional commands, they are described in the appropriate
chapter.
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Figure 3.23 Click the Red Triangle Icon to Reveal Commands
The following sections describe these menu commands and how to use them.
Save Responses and Save Factors
If you plan to do further experiments with factors and/or responses to which you have given meaningful
names and values, you can save them for later use.
To save factors or responses:
1. Select a design type from the DOE menu.
2. Enter the factors and responses into the appropriate panels (see “Enter Responses and Factors into the
Custom Designer,” p. 65, for details).
3. Click the red triangle icon on the title bar and select Save Responses or Save Factors.
Save Responses creates a data table containing a row for each response with a column called
Response Name that identifies the responses. Four additional columns identify more information
about the responses: Lower Limit, Upper Limit, Response Goal, and Importance.
Save Factors creates a data table containing a column for each factor and a row for each factor level.
The columns have two column properties (noted with asterisks icons in the column panel). These
properties include:
Design Role that identifies the factor as a DOE factor and lists its type (continuous, categorical,
blocking, and so on).
Factor Changes that identifies how difficult it is to change the factor level. Factor Changes options
are Easy, Hard, and Very Hard.
4. Save the data table.
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Load Responses and Load Factors
If you have saved responses and factors, you can quickly apply them to your design and avoid retyping this
information each time you run an experiment.
To design an experiment using responses or factors you have previously saved:
1. Open the data table that contains the factor names and levels.
2. Select a design type from the DOE menu.
3. Click the red triangle icon on the title bar and select Load Responses or Load Factors.
Tip: It is possible to create a factors table by keying data into an empty table, but remember to assign each
column a factor type. Do this by right-clicking the column name, selecting Column Info, and then selecting
Column Properties > Design Role. Lastly, click the button in the Design Role area and select the
appropriate role.
Save Constraints and Load Constraints
In custom, augment, and mixture designs, if you set up factor constraints and plan to do further
experiments with them, you can save them for later use. You can quickly apply these constraints to your
design and avoid retyping this information each time you run an experiment.
To save factor constraints:
1. Select a design type from the DOE menu.
2. Enter the factor constraints into the appropriate panels (see “Enter Responses and Factors into the
Custom Designer,” p. 65, for details).
3. Click the red triangle icon on the title bar and select Save Constraints. Save Constraints creates a data
table that contains the information you enter into a constraints panel. There is a column for each
constraint. Each has a column property called Constraint State that identifies it as a ‘less than’ or a
‘greater than’ constraint. There is a row for each variable and an additional row that has the inequality
condition for each variable.
4. Save the data table.
To design an experiment using factor constraints you have previously saved:
1. Open the data table that contains the constraints.
2. Select a design type from the DOE menu.
3. Click the red triangle icon on the title bar and select Load Constraints.
Set Random Seed: Setting the Number Generator
The design process begins with a random starting design. To set the random seed that the custom designer
uses to create this starting design, click the red triangle icon in the design title bar and select Set Random
Seed.
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The window that appears shows the generating seed for that design (Figure 3.24). From this window, you can
set a new random number and then run the design again.
If you use the same seed as a previous design, you will get the same design again.
Figure 3.24 Setting the Random Seed
Note: The random seed is also used when you simulate responses to be used with a design, as described
next.
Simulate Responses
Often, when you define a custom design (or any standard design), it may be useful to look at properties of
the design with response data before you have collected data. The Simulate Responses command adds
random response values to the JMP table that the custom designer creates. To use the command, select it
before you click Make Table. When you click Make Table to create the design table, the Y column contains
values for simulated responses.
For custom and augment designs, an additional window appears with the design data table that lists
coefficients for the design you described in the designer panels. You can enter any coefficient values you
want and click Apply to see new Y values in the data table. An example of an equation for a model with two
factors and interaction (Figure 3.25) would be,
y = 21 + 4X1 + 6X2 – 5X1X2 + random noise,
where the random noise is distributed with mean zero and standard deviation one.
Figure 3.25 Example of a Custom Design with Simulated Responses
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Save X Matrix: Viewing the Number of Rows in the Moments Matrix and the
Design Matrix (X) in the Log
To create scripts for the Moments Matrix and the Design Matrix, and to save these matrices as table
properties in the data table that will be generated from the design, click the red triangle icon in the Custom
Design title bar (Figure 3.23) and select Save X Matrix. After the design and table are created, two scripts
are saved as table properties and are called the Moments Matrix and the Design Matrix. Each script can be
selected and run from the upper left panel of the resulting data table. Results from running each of the
scripts are shown in the Log. When you run the script for the Moments Matrix, JMP shows the number of
rows for the global matrix, called Moments, in the Log. Similarly, when you run the script for the Design
Matrix, JMP displays the number of rows for the global matrix, called X, in the Log. The Moments Matrix
and the Design Matrix are used to calculate the Average Variance of Prediction, shown in the Design
Diagnostics section of the Design Evaluation. Saving these scripts to the data table provides an easy way to
remember and recreate your design at a later time and to compare the matrices values with alternate designs.
If you do not have the log visible, select View > Log (Window > Log on the Macintosh). To illustrate these
features:
1. Select DOE > Custom Design.
2. Add 3 continuous factors and click Continue.
3. Click on Interactions > 2nd and select Save X Matrix from the drop-down menu of Custom Design.
4. Using the Default Number of Runs (8), click Make Design and then Make Table.
5. If it is not already open, select View > Log (Window > Log on the Macintosh).
6. Click on the Moments Matrix red triangle in the upper left panel of the data table under Custom Design
and select Run Script. The result shows in the log as N Row(::Moments):7, which is the number of
rows in the global matrix called Moments. The Moments Matrix is dependent upon the model effects
but is independent of the design. (The model effects can be viewed by clicking the red triangle by Model
in the upper left panel of the data table and clicking on Run Script.) The Moments Matrix script for
this example displays the value of each moment and is shown by clicking on the red triangle of the
Moments Matrix and selecting Edit:
Moments = [1 0 0 0 0 0 0,
0 0.333333333333333 0 0 0 0 0,
0 0 0.333333333333333 0 0 0 0,
0 0 0 0.333333333333333 0 0 0,
0 0 0 0 0.111111111111111 0 0,
0 0 0 0 0 0.111111111111111 0,
0 0 0 0 0 0 0.111111111111111];
7. Click on the Design Matrix red triangle in the upper left panel of the data table under Custom Design
and select Run Script. The result shows in the log as N Row(::X):8, which is the number of rows in
the global matrix called X. The X Matrix is dependent upon the design for the experiment. The script
for this example shows the underlying design of the X matrix and is viewed by clicking on the red
triangle of the Design Matrix and selecting Edit:
X
1
1
1
= [1 -1 1 -1 -1 1 -1,
1 1 1 1 1 1,
-1 -1 -1 1 1 1,
1 -1 -1 -1 -1 1,
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1
1
1
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1 1 -1 1 -1 -1,
1 -1 1 -1 1 -1,
-1 1 1 -1 -1 1,
-1 -1 1 1 -1 -1];
Note that the Moments Matrix is defined as:
M =
∫ f ( x )f ( x )' dx
R
where M is a moments matrix of the parameter space that is independent of the design and can be
computed in advance, and where f(x)' denotes a row of the design matrix corresponding to factor
combinations of x. For additional details concerning moments and design matrices, see Myers,
Montgomery, and Anderson-Cook (2009, pp. 365-371). Note that the moment matrix is called a matrix of
region moments in this book.
Optimality Criterion
To change the design criterion, click the red triangle icon in the Custom Design title bar (Figure 3.23) and
select Optimality Criterion, then choose one of the options:
•
Make D-Optimal Design
•
Make I-Optimal Design
•
Make Alias Optimal Design
The default criterion for Recommended is D-optimal for all design types unless you have used the RSM
button in the Model panel to add effects that make the model quadratic. For specific information about
optimality criterion, see “Technical Discussion,” p. 60.
Number of Starts: Changing the Number of Random Starts
To override the default number of random starts, click the red triangle icon in the Custom Design title bar
(Figure 3.23) and select Number of Starts. When you select this command, the window shown in
Figure 3.26 appears with an edit box for you to enter the number of random starts for the design you want
to build. The number you enter overrides the default number of starts, which varies depending on the
design.
Figure 3.26 Selecting the Number of Starts
Note: If the design iterations are taking too long, click the Cancel button. The Custom Designer stops and
gives the best design found at that point.
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Why Change the Number of Starts?
One difficulty with the creation of optimal designs is that the methods used do not always find the globally
optimal design in cases where the optimal design is known from theory. For example, orthogonal designs are
D-optimal with respect to a linear additive model and a cubic design space.
As the number of factors and sample size increase, the optimization problem becomes harder. It is easy for
an optimizer to converge to a local optimum instead of a global optimum.
It is useful to know that:
•
If random starts are used for the optimization, the design produced at the end is not always the same.
Increasing the number of random starts tends to improve the optimality of the resulting design.
•
For designs with all two-level factors, there is a formula for the optimal determinant:
If D is the determinant, n is the sample size, and c is the number of columns in the design matrix, the
LogD = cLogn.
If the determinants that result from the random starts match the formula above, the algorithm stops.
The design is D-optimal and orthogonal.
Default Choice of Number of Random Starts: Technical Information
JMP does not start over with random designs until a jackpot is hit. The time it takes for one iteration of the
algorithm (coordinate exchange) increases roughly as the product of the sample size and the number of
terms in the model increases. By doing a large number of random starts for small sample sizes and reducing
this number proportional to the square of the sample size as the designs get larger, the total time it takes to
generate a design is kept roughly constant over the range of usual sample sizes.
The Custom Designer always attempts to find globally optimal designs when such designs are known from
theory. For example,
•
2-level fractional factorial designs are globally D-optimal for all main effect and two-factor interaction
models
•
Latin-Square designs are D-optimal for main effect models assuming the right sample size and numbers
of levels of the factors.
•
Plackett-Burman designs are D-optimal for main effect models.
If the custom designer can identify one of these special cases, it does many more random starts. In general,
however, the default number of random starts is controlled by the sample size, n, as follows:
Table 3.2 Sample Size and Random Starts
Sample Size
Number of Starts
9 or fewer
80
from 9 to 16
40
from 17 to 24
10
from 25 to 32
5
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Table 3.2 Sample Size and Random Starts (Continued)
Sample Size
Number of Starts
more than 32
2
Note the following exceptions:
•
If each factor has only two levers, the number of terms in the model is one greater than the number of
factors, and the sample size is a multiple of 4, then multiply the default number of starts shown in the
table above by 40.
•
If the design is I-optimal divide all the Number of Starts in the table above by 2 and add 1.
•
If the design could be a Latin Square, the Custom Designer does 1000 random starts.
•
If the number of terms in the model is greater than 100, though, the number of random starts is 1.
After each random start, the design is checked to see if it is globally optimal, and if so, the iterations stop.
Therefore, even if the default number of starts is large, it may only take a small fraction of the default
number to find the globally optimal design. Again, if the process seems to be taking too long, use the
Cancel button to see the best design found at that point.
Sphere Radius: Constraining a Design to a Hypersphere
You can constrain custom and augmented designs to a hypersphere by editing the sphere radius. Before
making the design, click the red triangle icon in the Custom Design title bar (Figure 3.23) and select
Sphere Radius. Enter the appropriate value and click OK.
Note that hypersphere constraints do not work with other constraints. Also, split plot designs cannot be
generated with hypersphere constraints.
If you have designed any factor’s changes as Hard (see “Factors that are Easy, Hard, or Very Hard, to
Change: Creating Optimal Split-Plot and Split-Split-Plot Designs,” p. 68, and “Creating Random Block
Designs,” p. 79), the sphere radius item becomes unavailable. Conversely, once you set the sphere radius,
you cannot make a factor Hard to change.
Disallowed Combinations: Accounting for Factor Level Restrictions
JMP gives you the flexibility to disallow particular combinations of levels of factors. You can do this for
custom and augmented designs except for experiments with mixture or blocking factors. This feature can
also be used with continuous factors or mixed continuous and categorical factors.
For example, in a market research choice experiment, you might want to exclude a choice that allows all the
best features of a product at the lowest price. In this case, the factor Feature has levels of worst (1), medium
(2), and best (3), and the factor Price has levels of high (1), medium (2), and low (3). You want to exclude
the third Feature level (best) and the third Price level (low).
To disallow a combination of factor levels:
1. Begin by adding the factors.
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2. Click the red triangle icon in the title bar (Figure 3.23) of the designer window and select Disallowed
Combinations. Note that this menu item is not available if you have already defined linear inequality
constraints.
3. Enter a Boolean expression that identifies what you do not want allowed (Figure 3.27). JMP evaluates
your expression, and when it sees it as true, it disallows the specified combination.
Note: When forming the expression, use the ordinal value of the level instead of the name of the level. If
the level names of the factor called Price are high, medium, and low, their associated ordinal values are 1, 2,
and 3.
For example, in Figure 3.27, Feature==3 & Price==3 will not allow a run containing the best
features at the lowest price. If there were two disallowed combinations in this example, you would use
Feature==3 & Price==3 | Quality==3 & Price==3, which tells JMP to disallow a run with the
best features at the lowest price or a run with the best quality and lowest price.
Figure 3.27 Enter a Boolean Expression
4. Make the design. It excludes the combination of factors you specified, as shown in Figure 3.28.
Figure 3.28 No Row Contains L3 for Both Price and Feature
Advanced Options for the Custom Designer
The following options are for advanced design of experiment users.
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Changing the Search Points Per Factor
For a main effects model, the coordinate exchange algorithm in the custom designer only considers the high
and low values. However, you can change this so the algorithm considers more search points. For example, if
you enter 5, then JMP considers five equally spaced settings for each factor. The 5 levels are considered, but
all 5 levels may not appear in the output table. The Custom Designer finds a D- or I-Optimal design, which
might not need to include all 5 levels.
To change the search points:
1. Select DOE > Custom Design.
2. Click the red triangle icon in the title bar (Figure 3.23) of the designer window and select Advanced
Options > Search Points Per Factor.
3. Enter a positive integer and click OK.
4. Make the design.
Altering the Mixture Sum
If you want to keep a component of a mixture constant throughout an experiment, then the sum of the
other mixture components must be between 0.001 and 1. You may have one or more fixed ingredients so
that the sum of the remaining add to less than one but more than zero. To alter the mixture sum:
1. Select DOE > Custom Design.
2. Click the red triangle icon in the title bar (Figure 3.23) of the designer window and select Advanced
Options > Mixture Sum.
3. Enter a positive number and click OK.
4. Make the design.
Split Plot Variance Ratio
The optimal split plot design depends on the ratio of the variance of the random whole plot variance to the
error variance. By default, this variance is one. If you have some prior knowledge of this variance ratio, you
can supply it by following these steps:
1. Select DOE > Custom Design.
2. Click the red triangle icon in the title bar (Figure 3.23) of the designer window and select Advanced
Options > Split Plot Variance Ratio.
3. Enter a positive number in the resulting entry field and click OK.
4. Make the design.
Prior Parameter Variance
If you have specified If Possible as the Estimability for any factors in your model, then you can use this
option to also specify the weight used for these terms. Default values are one. Larger values represent more
prior information and a smaller variance. Variances are the reciprocals of the entered values.
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1. Select DOE > Custom Design.
2. Click the red triangle icon in the title bar (Figure 3.23) of the designer window and select Advanced
Options > Prior Parameter Variance.
3. Enter a positive number for each of the terms for which you want to specify a weight and click OK.
4. Make the design.
D Efficiency Weight
Specify the relative importance of D-Efficiency (reducing the variance of the coefficients) versus aliasing
reduction. Values should be between 0 and 1, with larger values weighting more toward D-Efficiency.
Save Script to Script Window
This command creates the script for the design you described in the Custom Designer and saves it in an
open script window.
Assigning Column Properties
Columns in a data table can contain special column properties. Figure 3.29 shows that a column called
Stretch has two special properties: Role and Response Limits, that were assigned by the Custom Designer
when the table was created. To see the example in Figure 3.29, open Bounce Data.jmp from the Design
Experiment folder found in the sample data installed with JMP. Then, right-click the column name in the
data table and select Column Info. When the Column Info dialog appears, click on the property you want
to see.
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Figure 3.29 Column Properties Menu in the Column Info Dialog
All special column properties are covered in the Using JMP. The following discussion gives details about
properties specific to DOE and that are useful for analyzing DOE data.
Define Low and High Values (DOE Coding) for Columns
For continuous variables, the Coding column property transforms data in the range you specify from –1 to
+1. When you analyze the coded variable, JMP uses those transformed data values to compute meaningful
parameter estimates. You can specify the range in which the low and high values of the column are
transformed.
By default, when JMP generates a design table from values entered in the Factors panel, it uses those values
as the low and high values of the coding property. If a column has one or more limits missing, JMP
substitutes the data’s minimum and maximum for the high and low values.
You can use the Column Info dialog to manually add or delete a coding property, or change the range in
which the low and high values are transformed. Figure 3.30 shows the coding values for the Temperature
variable in the Reactor 8 Runs data table from the Design Experiment Sample Data.
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Figure 3.30 Coding Column Property in Column Info Dialog
Set Columns as Factors for Mixture Experiments
You might have a column in a data table that is one of several factors that form 100% of a mixture. You can
set up the column so JMP uses it to automatically generate a no-intercept model when you analyze the data
with the Fit Model platform. The following example uses the Donev Mixture Data from the Design
Experiment sample data
To set up the CuS04 column as a mixture factor, first select Cols > Column Info to see the Column Info
dialog for CuS04. Then continue as follows:
1. Select Mixture from the Column Properties drop-down menu. Upper and Lower limits, and the sum of
the limits appear in a panel on the dialog, as shown in Figure 3.31. You can use these limits, or enter
your own values.
2. Optionally, check the boxes beside L PseudoComponent Coding, U PseudoComponent Coding, or
both L and U PseudoComponent Coding. Using the example in Figure 3.31, where the mixture sum
value is 1, the terms are coded as:
X i L = ( X i – L i ) ⁄ ( 1 – L ) for the L pseudocomponent
X i U = ( U i – X i ) ⁄ ( U – 1 ) for the U pseudocomponent
where Li and Ui are the lower and upper bounds, L is the sum of Li and U is the sum of Ui.
Note: If you check either L PseudoComponent Coding or U PseudoComponent Coding for the mixture
coding of one mixture factor and you check the other alternative for one or more other mixture factors in
the model, of if you check both boxes for one or more of the mixture factors, the Fit Model platform uses
the L coding if (1 – L) < (U – 1), and the U coding otherwise. If only one coding box is checked consistently
for all mixture factors in the model, then only that one pseudocomponent coding is used.
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In the Fit Model report, the main effects are labeled with the coding transformation. Crossed effects are
not labeled, but coding values are used. All the features of fitting, such as the profilers and saved
formulas, respect the pseudocomponent coding but present the un-coded values in the tables and plots.
3. Select the Design Role Column Property, and choose Mixture from its drop down menu.
4. Click OK. The properties icon ( ) now appears next to the column name in the columns panel,
indicating the column contains one or more column properties.
Figure 3.31 Column Info to Create Mixture Column For Analysis
5. Repeat the above steps for any other mixture factors that will be included in the model.
Define Response Column Values
You can save response limits in a column, which means you can run analyses without having to re-specify
response limits each time. Saving these limits in a column facilitates consistency. For example, if you run an
analysis that employs these limits, then come back later and change the data, you can run a new analysis
using the same limits without having to reenter them. To see the example in Figure 3.32, open Bounce
Data.jmp from the Design Experiment folder in the sample data installed with JMP.
Figure 3.32 shows the panel with values that specify lower, middle, and upper limits, and a desirability
value. You can also select a possible goal for a DOE response variable: Maximize, Match Target, Minimize,
or None. If you have more than one response, you can enter an importance value, which lets JMP know
how to weigh the importance of one response against another.
To enter response limits:
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1. Double-click the column name Stretch in the data grid. The Column Info dialog appears.
2. Select Response Limits from the Column Properties drop-down menu.
3. Select a goal for the response variable. For example, if you are in the prediction profiler and want the
desired value to be close to 450, select Match Target.
4. When you have two responses, enter a number in the Importance box to indicate the amount of weight
you want this response to have when JMP computes the overall desirability.
5. Enter the lower, middle and upper limits as well as the desirability values.
Figure 3.32 Completed Response Limits
6. Click OK. The properties icon ( ) now appears next to the column name in the column panel of the
data table to indicate that the column contains a property.
Assign Columns a Design Role
The Custom designer in JMP assigns design roles to factors when you create the design. However, you can
assign a property to a column that identifies a factor column as a continuous, categorical, blocking,
covariate, mixture, constant, signal, or noise factor. The example in Figure 3.33 shows the Whole Plots
factor in the Vinyl Data.jmp table from the Design Experiment sample data assigned the Random Block
design role.
To give a column a design role:
1. Double-click the column name in the data grid. The Column Info window appears.
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2. Select Design Role from the Column Properties drop-down menu, as shown in Figure 3.33. Design
role information appears on the right.
3. Click the Design Role drop-down menu and select how you want JMP to use the factor column:
Continuous, Categorical, Blocking, Covariate, Mixture, Constant, Signal, Noise, Uncontrolled, or
Random Block.
4. Click OK to see the property icon (
) next to the column name in the data table’s column panel.
Figure 3.33 Assign a Design Role to a Factor Column
Note: Although you can save design roles for factors, which are then automatically used each time those
factors are loaded, you must always verify that the model for the design you create is correctly entered into
the DOE custom designer.
Identify Factor Changes Column Property
To create split plot or split-split plot designs, you must identify a factor as having values that are hard to
change, or very hard to change. This is done in the DOE design panel (see “Creating Split Plot Designs,”
p. 80, for details) each time you design an experiment. However, if you know that every time you use that
factor, you want it to be considered hard or very hard to change, you can save yourself steps by setting up a
column property to be used in all experiments using that factor. To do this:
1. Double-click the column name in the data grid to see the Column Info dialog for that column.
2. Select Factor Changes from the Column Properties drop-down menu, as shown in Figure 3.34.
3. Click the Factor Changes button and select Easy, Hard, or Very Hard from the Factor Changes
drop-down menu.
4. Click OK. The properties icon (
data table.
) now appears next to the column name in the column panel of the
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Figure 3.34 Factor Changes Column Property
How Custom Designs Work: Behind the Scenes
The custom designer starts with a random set of points inside the range of each factor. The computational
method is an iterative algorithm called coordinate exchange (Meyer and Nachtsheim, 1995). Each iteration of
the algorithm involves testing every value of every factor in the design to determine if replacing that value
increases the optimality criterion. If so, the new value replaces the old. This process continues until no
replacement occurs for an entire iteration.
To avoid converging to a local optimum, the whole process is repeated several times using a different
random start. The custom designer displays the best of these designs. For more details, see the section
“Optimality Criterion,” p. 88.
Sometimes a design problem can have several equivalent solutions. Equivalent solutions are designs with
equal precision for estimating the model coefficients as a group. When this is true, the design algorithm may
generate different (but equivalent) designs when you click the Back and Make Design buttons repeatedly.
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Chapter 4
Screening Designs
Screening designs are some of the most popular designs for industrial experimentation. They examine many
factors to see which have the greatest effect on the results of a process.
Screening designs generally require fewer experimental runs, which is why they cost less. Thus, they are
attractive because they are a relatively inexpensive and efficient way to begin improving a process.
Often screening designs are a prelude to further experiments. It is wise to spend only about a quarter of your
resource budget on an initial screening experiment. You can then use the results to guide further study.
The efficiency of screening designs depends on the critical assumption of effect sparsity. Effect sparsity results
because real-world processes usually have only a few driving factors; other factors are relatively unimportant.
To understand the importance of effect sparsity, you can contrast screening designs to full factorial designs:
•
Full factorial designs consist of all combinations of the levels of the factors. The number of runs is the
product of the factor levels. For example, a factorial experiment with a two-level factor, a three-level
factor, and a four-level factor has 2 x 3 x 4 = 24 runs.
•
By contrast, screening designs reduce the number of runs by restricting the factors to two (or three)
levels and by performing only a fraction of the full factorial design.
Each factor in a screening design is usually set at two levels to economize on the number of runs needed, and
response measurements are taken for only a fraction of the possible combinations of levels. In the case
described above, you can restrict the factors to two levels, which yield 2 x 2 x 2 = 8 runs. Further, by doing
half of these eight combinations you can still assess the separate effects of the three factors. So the screening
approach can reduce the original 24-run experiment to four runs.
Of course, there is a price for this reduction. This chapter discusses the screening approach in detail,
showing both pros and cons. It also describes how to use JMP’s screening designer, which supplies a list of
popular screening designs for two or more factors. These factors can be continuous or categorical, with two
or three levels. The list of screening designs you can use includes designs that group the experimental runs
into blocks of equal sizes where the size is a power of two.
Contents
Screening Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Using Two Continuous Factors and One Categorical Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Using Five Continuous Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Creating a Screening Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
Enter Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
Enter Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Choose a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Display and Modify a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Specify Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
View the Design Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120
Create a Plackett-Burman design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120
Analysis of Screening Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Using the Screening Analysis Platform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Using the Fit Model Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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Screening Design Examples
This chapter is divided into two sections. The first section consists of two examples using screening designs.
The second section outlines the procedures to follow to create a screening design to match your needs.
Using Two Continuous Factors and One Categorical Factor
Suppose an engineer wants to investigate a process that uses an electron beam welding machine to join two
parts. The engineer fits the two parts into a welding fixture that holds them snugly together. A voltage
applied to a beam generator creates a stream of electrons that heats the two parts, causing them to fuse. The
ideal depth of the fused region is 0.17 inches. The engineer wants to study the welding process to determine
the best settings for the beam generator to produce the desired depth in the fused region.
For this study, the engineer wants to explore the following three inputs, which are the factors for the study:
•
Operator, who is the technician operating the welding machine
•
Rotation Speed, which is the speed at which the part rotates under the beam
•
Beam Current, which is a current that affects the intensity of the beam
After each processing run, the engineer cuts the part in half. This reveals an area where the two parts have
fused. The Length of this fused area is the depth of penetration of the weld. This depth of penetration is the
response for the study.
The goals of the study are to:
•
find which factors affect the depth of the weld
•
quantify those effects
•
find specific factor settings that predict a weld depth of 0.17 inches
To begin this example, select DOE > Screening Design from the main menu. Note that in the Responses
panel, there is a single default response called Y. Change the default response as follows:
1. Double-click the response name and change it to Depth (In.).
2. The default goal for the single default response is Maximize, but the goal of this process is to get a target
value of 0.17 inches with a lower bound of 0.12 and an upper bound of 0.22. Click the Goal box and
choose Match Target from the drop-down menu, as shown in Figure 4.1.
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Figure 4.1 Screening Design Response With Match Target Goal
Click the Goal box and
choose Match Target
3. Click the Lower Limit text edit area and enter 0.12 as the lower limit (minimum acceptable value).
Then click the Upper Limit text edit area and enter 0.22 as the upper limit (maximum acceptable value).
This example has one categorical factor (Operator) and two continuous factors (Speed and Current).
4. Add the categorical factor by clicking the Add button beside 2-Level Categorical.
5. Add two continuous factors by typing 2 in the Continuous box and clicking the associated Add button.
6. Double-click the factor names and rename them Operator, Speed, and Current.
7. Set high and low values for Speed to 3 and 5 rpm. Set high and low values for Current to 150 and 165
amps, and assign John and Mary as values for the categorical factor called Operator, as shown in
Figure 4.2.
Figure 4.2 Screening Design with Two Continuous and One Categorical Factor
8. Click Continue.
9. Select Full Factorial in the list of designs, as shown in Figure 4.3, and then click Continue.
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Figure 4.3 List of Screening Designs for Two Continuous and One Categorical Factors
In the Output Options section of the Display and Modify Design panel that appears, click on the Run
Order menu and select Sort Left to Right, which arranges the runs in the JMP design data table (see
Figure 4.7). Then click Make Table to create the JMP table that contains the specified design.
The table in Figure 4.4 appears. The table uses the names for responses, factors, and levels you specified.
The Pattern variable shows the coded design runs. You can also see the table produced in this example by
selecting Help > Sample Data > Design of Experiments > DOE Example 1. (You can also open DOE
Example 1.jmp from the sample data directory.)
Figure 4.4 The Design Data Table
Using Five Continuous Factors
As illustrated in the previous section, experiments for screening the effects of many factors usually consider
only two levels of each factor. This allows the examination of many factors with a minimum number of
runs.
The following example, adapted from Meyer, et al. (1996), demonstrates how to use JMP’s screening
designer when you have many factors. In this study, a chemical engineer investigates the effects of five
factors on the percent reaction of a chemical process. The factors are:
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•
feed rate, the amount of raw material added to the reaction chamber in liters per minute
•
percentage of catalyst
•
stir rate, the RPMs of a propeller in the chamber
•
reaction temperature in degrees Celsius
•
concentration of reactant
Chapter 4
To start the example:
1. Select DOE > Screening Design.
2. You see one default response called Y. Change the default response name (Y) to Percent Reacted.
3. The Goal is to maximize the response, but change the minimum acceptable reaction percentage to 90
(Lower Limit), and upper limit to 99 (Upper Limit), as shown in Figure 4.5.
4. Add five continuous factors.
5. Change the default factor names (X1-X5) to Feed Rate, Catalyst, Stir Rate, Temperature, and
Concentration.
6. Enter the high and low values, as shown in Figure 4.5.
Figure 4.5 Screening for Many Factors
7. Click Continue. Now, JMP lists the designs for the number of factors you specified, as shown to the left
in Figure 4.6.
8. Select the first item in the list, which is an 8-run fractional factorial design with no blocks.
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9. Click Continue to see the Output Options outline on the right in Figure 4.6.
Figure 4.6 Two-level Screening Design (left) and design output options (right)
The design dialog has options shown in Figure 4.7 that can modify the final design table.
10. Select Sort Left to Right from the Run Order Menu.
Figure 4.7 Output Options for Design Table
11. Click Make Table to create the data table shown in Figure 4.8 that lists the runs for the design you
selected. Note that it also has a column called Percent Reacted for recording experimental results,
showing as the rightmost column of the data table.
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Figure 4.8 JMP Table of Runs for Screening Example
Of the five factors in the reaction percentage experiment, you expect a few to stand out in comparison to the
others. Let’s take an approach to the analysis that looks for active effects.
12. To run the model generated by the data shown in Figure 4.8, open Reactor 8 Runs.jmp from the
Design Experiment folder found in the sample data that was installed with JMP. This table has the
design runs and the results of the experiment.
13. In the design data table, click the Screening script that shows on the upper left of the data table, and
select Run Script. Or, you can choose Analyze > Modeling > Screening to analyze the data. Select
Percent Reacted as Y and all other continuous variables as X. Click OK.The report is shown in
Figure 4.9.
Figure 4.9 Report for Screening Example
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Note: Analysis of the screening data is covered in the section “Create a Plackett-Burman design,” p. 120 at
the end of this chapter.
Creating a Screening Design
To begin, select DOE > Screening Design, or click the Screening Design button on the JMP Starter DOE
page. Then, see the following sections for each step to create a screening design:
1. “Enter Responses,” p. 109
2. “Enter Factors,” p. 110
3. “Choose a Design,” p. 111
4. “Display and Modify a Design,” p. 115
5. “Specify Output Options,” p. 119
6. “View the Design Table,” p. 120
Enter Responses
To enter responses, follow the steps in Figure 4.10.
1. To enter one response at a time, click and then select a goal type. Possible goal types are Maximize,
Match Target, Minimize, or None.
2. (Optional) Double-click to edit the response name.
3. (Optional) Click to change the response goal.
4. Click to enter lower and upper limits and importance weights.
Figure 4.10 Entering Responses
4
1
2
3
Tip: To quickly enter multiple responses, click the Number of Responses button and enter the number of
responses you want.
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Specifying Goal Types and Lower and Upper Limits
When entering responses, you can tell JMP that your goal is to obtain the maximum or minimum value
possible, to match a specific value, or that there is no goal.
The following description explains the relationship between the goal type (step 3 in Figure 4.10) and the
lower and upper limits (step 4 in Figure 4.10):
•
For responses such as strength or yield, the best value is usually the largest possible. A goal of Maximize
supports this objective.
•
The Minimize goal supports an objective of having the smallest value be the most desirable, such as
when the response is impurity or defects.
•
The Match Target goal supports the objective when the best value for a response is a specific target
value, such as dimensions of a manufactured part. The default target value is assumed to be midway
between the lower and upper limits.
Note: If your target range is not symmetric around the target value, you can alter the default target after
you make a table from the design. In the data table, open the response’s Column Info dialog by
double-clicking the column name, and enter an asymmetric target value.
Understanding Importance Weights
When computing overall desirability, JMP uses the value you enter as the importance weight (step 4 in
Figure 4.10) as the weight of each response. If there is only one response, then specifying importance is
unnecessary. With two responses you can give greater weight to one response by assigning it a higher
importance value.
Enter Factors
Next, you enter factors. The Factors panel’s appearance depends on the design you select. Entering factors is
the same in Screening Design, Space Filling Design, Mixture Design, and Response Surface Design.
This process is described below, in Figure 4.11.
1. To enter factors, type the number of factors and click Add.
2. Highlight the factor and click the Remove Selected button to remove a factor in the list.
3. Double-click to edit the factor name.
4. Click to enter factor values. To remove a level, click it, press the delete key on your keyboard, then press
the Return or Enter key on your keyboard.
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Figure 4.11 Entering Factors
1
2
3
4
Types of Factors
In general, when designing experiments, you can enter different types of factors in the model. Below is a
description of factor types from which you can choose when creating screening designs:
Continuous Continuous factors have numeric data types only. In theory, you can set a continuous
factor to any value between the lower and upper limits you supply.
Categorical factors (either numeric or categorical data types) have no implied order. If
the values are numbers, the order is the numeric magnitude. If the values are character, the order is
the sorting sequence. The settings of a categorical factor are discrete and have no intrinsic order.
Examples of categorical factors are machine, operator, and gender.
Categorical
After you enter responses and factors, click Continue.
Choose a Design
The list of screening designs you can use includes designs that group the experimental runs into blocks of
equal sizes where the size is a power of two. Highlight the type of screening design you want to use and click
Continue.
Figure 4.12 Choosing a Type of Screening Design
The screening designer provides the following types of designs:
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Two-Level Full Factorial
A full factorial design has runs for all combinations of the levels of the factors. The samples size is the
product of the levels of the factors. For two-level designs, this is 2k where k is the number of factors. This
can be expensive if the number of factors is greater than 3 or 4.
These designs are orthogonal. This means that the estimates of the effects are uncorrelated. If you remove an
effect in the analysis, the values of the other estimates remain the same. Their p-values change slightly,
because the estimate of the error variance and the degrees of freedom are different.
Full factorial designs allow the estimation of interactions of all orders up to the number of factors. Most
empirical modeling involves first- or second-order approximations to the true functional relationship
between the factors and the responses. The figure to the left in Figure 4.13 is a geometric representation of a
two-level factorial.
Two-Level Fractional Factorial
A fractional factorial design also has a sample size that is a power of two. If k is the number of factors, the
number of runs is 2k – p where p < k. The fraction of the full factorial is 2-p. Like the full factorial, fractional
factorial designs are orthogonal.
The trade-off in screening designs is between the number of runs and the resolution of the design. If price is
no object, you can run several replicates of all possible combinations of m factor levels. This provides a good
estimate of everything, including interaction effects to the mth degree. But because running experiments
costs time and money, you typically only run a fraction of all possible levels. This causes some of the
higher-order effects in a model to become nonestimable. An effect is nonestimable when it is confounded
with another effect. In fact, fractional factorials are designed by deciding in advance which interaction
effects are confounded with the other interaction effects.
Resolution Number: The Degree of Confounding
In practice, few experimenters worry about interactions higher than two-way interactions. These
higher-order interactions are assumed to be zero.
Experiments can therefore be classified by resolution number into three groups:
•
Resolution = 3 means that main effects are confounded with one or more two-way interactions, which
must be assumed to be zero for the main effects to be meaningful.
•
Resolution = 4 means that main effects are not confounded with other main effects or two-factor
interactions. However, two-factor interactions are confounded with other two-factor interactions.
•
Resolution ≥ 5 means there is no confounding between main effects, between two-factor interactions, or
between main effects and two-factor interactions.
A minimum aberration design is one in which there are a minimum number of confoundings for a given
resolution. For DOE experts, the minimum aberration design of a given resolution minimizes the number
of words in the defining relation that are of minimum length.
The figure on the right in Figure 4.13 is geometric representation of a two-level fractional factorial design.
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Figure 4.13 Representation of Full Factorial (Left) and Two-Level Fractional Factorial (Right) Designs
-1, 1, -1
-1, -1, -1
1, -1, -1
1, 1, -1
1, 1, 1
-1, -1, 1
Plackett-Burman Designs
Plackett-Burman designs are an alternative to fractional factorials for screening. One useful characteristic is
that the sample size is a multiple of four rather than a power of two. There are no two-level fractional
factorial designs with sample sizes between 16 and 32 runs. However, there are 20-run, 24-run, and 28-run
Plackett-Burman designs.
The main effects are orthogonal and two-factor interactions are only partially confounded with main effects.
This is different from resolution-three fractional factorial where two-factor interactions are indistinguishable
from main effects.
In cases of effect sparsity, a stepwise regression approach can allow for removing some insignificant main
effects while adding highly significant and only somewhat correlated two-factor interactions. The new
Screening platform in JMP, Analyze > Modeling > Screening, is a streamlined approach for looking at
sparse data. This platform can accept multiple responses and multiple factors, then automatically fits a
two-level design and shows significant effects with plots and statistics. See the chapter in the Modeling and
Multivariate Methods book on the Screening platform for more information.
Mixed-Level Designs
If you have qualitative factors with three values, then none of the classical designs discussed previously are
appropriate. For pure three-level factorials, JMP offers fractional factorials. For mixed two-level and
three-level designs, JMP offers complete factorials and specialized orthogonal-array designs, listed in Table
4.1
If you have fewer than or equal to the number of factors for a design listed in the table, you can use that
design by selecting an appropriate subset of columns from the original design. Some of these designs are not
balanced, even though they are all orthogonal.
Table 4.1 Table of Mixed-Level Designs
Design
Two–Level Factors
Three–Level Factors
L18 John
1
7
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Table 4.1 Table of Mixed-Level Designs (Continued)
Design
Two–Level Factors
Three–Level Factors
L18 Chakravarty
3
6
L18 Hunter
8
4
L36
11
12
Cotter Designs
Cotter designs are used when you have very few resources and many factors, and you believe there may be
interactions. Suppose you believe in effect sparsity— that very few effects are truly nonzero. You believe in
this so strongly that you are willing to bet that if you add up a number of effects, the sum will show an effect
if it contains an active effect. The danger is that several active effects with mixed signs will cancel and still
sum to near zero and give a false negative.
Cotter designs are easy to set up. For k factors, there are 2k + 2 runs. The design is similar to the “vary one
factor at a time” approach many books call inefficient and naive.
A Cotter design begins with a run having all factors at their high level. Then follow k runs each with one
factor in turn at its low level, and the others high. The next run sets all factors at their low level and
sequences through k more runs with one factor high and the rest low. This completes the Cotter design,
subject to randomizing the runs.
When you use JMP to generate a Cotter design, the design also includes a set of extra columns to use as
regressors. These are of the form factorOdd and factorEven where factor is a factor name. They are
constructed by adding up all the odd and even interaction terms for each factor. For example, if you have
three factors, A, B, and C:
Table 4.2 Cotter Design Table
AOdd = A + ABC
AEven = AB + AC
BOdd = B + ABC
BEven = AB + BC
COdd = C + ABC
CEven = BC + AC
Because these columns in a Cotter design make an orthogonal transformation, testing the parameters on
these combinations is equivalent to testing the combinations on the original effects. In the example of
factors listed above, AOdd estimates the sum of odd terms involving A. AEven estimates the sum of the even
terms involving A, and so forth.
Because Cotter designs have a false-negative risk, many statisticians discourage their use.
How to Run a Cotter Design
By default, JMP does not include a Cotter design in the list of available screening designs (Figure 4.12).
However, if you want to make a Cotter design:
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1. Immediately after entering responses and factors (and before clicking Continue), click the red triangle
icon in the Screening Design title bar.
2. Select Supress Cotter Designs (to uncheck it).
Changing the setting via the red triangle menu applies only to the current design. To alter the setting for all
screening designs:
1. Select File > Preferences.
2. Click the Platforms icon.
3. Click DOE to highlight it.
4. Uncheck the box beside Suppress Cotter Designs.
Display and Modify a Design
After you select a design type, open the Display and Modify Design outline.
Figure 4.14 Display and Modification Options
Change Generating Rules Controls the choice of different fractional factorial designs for a given
number of factors.
Aliasing of Effects Shows the confounding pattern for fractional factorial designs.
Coded Design Shows the pattern of high and low values for the factors in each run.
Aliasing of Effects
To see which effects are confounded with which other effects, open the Aliasing of Effects outline. It shows
effects and confounding up to two-factor interactions.
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Figure 4.15 Generating Rules and Aliasing of Effects Panel
For example, a full factorial with five factors requires 25 = 32 runs. Eight runs can only accommodate a full
factorial with three two-level factors. It is necessary to construct the two additional factors in terms of the
first three factors.
The price of reducing the number of runs from 32 to eight is effect aliasing (confounding). Confounding is
the direct result of the assignment of new factor values to products of the coded design columns.
In the example above, the values for Temperature are the product of the values for Feed Rate and
Concentration. This means that you can’t tell the difference of the effect of Temperature and the synergistic
(interactive) effect of Feed Rate and Concentration.
In the example shown in Figure 4.15, all the main effects are confounded with two-factor interactions. This
is characteristic of resolution-three designs.
Look at the Confounding Pattern
JMP can create a data table that shows the aliasing pattern for a specified level. To create this table:
1. Click the red triangle at the bottom of the Aliasing of Effects area.
2. Select Show Confounding Pattern (Figure 4.16).
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Figure 4.16 Show Confounding Patterns
3. Enter the order of confounding you want to see (Figure 4.17).
Figure 4.17 Enter Order of Confounding in Text Edit Box
4. Click OK.
Figure 4.18 shows the third order aliasing for the five-factor reactor example. The effect names begin with C
(Constant) and are shown by their order number in the design. Thus, Temperature appears as “4”, with
second order aliasing as “1 5” (Feed Rate and Concentration), and third order confounding as “1 2 3”
(Feed Rate, Catalyst, and Stir Rate).
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Figure 4.18 The Third Level Alias for the Five-Factor Reactor Example
Understanding Design Codes
In the Coded Design panel, each row represents a run. Plus signs designate high levels and minus signs
represent low levels. As shown in Figure 4.19, rows for the first three columns of the coded design, which
represent Feed Rate, Catalyst, and Stir Rate are all combinations of high and low values (a full factorial
design). The fourth column (Temperature) of the coded design is the element-by-element product of the
first three columns. Similarly, the last column (Concentration) is the product of the second and third
columns.
Figure 4.19 Default Coded Designs
Feed Rate
Catalyst
Temperature
Concentration
Stir Rate
Changing the Coded Design
In the Change Generating Rules panel, changing the check marks and clicking Apply changes the coded
design; it changes the choice of different fractional factorial designs for a given number of factors. The
Coded Design table in Figure 4.19 shows how the last two columns are constructed in terms of the first
three columns. The check marks in the Change Generating Rules table shown in Figure 4.20 for
Temperature now show it is a function of Feed Rate, and Catalyst. The check marks for Concentration
show it is a function of Feed Rate and Stir Rate.
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If you check the options as shown in Figure 4.20 and click Apply, the Coded Design panel changes. The
first three columns of the coded design remain a full factorial for the first three factors (Feed Rate, Catalyst,
and Stir Rate). Temperature is now the product of Feed Rate and Catalyst, so the fourth column of the
coded design is the element by element product of the first two columns. Concentration is a function of
Feed Rate and Stir Rate.
Figure 4.20 Modified Coded Designs and Generating Rules
Specify Output Options
Use the Output Options panel (Figure 4.21) to specify how you want the output data table to appear. When
the options are the way you want them, click Make Table.
Figure 4.21 Select the Output Options
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Run Order Lets you designate the order you want the runs to appear in the data table when it is
created. Choices are:
Keep the Same
Sort Left to Right
Randomize
the rows (runs) in the output table appear as they do in the Coded Design panel.
the rows (runs) in the output table appear sorted from left to right.
the rows (runs) in the output table appear in a random order.
Sort Right to Left
the rows (runs) in the output table appear sorted from right to left.
Randomize within Blocks
the rows (runs) in the output table will appear in random order within
the blocks you set up.
Number of Center Points Specifies additional runs placed at the center points.
Number of Replicates Specify the number of times to replicate the entire design, including
centerpoints. Type the number of times you want to replicate the design in the associated text box.
One replicate doubles the number of runs.
View the Design Table
Click Make Table to create a data table that contains the runs for your experiment. In the table, the high
and low values you specified are displayed for each run.
Figure 4.22 The Design Data Table
The name of the table is the design type that generated it. Run the Screening script to screen for active
effects. The column called Pattern shows the pattern of low values denoted “–” and high values denoted
“+”. Pattern is especially useful as a label variable in plots.
Create a Plackett-Burman design
The previous example shows an 8-run fractional factorial design for five continuous factors. But suppose
you can afford 4 additional runs. First, repeat the steps shown in the previous sections. This time, use the
Load Responses and Load Factors commands to define the design, as follows:
1. Select DOE > Screening Design.
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2. Select Load Responses from the red triangle menu on the Screening Design title bar. Navigate to the
Design Experiment folder in the Sample Data installed with JMP and open the file called
Reactor Response.jmp.
3. Next, select Load Factors from red triangle menu on the Screening Design title bar. Navigate to the
Design Experiment folder in the Sample Data installed with JMP and open the file called
Reactor Factors.jmp.
These two commands complete the DOE screening dialog for you, with the correct response and factor
names, goal and limits for the response, and the values for the factors.
4. Click Continue on the completed Screening design dialog to see the list of designs in Figure 4.23, and
chose the Plackett-Burman, as shown.
Figure 4.23 Design List for 5-factor Plackett-Burman Screening Design
5. Click Continue.
After you select the model from the Design list, the outlines for modifying and evaluating the model appear.
In the Custom designer, you have the ability to form any model effects you want. The Screening designer
creates the design effects based on the design you choose. In particular, the full factorial with all two-factor
interactions has no aliasing of the included interactions, as shown in Figure 4.24.
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Figure 4.24 No aliasing of the included interactions
A complete discussion of the Design Evaluation options is found in Chapter 3, “Building Custom Designs.”
To continue with this example, do the following:
6. Choose Sort Left to Right in the Output Options panel.
7. Click Make Table to see the design runs shown in Figure 4.25.
Examine the data table and note the Pattern variable to see the arrangement of plus and minus signs that
define the runs. This table is used in the analysis sections that follow.
Figure 4.25 Listing of a 5-factor Placket-Burman Design Table
Analysis of Screening Data
After creating and viewing the data table, you can now run analyses on the data. As an example, open the
data table called Plackett-Burman.jmp, found in Design Experiment folder in the Sample Data installed
with JMP. This table contains the design runs and the Percent Reacted experimental results for the 12-run
Plackett-burman design created in the previous section.
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Using the Screening Analysis Platform
The data table has two scripts called Screening and Model, showing in the upper-left corner of the table,
that were created by the DOE Screening designer. You can use these scripts to analyze the data, however it is
simple to run the analyses yourself.
1. Select Analyze > Modeling > Screening to see the completed launch dialog shown in Figure 4.26.
When you create a DOE design table, the variable roles are saved with the data table and used by the
launch platform to complete the dialog.
Figure 4.26 Launch Dialog for the Screening Platform
2. Click OK to see the Screening platform result shown in Figure 4.27.
The Contrasts section of the Screening platform results lists all possible model effects, a contrast value for
each effect, Lenth t-ratios (calculated as the contrast value divided by the Lenth PSE (pseudo-standard
error), individual and simultaneous p-values, and aliases if there are any. Significant and marginally
significant effects are highlighted. See the chapter on analyzing Screening designs in the Modeling and
Multivariate Methods book for complete documentation of the Screening analysis platform.
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Figure 4.27 Results of the Screening Analysis
3. Examine the Half Normal plot in Figure 4.27.
Using the Fit Model Platform
The Make Model button beneath the Half Normal Plot creates a Fit Model dialog that includes all the
highlighted effects. However, note that the Catalyst*Stir Rate interaction is highlighted, but the Stir Rate
main effect is not. Therefore, that interaction shouldn’t be in the model.
4. Click the Make Model Button beneath the Half Normal Plot.
5. Highlight the Catalyst*Stir Rate interaction and click Remove on the Fit Model dialog. The dialog is
shown in Figure 4.28.
6. Then click Run to see the analysis results.
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Figure 4.28 Create Fit Model Dialog and Remove Unwanted Effect
The Actual-by-Predicted Plot
The Whole Model actual-by-predicted plot, shown in Figure 4.29, appears at the top of the Fit Model
report. You see at a glance that this model fits well. The blue line falls outside the bounds of the 95%
confidence curves (red-dotted lines), which tells you the model is significant. The model p-value (p =
0.0178), R2, and RMSE appear below the plot. The RMSE is an estimate of the standard deviation of the
process noise, assuming that the unestimated effects are negligible.
Figure 4.29 An Actual-by-Predicted Plot
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The Scaled Estimates Report
To see a scaled estimates report, use Effect Screening > Scaled Estimates found in the red triangle menu
on the Response Percent Reacted title bar. When there are quadratic or polynomial effects, the
coefficients and the tests for them are more meaningful if effects are scaled and coded. The Scaled Estimates
report includes a bar chart of the individual effects embedded in a table of parameter estimates. The last
column of the table has the p-values for each effect.
Figure 4.30 Example of a Scaled Estimates Report
A Power Analysis
The Fit Model report has outline nodes for the Catalyst and Temperature effects. To run a power analysis
for an effect, click the red triangle icon on its title bar and select Power Analysis.
This example shows a power analysis for the Catalyst variable, using default value for α (0.05), the root
mean square error and parameter estimate for Catalyst, for a sample size of 12. The resulting power is
0.802, which means that in similar experiments, you can expect an 89% chance of detecting a significant
effect for Catalyst.
Figure 4.31 Example of a Power Analysis
Refer to Modeling and Multivariate Methods for details.
Chapter 5
Response Surface Designs
Response surface designs are useful for modeling a curved quadratic surface to continuous factors. A
response surface model can pinpoint a minimum or maximum response, if one exists inside the factor
region. Three distinct values for each factor are necessary to fit a quadratic function, so the standard
two-level designs cannot fit curved surfaces.
The most popular response surface design is the central composite design, illustrated in the figure to the left
below. It combines a two-level fractional factorial and two other kinds of points:
•
Center points, for which all the factor values are at the zero (or midrange) value.
•
Axial (or star) points, for which all but one factor are set at zero (midrange) and that one factor is set at
outer (axial) values.
The Box-Behnken design, illustrated in the figure on the right below, is an alternative to central composite
designs. One distinguishing feature of the Box-Behnken design is that there are only three levels per factor.
Another important difference between the two design types is that the Box-Behnken design has no points at
the vertices of the cube defined by the ranges of the factors. This is sometimes useful when it is desirable to
avoid these points due to engineering considerations. The price of this characteristic is the higher
uncertainty of prediction near the vertices compared to the central composite design.
Figure 5.1 Response Surface Designs
Central Composite Design
Box-Behnken Design
fractional factorial points
axial points
center points
Contents
A Box-Behnken Design: The Tennis Ball Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
The Prediction Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A Response Surface Plot (Contour Profiler) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Geometry of a Box-Behnken Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Creating a Response Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Enter Responses and Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Choose a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Specify Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
View the Design Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140
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A Box-Behnken Design: The Tennis Ball Example
The Bounce Data.jmp sample data file has response surface data inspired by the tire tread data described in
Derringer and Suich (1980). The objective of this experiment is to match a standardized target value (450)
of tennis ball bounciness. The bounciness varies with amounts of Silica, Silane, and Sulfur used to
manufacture the tennis balls. The experimenter wants to collect data over a wide range of values for these
variables to see if a response surface can find a combination of factors that matches a specified bounce target.
To follow this example:
1. Select DOE > Response Surface Design.
2. Load factors by clicking the red triangle icon on the Response Surface Design title bar and selecting
Load Factors. Navigate to the Sample Data folder installed with JMP, and open Bounce Factors.jmp
from the Design Experiment folder.
3. Load the responses by clicking the red triangle icon on the Response Surface Design title bar and
selecting Load Responses. Navigate to the Sample Data folder, and open Bounce Response.jmp from
the Design Experiment folder. Figure 5.2 shows the completed Response panel and Factors panel.
Figure 5.2 Response and Factors For Bounce Data
After the response data and factors data are loaded, the Response Surface Design Choice dialog lists the
designs in Figure 5.3. (Click Continue to see the choices on the right.)
Figure 5.3 Response Surface Design Selection
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The Box-Behnken design selected for three effects generates the design table of 15 runs shown in Figure 5.4.
In real life, you would conduct the experiment and then enter the responses into the data table. Suppose you
completed the experiment and the final data table is Bounce Data.jmp.
1. Open Bounce Data.jmp from the Design Experiment folder found in the sample data installed with
JMP (Figure 5.4).
Figure 5.4 JMP Table for a Three-Factor Box-Behnken Design
After opening the Bounce Data.jmp data table, run a fit model analysis on the data. The data table contains
a script labeled Model, showing in the upper left panel of the table.
2. Click the red triangle next to Model and select Run Script to start a fit model analysis.
3. When the Fit Model dialog appears, click Run.
The standard Fit Model analysis results appear in tables shown in Figure 5.5, with parameter estimates for
all response surface and crossed effects in the model.
The prediction model is highly significant with no evidence of lack of fit. All main effect terms are
significant as well as the two interaction effects involving Sulfur.
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Figure 5.5 JMP Statistical Reports for a Response Surface Analysis of Bounce Data
See Modeling and Multivariate Methods for more information about interpretation of the tables in
Figure 5.5.
Note: DOE response surface designs are available for up to eight factors only. In the DOE Response
Surface Design platform, an error message is generated if more than eight factors are specified with a
response surface design. Response surface designs with more than eight factors can be generated using DOE
> Custom Design, where either a D-optimal or an I-optimal design can be specified. See “Creating
Response Surface Experiments,” p. 34 of JMP Design of Experiments for how to use the custom designer to
create response surface designs. Curvature analysis is not shown (no error or warning message is given) for
response surface designs of more than 20 factors when using the custom designer or the Fit Model platform;
all other analyses are valid and are shown.
The Response Surface report also has the tables shown in Figure 5.6.
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Figure 5.6 Statistical Reports for a Response Surface Analysis
The Response Surface report shows a summary of the parameter estimates.
The Solution report lists the critical values of the surface factors and tells the kind of solution (maximum,
minimum, or saddle point). The solution for this example is a saddle point. The table also warns that the
critical values given by the solution are outside the range of data values.
The Canonical Curvature report shows eigenvalues and eigenvectors of the effects. The eigenvector values
show that the dominant negative curvature (yielding a maximum) is mostly in the Sulfur direction. The
dominant positive curvature (yielding a minimum) is mostly in the Silica direction. This is confirmed by
the prediction profiler in Figure 5.8.
See Modeling and Multivariate Methods for details about the response surface analysis tables in Figure 5.6.
The Prediction Profiler
Next, use the response Prediction Profiler to get a closer look at the response surface and help find the
settings that produce the best response target. The Prediction Profiler is a way to interactively change
variables and look at the effects on the predicted response.
1. If the Prediction Profiler is not already open, click the red triangle on the Response Stretch title bar and
select Factor Profiling > Profiler.
The first three plots in the top row of plots in the Prediction Profiler (Figure 5.7) display prediction traces for
each x variable. A prediction trace is the predicted response as one variable is changed while the others are
held constant at the current values (Jones 1991).
The current predicted value of Stretch, 396, is based on the default factor setting. It is represented by the
horizontal dotted line that shows slightly below the desirability function target value (Figure 5.7). The
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133
profiler shows desirability settings for the factors Silica, Silane, and Sulfur that give a value for Stretch of
396, which is quite different from the specified target of 450.
The bottom row has a plot for each factor, showing its desirability trace. The profiler also contains a
Desirability column, which graphs desirability on a scale from 0 to 1 and has an adjustable desirability
function for each y variable. The overall desirability measure is on the left of the desirability traces.
Figure 5.7 The Prediction Profiler
2. To adjust the prediction traces of the factors and find a Stretch value that is closer to the target, click the
red triangle on the Prediction Profiler title bar and select Maximize Desirability. This command adjusts
the profile traces to produce the response value closest to the specified target (the target given by the
desirability function). The range of acceptable values is determined by the positions of the upper and
lower handles.
Figure 5.8 shows the result of the most desirable settings. Changing the settings of Silica from 1.2 to 1.7,
Silane from 50 to 54, and Sulfur from 2.3 to 2.8 raised the predicted response from 396 to the target value
of 450. Finding maximum desirability is an iterative process so your results may differ slightly from those
shown below.
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Figure 5.8 Prediction Profiler with Maximum Desirability Set for a Response Surface Analysis
See Modeling and Multivariate Methods for further discussion of the Prediction Profiler.
A Response Surface Plot (Contour Profiler)
Another way to look at the response surface is to use the Contour Profiler. Click the red triangle on the
Response Stretch title bar and select Factor Profiling > Contour Profiler to display the interactive contour
profiler, as shown in Figure 5.9.
The contour profiler is useful for viewing response surfaces graphically, especially when there are multiple
responses. This example shows the profile to Silica and Sulfur for a fixed value of Silane.
Options on the Contour Profiler title bar can be used to set the grid density, request a surface plot (mesh
plot), and add contours at specified intervals, like those shown in the contour plot in Figure 5.9. The sliders
for each factor set values for Current X and Current Y.
Chapter 5
Response Surface Designs
A Box-Behnken Design: The Tennis Ball Example
Figure 5.9 Contour Profiler for a Response Surface Analysis
Enter the Lo limit and Hi limit values to shade the unacceptable regions in the contour plot
Figure 5.10 Contour Profiler with High and Low Limits
135
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The Prediction Profiler and the Contour Profiler are discussed in more detail in Modeling and Multivariate
Methods.
Geometry of a Box-Behnken Design
The geometric structure of a design with three effects is seen by using the Scatterplot 3D platform. The plot
shown in Figure 5.11 illustrates the three Box-Behnken design columns. You can clearly see the center
points and the 12 points midway between the vertices. For details on how to use the Scatterplot 3D
platform, see Basic Analysis and Graphing.
Figure 5.11 Scatterplot 3D Rendition of a Box-Behnken Design for Three Effects
Creating a Response Surface Design
Response Surface Methodology (RSM) is an experimental technique invented to find the optimal response
within specified ranges of the factors. These designs are capable of fitting a second-order prediction equation
for the response. The quadratic terms in these equations model the curvature in the true response function.
If a maximum or minimum exists inside the factor region, RSM can estimate it. In industrial applications,
RSM designs usually involve a small number of factors. This is because the required number of runs
increases dramatically with the number of factors. Using the response surface designer, you choose to use
well-known RSM designs for two to eight continuous factors. Some of these designs also allow blocking.
Response surface designs are useful for modeling and analyzing curved surfaces.
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Creating a Response Surface Design
137
To start a response surface design, select DOE > Response Surface Design, or click the Response
Surface Design button on the JMP Starter DOE page. Then, follow the steps described in the following
sections.
•
“Enter Responses and Factors,” p. 137
•
“Choose a Design,” p. 138
•
“Specify Axial Value (Central Composite Designs Only),” p. 138
•
“Specify Output Options,” p. 139
•
“View the Design Table,” p. 140
Enter Responses and Factors
The steps for entering responses are the same in Screening Design, Space Filling Design, Mixture
Design, Response Surface Design, Custom Design, and Full Factorial Design. These steps are outlined
in “Enter Responses and Factors into the Custom Designer,” p. 65
Factors in a response surface design can only be continuous. The Factors panel for a response surface design
appears with two default continuous factors. To enter more factors, type the number you want in the
Factors panel edit box and click Add, as shown in Figure 5.12.
Figure 5.12 Enter Factors into a Response Surface Design
Click Continue to proceed to the next step.
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Choose a Design
Highlight the type of response surface design you want and click Continue. The next sections describe the
types of response surface designs shown in Figure 5.13.
Figure 5.13 Choose a Design Type
Box-Behnken Designs
The Box-Behnken design has only three levels per factor and has no points at the vertices of the cube
defined by the ranges of the factors. This is sometimes useful when it is desirable to avoid extreme points
due to engineering considerations. The price of this characteristic is the higher uncertainty of prediction
near the vertices compared to the central composite design.
Central Composite Designs
The response surface design list contains two types of central composite designs: uniform precision and
orthogonal. These properties of central composite designs relate to the number of center points in the design
and to the axial values:
•
Uniform precision means that the number of center points is chosen so that the prediction variance near
the center of the design space is very flat.
•
For orthogonal designs, the number of center points is chosen so that the second order parameter
estimates are minimally correlated with the other parameter estimates.
Specify Axial Value (Central Composite Designs Only)
When you select a central composite (CCD-Uniform Precision) design and then click Continue, you see the
panel in Figure 5.14. It supplies default axial scaling information. Entering 1.0 in the text box instructs JMP
to place the axial value on the face of the cube defined by the factors, which controls how far out the axial
points are. You have the flexibility to enter the values you want to use.
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Creating a Response Surface Design
139
Figure 5.14 Display and Modify the Central Composite Design
makes the variance of prediction depend only on the scaled distance from the center of the
design. This causes the axial points to be more extreme than the range of the factor. If this factor
range cannot be practically achieved, it is recommended that you choose On Face or specify your
own value.
Rotatable
Orthogonal makes the effects orthogonal in the analysis. This causes the axial points to be more
extreme than the –1 or 1 representing the range of the factor. If this factor range cannot be
practically achieved, it is recommended that you choose On Face or specify your own value.
On Face leaves the axial points at the end of the -1 and 1 ranges.
User Specified
uses the value you enter in the Axial Value text box.
If you want to inscribe the design, click the box beside Inscribe. When checked, JMP rescales the whole
design so that the axial points are at the low and high ends of the range (the axials are –1 and 1 and the
factorials are shrunken based on that scaling).
Specify Output Options
Use the Output Options panel to specify how you want the output data table to appear. When the options
are specified the way you want them, click Make Table. Note that the example shown in Figure 5.15 is for a
Box-Behnken design. The Box-Behnken design from the design list and the Output Options request 3
center points for a single replicate.
Figure 5.15 Select the Output Options
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Creating a Response Surface Design
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Run Order provides a menu with options for designating the order you want the runs to appear in the data
table when it is created. Menu choices are:
Keep the Same
the rows (runs) in the output table will appear in the standard order.
Sort Left to Right the rows (runs) in the output table will appear sorted from left to right.
Randomize the rows (runs) in the output table will appear in a random order.
Sort Right to Left the rows (runs) in the output table will appear sorted from right to left.
Randomize within Blocks
the rows (runs) in the output table will appear in random order within the
blocks you set up.
Add additional points with options given by Make JMP Table from design plus:
Number of Center Points Specifies additional runs placed at the center points.
Number of Replicates Specify the number of times to replicate the entire design, including
centerpoints. Type the number of times you want to replicate the design in the associated text box.
One replicate doubles the number of runs.
View the Design Table
Now you have a data table that outlines your experiment, as described in Figure 5.16.
Figure 5.16 The Design Data Table
The name of the table is the design type that generated it.
Run the Model script to fit a model using the values in the design table.
The column called Pattern identifies the coding of the factors. It shows all the codings with “+” for high,
“–” for low factor, “a” and “A” for low and high axial values, and “0” for midrange. Pattern is suitable to use
as a label variable in plots because when you hover over a point in a plot of the factors, the pattern value
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Creating a Response Surface Design
141
shows the factor coding of the point.The three rows whose values in the Pattern column are 000 are three
center points.
The runs in the Pattern column are in the order you selected from the Run Order menu.
The Y column is for recording experimental results.
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Chapter 6
Full Factorial Designs
A full factorial design contains all possible combinations of a set of factors. This is the most fool proof
design approach, but it is also the most costly in experimental resources. The full factorial designer supports
both continuous factors and categorical factors with up to nine levels.
In full factorial designs, you perform an experimental run at every combination of the factor levels. The
sample size is the product of the numbers of levels of the factors. For example, a factorial experiment with a
two-level factor, a three-level factor, and a four-level factor has 2 x 3 x 4 = 24 runs.
Factorial designs with only two-level factors have a sample size that is a power of two (specifically 2f where f
is the number of factors). When there are three factors, the factorial design points are at the vertices of a
cube as shown in the diagram below. For more factors, the design points are the vertices of a hypercube.
Full factorial designs are the most conservative of all design types. There is little scope for ambiguity when
you are willing to try all combinations of the factor settings.
Unfortunately, the sample size grows exponentially in the number of factors, so full factorial designs are too
expensive to run for most practical purposes.
Figure 6.1 Full Factorial Design
Contents
The Five-Factor Reactor Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Analyze the Reactor Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
Creating a Factorial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Enter Responses and Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Select Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Make the Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Chapter 6
Full Factorial Designs
The Five-Factor Reactor Example
145
The Five-Factor Reactor Example
The following example, adapted from Meyer et al. (1996) and Box, Hunter, and Hunter (1978), shows a
five-factor reactor example.
Previously, the screening designer was used to investigate the effects of five factors on the percent reaction of
a chemical process (see “Screening Designs,” p. 101). The factors (Feed Rate, Catalyst, Stir Rate,
Temperature, and Concentration) are all two-level continuous factors. The next example studies the same
system using a full factorial design.
1. Select DOE > Full Factorial Design.
2. Click the red triangle icon on the Full Factorial Design title bar and select Load Responses.
3. In the Sample Data folder (installed with JMP), open Reactor Response.jmp found in the Design
Experiment folder.
4. Click the red triangle icon on the Full Factorial Design title bar and select Load Factors.
5. In the Sample Data folder (installed with JMP), open Reactor Factors.jmp found in the Design
Experiment folder.
The completed dialog should look like the one shown in Figure 6.2.
Figure 6.2 Full Factorial Example Response and Factors Panels
6. Click Continue to see the Output Options panel. In the Output Options panel, select Sort Left to
Right from the Run Order menu, as shown to the right. This command defines the order of runs as they
will be in the final JMP design table.
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Chapter 6
7. Click Make Table.
The design data table (Figure 6.3) contains a run for every combination of high and low values for the five
variables, which covers all combinations of five factors with two levels each. Since there are five variables,
there are 25=32 runs. Initially, the table has an empty Y column named Percent Reacted for entering
response values when the experiment is complete.
To see the completed experiment and continue this example, open Reactor 32 Runs.jmp found in the
Design Experiment Sample Data folder.
Figure 6.3 Partial Listing of Reactor 32 Runs.jmp from the Sample Data Folder
Analyze the Reactor Data
Begin the analysis with a quick look at the response data before fitting the factorial model.
1. Select Analyze > Distribution.
2. Highlight Percent Reacted and click Y, Columns. Then click OK.
3. Click the red triangle icon on the Percent Reacted title bar and select Normal Quantile Plot. The results
are shown in Figure 6.4.
Chapter 6
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The Five-Factor Reactor Example
147
Figure 6.4 Distribution of Response Variable for Reactor Data
This initial analysis shows some experimental runs with a very high percent reacted response.
Start the formal analysis with a stepwise regression. The data table has a script stored with it that
automatically defines an analysis of the model with main effects and all two-factor interactions.
4. Click the red triangle icon next to the Fit Model script and select Run Script. The stepwise analysis
begins with the Stepwise Regression Control panel shown in Figure 6.5.
5. Select P-value Threshold from the Stopping Rule list.
6. The probability to enter a factor (Prob to Enter) in the model should be 0.05.
7. The probability to remove a factor (Prob to Leave) should be 0.1.
8. A useful way to use the Stepwise platform is to check all the main effects in the Current Estimates table.
However, make sure that the menu beside Direction in the Stepwise Regression Control panel specifies
Mixed (see Figure 6.5).
Figure 6.5 Stepwise Control Panel
9. Check the boxes for the main effects of the factors as shown in Figure 6.6.
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Chapter 6
Figure 6.6 Starting Model For Stepwise Process
10. Click Go.
The mixed stepwise procedure removes insignificant main effects and adds important interactions. The end
result is shown in Figure 6.7. Note that the Feed Rate and Stir Rate factors are no longer in the model.
Figure 6.7 Model After Mixed Stepwise Regression
11. Click the Make Model button in the Stepwise Regression Control panel. The Model Specification
window that appears is automatically set up with the appropriate effects (Figure 6.8).
Chapter 6
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The Five-Factor Reactor Example
149
Figure 6.8 Fitting a Prediction Model
12. Click Run to see the analysis for a candidate prediction model (Figure 6.9).
The figure on the left in Figure 6.9 shows the actual by predicted plot for the model. The predicted model
covers a range of predictions from 40% to 95% reacted. The size of the random noise as measured by the
RMSE is only 3.3311%, which is more than an order of magnitude smaller than the range of predictions.
This is strong evidence that the model has good predictive capability.
The figure on the right in Figure 6.9 shows a table of model coefficients and their standard errors (labeled
Parameter Estimates). All effects selected by the stepwise process are highly significant.
Figure 6.9 Actual by Predicted Plot and Prediction Parameter Estimates Table
The factor Prediction Profiler also gives you a way to compare the factors and find optimal settings.
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Chapter 6
1. To open the Prediction Profiler, click the red triangle on the Response Percent Reacted title bar and
select Factor Profiling > Profiler.
Figure 6.10 shows the profiler’s initial display. The Prediction Profiler is discussed in more detail in the
chapter “Response Surface Designs,” p. 127, and in Modeling and Multivariate Methods.
Figure 6.10 Viewing the Profiler
2. Click the red triangle on the Prediction Profiler title bar and select Maximize Desirability to see the
profiler in Figure 6.11.
Figure 6.11 Viewing the Prediction Profiles at the Optimum Settings
The goal is to maximize Percent Reacted. The reaction is unfeasible economically unless the Percent
Reacted is above 90%. Percent Reacted increases from 65.5 at the center of the factor ranges to a
predicted maximum of 95.875 ± 2.96 at the most desirable settings. The best settings of all three factors are
Chapter 6
Full Factorial Designs
Creating a Factorial Design
151
at the ends of their ranges. Future experiments could investigate what happens as you continue moving
further in this direction.
Creating a Factorial Design
To start a full factorial design, select DOE > Full Factorial Design, or click the Full Factorial Design
button on the JMP Starter DOE page. Then, follow the steps below:
•
“Enter Responses and Factors,” p. 151
•
“Select Output Options,” p. 152
•
“Make the Table,” p. 152
Enter Responses and Factors
The steps for entering responses are outlined in “Enter Responses and Factors into the Custom Designer,”
p. 65
The steps for entering factors in a full factorial design are unique to this design. To add factors, see
Figure 6.12.
1. To enter factors, click either the Continuous button or the Categorical button and select a factor type,
level 2 - 9.
2. Double-click to edit the factor name.
3. Click to enter values or change the level names.
Figure 6.12 Entering Factors in a Full Factorial Design
1
2
3
When you finish adding factors, click Continue.
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Chapter 6
Select Output Options
Use the Output Options panel to specify how you want the output data table to appear, as illustrated in
Figure 6.13:
Figure 6.13 Output Options Panel
Run Order gives options to designate the order you want the runs to appear in the data table when it is
created. Choices are:
Keep the Same
the rows (runs) in the output table will appear in the standard order.
Sort Left to Right the rows (runs) in the output table will appear sorted from left to right.
Randomize the rows (runs) in the output table will appear in a random order.
Sort Right to Left the rows (runs) in the output table will appear sorted from right to left.
Add additional points to the data table with these options:
Number of Center Points Specifies additional runs placed at the center of each continuous factor’s
range.
Number of Replicates Specify the number of times to replicate the entire design, including
centerpoints. Type the number of times you want to replicate the design in the associated text box.
One replicate doubles the number of runs.
Make the Table
When you click Make Table, the table shown in Figure 6.14 appears.
Figure 6.14 Factorial Design Table
Chapter 6
Full Factorial Designs
Creating a Factorial Design
The name of the table is the design type that generated it.
Run the Model script to fit a model using values in the design table.
Values in the Pattern column describe the run each row represents.
•
For continuous factors, a plus sign represents high levels.
•
For continuous factors, a minus sign represents low levels.
•
Level numbers represent values of categorical factors.
153
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Chapter 6
Chapter 7
Mixture Designs
The mixture designer supports experiments with factors that are ingredients in a mixture. You can choose
among several classical mixture design approaches, such as simplex, extreme vertices, and lattice. For the
extreme vertices approach you can supply a set of linear inequality constraints limiting the geometry of the
mixture factor space.
The properties of a mixture are almost always a function of the relative proportions of the ingredients rather
than their absolute amounts. In experiments with mixtures, a factor's value is its proportion in the mixture,
which falls between zero and one. The sum of the proportions in any mixture recipe is one (100%).
Designs for mixture experiments are fundamentally different from those for screening. Screening
experiments are orthogonal. That is, over the course of an experiment, the setting of one factor varies
independently of any other factor. Thus, the interpretation of screening experiments is relatively simple,
because the effects of the factors on the response are separable.
With mixtures, it is impossible to vary one factor independently of all the others. When you change the
proportion of one ingredient, the proportion of one or more other ingredients must also change to
compensate. This simple fact has a profound effect on every aspect of experimentation with mixtures: the
factor space, the design properties, and the interpretation of the results.
Because the proportions sum to one, mixture designs have an interesting geometry. The feasible region for
the response in a mixture design takes the form of a simplex. For example, consider three factors in a 3-D
graph. The plane where the sum of the three factors sum to one is a triangle-shaped slice. You can rotate the
plane to see the triangle face-on and see the points in the form of a ternary plot.
Figure 7.1 Mixture Design
x3
triangular feasible region
x2
x1
Contents
Mixture Design Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
The Optimal Mixture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
The Simplex Centroid Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Creating the Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Simplex Centroid Design Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
The Simplex Lattice Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
The Extreme Vertices Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Creating the Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
An Extreme Vertices Example with Range Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
An Extreme Vertices Example with Linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166
Extreme Vertices Method: How It Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167
The ABCD Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Creating Ternary Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Fitting Mixture Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170
Whole Model Tests and Analysis of Variance Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Understanding Response Surface Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A Chemical Mixture Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Create the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Analyze the Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
The Prediction Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
The Mixture Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .176
A Ternary Plot of the Mixture Response Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Chapter 7
Mixture Designs
Mixture Design Types
157
Mixture Design Types
To create a mixture design, choose DOE > Mixture Design, enter the responses and factors into the initial
mixture designer panel, and click Continue. You then see the Choose Mixture Design Type panel shown in
Figure 7.2. Select one of the designs from the panel:
Optimal invokes the custom designer with all the mixture variables already defined.
Simplex Centroid
lets you specify the degree to which the factor combinations are made.
Simplex Lattice lets you specify how many levels you want on each edge of the grid.
Extreme Vertices lets you specify linear constraints or restrict the upper and lower bounds to be
within the 0 to 1 range.
ABCD Design
generates a screening design for mixtures devised by Snee (1975).
Figure 7.2 Mixture Design Selection Dialog
After you select the design type, choose the number of runs in the Design Generation panel and click Make
Design.
The following sections describe each mixture design type and show examples.
The Optimal Mixture Design
The Optimal mixture design choice invokes the custom designer with the mixture variables entered into the
response and factors panels. To create an optimal mixture design:
1. Select DOE > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Enter Responses and
Factors into the Custom Designer,” p. 65.
3. After you enter responses and factors, click Continue.
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Chapter 7
4. Click Optimal on the Choose Mixture Design Type panel.
5. Add effects to the model using the instructions below.
Adding Effects to the Model
Initially, the Model panel lists only the main effects corresponding to the factors you entered, as shown in
Figure 7.3.
Figure 7.3 The Model Panel
However, you can add factor interactions, specific crossed factor terms, powers, or Scheffe Cubic terms to
the model.
•
To add interaction terms to a model, click the Interactions button and select 2nd, 3rd, 4th, or 5th. For
example, if you have factors X1 and X2, click Interactions > 2nd and X1*X2 is added to the list of model
effects.
•
To add crossed effects to a model, highlight the factors and effects you want to cross and click the Cross
button.
•
To add powers of continuous factors to the model, click the Powers button and select 2nd, 3rd, 4th, or
5th.
•
When you want a mixture model with third-degree polynomial terms, the Scheffe Cubic button
provides a polynomial specification of the surface by adding terms of the form X1*X2*(X1-X2).
The Simplex Centroid Design
A simplex centroid design of degree k with n factors is composed of mixture runs with
•
all one factor
•
all combinations of two factors at equal levels
•
all combinations of three factors at equal levels
•
and so on up to k factors at a time combined at k equal levels.
A center point run with equal amounts of all the ingredients is always included.
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Creating the Design
To create a simplex centroid design:
1. Select DOE > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Enter Responses and
Factors into the Custom Designer,” p. 65.
3. After you enter responses and factors, click Continue.
4. Enter the number of ingredients in the box under K. JMP will create runs for each ingredient without
mixing, then create runs that mix equal proportions of K ingredients at a time to the specified limit.
5. Click the Simplex Centroid button.
6. View factor settings and Output Options, as illustrated in Figure 7.4.
Figure 7.4 Example of Factor Settings and Output Options
7. Specify Run Order, which is the order you want the runs to appear in the data table when it is created.
Run order choices are:
Keep the Same the rows (runs) in the output table will appear as they do in the Factor Settings panel.
Sort Left to Right
the rows (runs) in the output table will appear sorted from left to right.
Randomize the rows (runs) in the output table will appear in a random order.
Sort Right to Left the rows (runs) in the output table will appear sorted from right to left.
Randomize within Blocks the rows (runs) in the output table will appear in random order within
the blocks you set up.
8. Specify Number of Replicates. The number of replicates is the number of times to replicate the entire
design, including centerpoints. Type the number of times you want to replicate the design in the
associated text box. One replicate doubles the number of runs.
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9. Click Make Table.
Simplex Centroid Design Examples
The table of runs for a design of degree 1 with three factors (left in Figure 7.5) shows runs for each single
ingredient followed by the center point. The table of runs to the right is for three factors of degree 2. The
first three runs are for each single ingredient, the second set shows each combination of two ingredients in
equal parts, and the last run is the center point.
Figure 7.5 Three-Factor Simplex Centroid Designs of Degrees 1 and 2
To generate the two sets of runs in Figure 7.5:
1. Choose DOE > Mixture Design.
2. Enter three mixture factors.
3. Click Continue.
4. Enter 1 in the K box, and click Simplex Centroid to see the design on the left in Figure 7.6.
5. Click the Back button, then click Continue, and enter 2 in the K box. Then click Simplex Centroid to
see the design on the right in Figure 7.6.
Figure 7.6 Create Simplex Centroid Designs of Degrees 1 and 2
As another example:
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1. Choose DOE > Mixture Design.
2. Enter five factors and click Continue.
3. Use the default value, 4, in the K box.
4. Click Simplex Centroid.
5. Click Make Table to see the 31-run JMP data table shown in Figure 7.7.
Figure 7.7 Partial Listing of Factor Settings for Five-Factor Simplex Centroid Design
The Simplex Lattice Design
The simplex lattice design is a space filling design that creates a triangular grid of runs. The design is the set
of all combinations where the factors’ values are i / m, where i is an integer from 0 to m such that the sum of
the factors is 1.
To create simplex lattice designs:
1. Select DOE > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Enter Responses and
Factors into the Custom Designer,” p. 65.
3. Click Continue.
4. Specify the number of levels you want in the Mixture Design Type dialog (Figure 7.2) and click Simplex
Lattice.
Figure 7.8 shows the runs for three-factor simplex lattice designs of degrees 3, 4, and 5, with their
corresponding geometric representations. In contrast to the simplex centroid design, the simplex lattice
design does not necessarily include the centroid.
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Figure 7.8 Three-Factor Simplex Lattice Designs for Factor Levels 3, 4, and 5
Figure 7.9 lists the runs for a simplex lattice of degree 3 for five effects. In the five-level example, the runs
creep across the hyper-triangular region and fill the space in a grid-like manner.
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Figure 7.9 JMP Design Table for Simplex Lattice with Five Variables, Order (Degree) 3
The Extreme Vertices Design
The extreme vertices design accounts for factor limits and selects vertices and their averages (formed by
factor limits) as design points. Additional limits are usually in the form of range constraints, upper bounds,
and lower bounds on the factor values.
The extreme vertices design finds the corners (vertices) of a factor space constrained by limits specified for
one or more of the factors. The property that the factors must be non-negative and must add up to one is
the basic mixture constraint that makes a triangular-shaped region.
Sometimes other ingredients need range constraints that confine their values to be greater than a lower
bound or less than an upper bound. Range constraints chop off parts of the triangular-shaped (simplex)
region to make additional vertices. It is also possible to have a linear constraint, which defines a linear
combination of factors to be greater or smaller than some constant.
The geometric shape of a region bound by linear constraints is called a simplex, and because the vertices
represent extreme conditions of the operating environment, they are often the best places to use as design
points in an experiment.
You usually want to add points between the vertices. The average of points that share a constraint boundary
is called a centroid point, and centroid points of various degrees can be added. The centroid point for two
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neighboring vertices joined by a line is a second degree centroid because a line is two dimensional. The
centroid point for vertices sharing a plane is a third degree centroid because a plane is three dimensional,
and so on.
If you specify an extreme vertices design but give no constraints, a simplex centroid design results.
Creating the Design
Follow these steps to create an extreme vertices design. The next sections show examples with specific
constraints.
1. Select DOE > Mixture Design.
2. Enter factors and responses. These steps are outlined in “Enter Responses and Factors into the Custom
Designer,” p. 65. Remember to enter the upper and lower limits in the factors panel (see Figure 7.10).
3. Click Continue.
4. In the Degree text box, enter the degree of the centroid point you want to add. The centroid point is
the average of points that share a constraint boundary.
5. If you have linear constraints, click the Linear Constraints button for each constraint you want to add.
Use the text boxes that appear to define a linear combination of factors to be greater or smaller than
some constant.
6. Click Extreme Vertices to see the factor settings.
7. (Optional) Change the order of the runs in the data table when it is created. Run order choices are:
Keep the Same—the rows (runs) in the output table will appear as they do in the Design panel.
Sort Left to Right—the rows (runs) in the output table will appear sorted from left to right.
Randomize—the rows (runs) in the output table will appear in a random order.
Sort Right to Left—the rows (runs) in the output table will appear sorted from right to left.
Randomize within Blocks—the rows (runs) in the output table will appear in random order within
the blocks you set up.
8. (Optional) Type the sample size you want in the Choose desired sample size text box.
9. (Optional) Click Find Subset to generate the optimal subset having the number of runs specified in
sample size box described in Step 8. The Find Subset option uses the row exchange method (not
coordinate exchange) to find the optimal subset of rows.
10. Click Make Table.
An Extreme Vertices Example with Range Constraints
The following example design table is for five factors with the range constraints shown in Figure 7.10, where
the ranges are smaller than the default 0 to 1 range.
1. Select DOE > Mixture Design.
2. Add two additional factors (for a total of 5 factors) and give them the values shown in Figure 7.10.
3. Click Continue.
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4. Enter ‘4’ in the Degree text box (Figure 7.10).
Figure 7.10 Example of Five-factor Extreme Vertices
5. Click Extreme Vertices.
6. Select Sort Left to Right from the Run Order menu.
7. Click Make Table. Figure 7.11 shows a partial listing of the resulting table. Note that the Rows panel in
the data table shows that the table has the default 116 runs.
Figure 7.11 JMP Design Table for Extreme Vertices with Range Constraints
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Suppose you want fewer runs. You can go back and enter a different sample size (number of runs).
8. Click Back, then click Continue.
9. Enter ‘4’ in the Degree text box and click Extreme Vertices.
10. In the sample size text box, enter ‘10’ as the sample size.
11. Click Find Subset to generate the optimal subset having the number of runs specified. The resulting
design (Figure 7.12) is the optimal 10-run subset of the 116 current runs. This is useful when the
extreme vertices design generates a large number of vertices.
Figure 7.12 JMP Design Table for 10-Run Subset of the 116 Current Runs
Note: The Find Subset option uses the row exchange method (not coordinate exchange) to find the
optimal subset of rows.
An Extreme Vertices Example with Linear Constraints
Consider the classic example presented by Snee (1979) and Piepel (1988). This example has three factors,
X1, X2, and X3, with five individual factor bound constraints and three additional linear constraints:
Table 7.1 Linear Constraints for the Snee and Piepel Example
X1 ≥ 0.1
X1 ≤ 0.5
X2 ≥ 0.1
X2 ≤ 0.7
X3 ≤ 0.7
To enter these constraints:
90 ≤ 85*X1 + 90*X2 + 100*X3
85*X1 + 90*X2 + 100*X3 ≤ 95
.4 ≤ 0.7*X1 + X3
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1. Enter the upper and lower limits in the factors panel.
2. Click Continue.
3. Click the Linear Constraint button three times. Enter the constraints as shown in Figure 7.13.
4. Click the Extreme Vertices button.
5. Change the run order to Sort Right to Left, and keep the sample size at 13. See Figure 7.13 for the
default Factor Settings and completed Output Options.
6. Click Make Table.
Figure 7.13 Constraints
This example is best understood by viewing the design as a ternary plot, as shown at the end of this chapter,
in Figure 7.15. The ternary plot shows how close to one a given component is by how close it is to the vertex
of that variable in the triangle. See “Creating Ternary Plots,” p. 168, for details.
Extreme Vertices Method: How It Works
If there are linear constraints, JMP uses the CONSIM algorithm developed by R.E. Wheeler, described in
Snee (1979) and presented by Piepel (1988) as CONVRT. The method is also described in Cornell (1990,
Appendix 10a). The method combines constraints and checks to see if vertices violate them. If so, it drops
the vertices and calculates new ones. The method named CONAEV for doing centroid points is by Piepel
(1988).
If there are no linear constraints (only range constraints), the extreme vertices design is constructed using
the XVERT method developed by Snee and Marquardt (1974) and Snee (1975). After the vertices are
found, a simplex centroid method generates combinations of vertices up to a specified order.
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The XVERT method first creates a full 2nf – 1 design using the given low and high values of the nf – 1
factors with smallest range. Then, it computes the value of the one factor left out based on the restriction
that the factors’ values must sum to one. It keeps the point if it is in that factor’s range. If not, it increments
or decrements it to bring it within range, and decrements or increments each of the other factors in turn by
the same amount, keeping the points that still satisfy the initial restrictions.
The above algorithm creates the vertices of the feasible region in the simplex defined by the factor
constraints. However, Snee (1975) has shown that it can also be useful to have the centroids of the edges and
faces of the feasible region. A generalized n-dimensional face of the feasible region is defined by nf – n of the
boundaries and the centroid of a face defined to be the average of the vertices lying on it. The algorithm
generates all possible combinations of the boundary conditions and then averages over the vertices generated
on the first step.
The ABCD Design
This approach by Snee (1975) generates a screening design for mixtures. To create an ABCD design:
1. Select DOE > Mixture Design.
2. Enter factors and responses. The steps for entering responses are outlined in “Enter Responses and
Factors into the Custom Designer,” p. 65.
3. After you enter responses and factors, click Continue.
4. Click the ABCD Design button.
5. View factor settings and Output Options.
6. Specify Run Order, which is the order you want the runs to appear in the data table when it is created.
Run order choices are:
Keep the Same
The rows (runs) in the output table will appear as they do in the Factor Settings
panel.
Sort Left to Right
Randomize
The rows (runs) in the output table will appear sorted from left to right.
The rows (runs) in the output table will appear in a random order.
Sort Right to Left
The rows (runs) in the output table will appear sorted from right to left.
Randomize within Blocks
The rows (runs) in the output table will appear in random order
within the blocks you set up.
7. Specify Number of Replicates. The number of replicates is the number of times to replicate the entire
design, including centerpoints. Type the number of times you want to replicate the design in the
associated text box. One replicate doubles the number of runs.
8. Click Make Table.
Creating Ternary Plots
A mixture problem in three components can be represented in two dimensions because the third
component is a linear function of the others. The ternary plot in Figure 7.15 shows how close to one (1) a
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given component is by how close it is to the vertex of that variable in the triangle. The plot to the left in
Figure 7.14 illustrates a ternary plot.
Figure 7.14 Ternary Plot (left) and Tetrary Plot (right) for Mixture Design
X1
X1 (1, 0, 0)
(1/3, 1/3, 1/3)
(1/2, 1/2, 0)
(0.1, 0.1, 0.8)
X2 (0, 1, 0)
X3 (0, 0, 1)
X2
X3
The Piepel (1979) example referenced in “An Extreme Vertices Example with Linear Constraints,” p. 166 is
best understood by the ternary plot shown in Figure 7.15.
To view a mixture design as a ternary plot:
1. Create the Piepel mixture data as shown previously, or open the table called Peipel.jmp, found in the
Design Experiments folder of the Sample Data Library.
2. Choose Graph > Ternary Plot.
3. In the ternary plot launch dialog, specify the three mixture components and click OK.
The JMP Ternary plot platform recognizes the three factors as mixture factors, and also considers the upper
and lower constraints entered into the Factors panel when the design was created. The Ternary plot uses
shading to exclude the unfeasible areas excluded by those constraints.
The Piepel data had additional constraints, entered as linear constraints for the extreme vertices design.
There are six active constraints, six vertices, and six centroid points shown on the plot, as well as two
inactive (redundant) constraints. The feasible area is the inner white polygon delimited by the design points
and constraint lines.
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Figure 7.15 Diagram of Ternary Plot Showing Piepel Example Constraints
Fitting Mixture Designs
When fitting a model for mixture designs, you must take into account that all the factors add up to a
constant, and thus a traditional full linear model will not be fully estimable.
The recommended response surface model, called the Scheffé polynomial (Scheffé 1958), does the
following:
•
suppresses the intercept
•
includes all the linear main-effect terms
•
excludes all the square terms (such as X1*X1)
•
includes all the cross terms (such as X1*X2)
To fit a model:
1. Choose DOE > Mixture Design and make the design data table. Remember that to fit a model, the Y
column in the data table must contain values, so either assign responses or click the red triangle menu
and select Simulate Responses before you click Make Table.
2. The design data table stores the model in the data table as a table property. This table property is a JSL
script called Model, located in the left panel of the table.
3. Right-click the model and select Run Script to launch the Fit Model dialog, which is automatically
filled with the saved model.
4. Click Run on the Fit Model dialog.
In this model, the parameters are easy to interpret (Cornell 1990). The coefficients on the linear terms are
the fitted response at the extreme points where the mixture consists of a single factor. The coefficients on the
cross terms indicate the curvature across each edge of the factor space.
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The model report usually has several sections of interest, including the whole model tests, Analysis of
Variance reports, and response surface reports, which are described below.
Whole Model Tests and Analysis of Variance Reports
In a whole-model Analysis of Variance table, JMP traditionally tests that all the parameters are zero except
for the intercept. In a mixture model without an intercept, JMP looks for a hidden intercept, in the sense
that a linear combination of effects is a constant. If it finds a hidden intercept, it does the whole model test
with respect to the intercept model rather than a zero-intercept model. This test is equivalent to testing that
all the parameters are zero except the linear parameters, and testing that they are equal.
The hidden-intercept property also causes the R2 to be reported with respect to the intercept model rather
than reported as missing.
Understanding Response Surface Reports
When there are effects marked as response surface effects “&RS,” JMP creates additional reports that
analyze the fitted response surface. These reports were originally designed for full response surfaces, not
mixture models. However, if JMP encounters a no-intercept model and finds a hidden intercept with linear
response surface terms, but no square terms, then it folds its calculations, collapsing on the last response
surface term to calculate critical values for the optimum. This can be done for any combination that yields a
constant and involves the last response surface term.
A Chemical Mixture Example
Three plasticizers (p1, p2, and p3) comprise 79.5% of the vinyl used for automobile seat covers (Cornell,
1990). Within this 79.5%, the individual plasticizers are restricted by the following constraints: 0.409 ≤ x1
≤ 0.849, 0 ≤ x2 ≤ 0.252, and 0.151 ≤ x3 ≤ 0.274.
Create the Design
To create Cornell’s mixture design in JMP:
1. Select DOE > Mixture Design.
2. In the Factors panel, use the three default factors but name them p1, p2, and p3, and enter the high and
low constraints as shown in Figure 7.16. Or, load the factors with the Load Factors command in the
red triangle on the Mixture Design title bar. To import the factors, open Plastifactors.jmp, found in the
Design Experiment Sample Data folder that was installed with JMP.
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Figure 7.16 Factors and Factor Constraints for the Plasticizer Experiment
3. Click Continue.
4. Enter 3 in the Degree text box.
5. Click Extreme Vertices.
6. Click Make Table. JMP uses the 9 factor settings to generate a JMP table (Figure 7.17).
Figure 7.17 Extreme Vertices Mixture Design
7. Add an extra five design runs by duplicating the vertex points and center point, to give a total of 14 rows
in the table.
Note: To identify the vertex points and the center (or interior) point, use the sample data script called
LabelMixturePoints.jsl in the Sample Scripts folder installed with JMP.
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8. Run the LabelMixturePoints.jsl to see the results in Figure 7.18, and highlight the vertex points and the
interior point as shown.
Figure 7.18 Identify Vertices and Center Point with Sample Script
9. Select Edit > Copy, to copy the selected rows to the clipboard.
10. Select Rows > Add Rows and type 5 as the number of rows to add.
11. Click the At End radio button on the dialog, then click OK.
12. Highlight the new rows and select Edit > Paste to add the duplicate rows to the table.
The Plasticizer data with the results (Y values) that Cornell obtained are available in the Sample data. Open
Plasticizer.jmp in the Sample Data folder installed with JMP to see this table (Figure 7.19).
Figure 7.19 Plasticizer.jmp Data Table from the Sample Data Library
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Analyze the Mixture Model
Use the Cornell plasticizer data from the Sample Data library (Figure 7.19) to run the mixture model:
1. Click the table property named Model on the upper-left of the data table and select Run Script, which
runs a script that displays a completed Fit Model dialog. Click Run to see the response surface analysis.
2. Plasticizer.jmp contains a column called Pred Formula Y. This column was added after the analysis by
selecting Save Columns > Prediction Formula from the red triangle menu in the Response Y title bar
of the analysis report. To see the prediction formula, right-click (Ctrl+click on the Mac) the column
name and select Formula:
0–50.1465*p1 – 282.1982*p2 – 911.6484*p3 + p2*p1*317.363 +
p3*p1*1464.3298 + p3*p2*1846.2177
Note: These results correct the coefficients reported in Cornell (1990).
The Response Surface Solution report (Figure 7.20) shows that a maximum predicted value of 19.570299
occurs at point (0.63505, 0.15568, 0.20927).
Figure 7.20 Mixture Response Surface Analysis
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The Prediction Profiler
The report contains a prediction profiler.
1. If the profiler is not visible, click the red triangle in the Response Y title bar and select Factor Profiling
> Profiler. You should see the initial profiler shown in Figure 7.21.
The crossed effects show as curvature in the prediction traces. Drag one of the vertical reference lines, and
the other two move in the opposite direction maintaining their ratio.
Note: The axes of prediction profiler traces range from the upper and lower bounds of the factors, p1, p2,
and p3, entered to create the design and the design table. When you experiment moving a variable trace,
you see the other traces move such that their ratio is preserved. As a result, when the limit of a variable is
reached, it cannot move further and only the third variable changes.
2. To limit the visible profile curves to bounds that use all three variables, use the Profile at Boundary >
Stop at Boundaries command from the menu on the Prediction Profiler title bar.
3. If needed, select the Desirability Functions command to display the desirability function showing to
the right of the prediction profile plots in Figure 7.22.
4. Then select Maximize Desirability from the Prediction Profiler menu to see the best factor settings.
The profiler in Figure 7.22, displays optimal settings (rounded) of 0.6350 for p1, 0.1557 for p2, and
0.2093 for p3, which give an estimated response of 19.5703.
Figure 7.21 Initial Prediction Profiler
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Figure 7.22 Maximum Desirability in Profiler for Mixture Analysis Example
The Mixture Profiler
The Fit Model report also has a Mixture Profiler that is useful for visualizing and optimizing response
surfaces from mixture experiments.
Many of the features are the same as those of the Contour Profiler however some are unique to the Mixture
Profiler:
•
A ternary plot is used instead of a Cartesian plot, which enables you to view three mixture factors at a
time.
•
If you have more than three factors, radio buttons let you choose which factors to plot.
•
If the factors have constraints, you can enter their low and high limits in the Lo Limit and Hi Limit
columns. This shades non-feasible regions in the profiler.
Select Factor Profiling > Mixture Profiler from the menu on the Response Y title bar to see the mixture
profiler for the plasticizer data, shown in Figure 7.23.
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Figure 7.23 Mixture Profiler for Plasticizer Example
A Ternary Plot of the Mixture Response Surface
You can also plot the response surface of the plasticizer data as a ternary plot using the Ternary graph
platform and contour the plot with information from an additional variable:
1. Choose Graph > Ternary Plot.
2. Specify plot variables (p1, p2, and p3) and click X, Plottting, as shown in Figure 7.24. To identify the
contour variable (the prediction equation), select Pred Formula Y and click the Contour Formula
button. The contour variable must have a prediction formula to form the contour lines, as shown by the
ternary plots at the bottom in Figure 7.25. If there is no prediction formula, the ternary plot only shows
points.
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Figure 7.24 Launch Dialog for the Ternary Plot Platform
3. Click OK and view the results, as shown in Figure 7.25. By default, the ternary plot displays contour
lines only, but you can request a Fill Above or Fill Below, as shown, with the Contour Fill command
found in the red triangle menu on the Ternary Plot title bar.
Figure 7.25 Ternary Plot of a Mixture Response Surface
Chapter 8
Discrete Choice Designs
The Discrete Choice designer creates experiments with factors that are product attributes. A collection of
attributes is called a product profile. Respondents choose one in each set of product profiles.
Industrial experimentation deals with the question of how to improve processes to deliver better products.
Choice experiments help a company prioritize product features for their market. The purpose of a choice
experiment is to define a product that people want to buy.
Choice experiments always involve people comparing prospective products and picking the one they prefer.
For example, suppose a computer company wants to update its high-end laptop. Laptops have many
features that are important to customers such as processor speed, hard disk size, screen size, battery life, and
price. To build a laptop that customers want, the computer company needs to know the relative importance
of each feature. Most people prefer a faster computer with more storage, longer battery life, and a low price.
What the company does not know is how much more an extra hour of battery life is worth to a customer or
whether doubling the hard disk size is as important as doubling the processor speed. A choice experiment
can answer these questions and indicate the optimal set of trade-offs among product features.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Create an Example Choice Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Analyze the Example Choice Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Design a Choice Experiment Using Prior Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Administer the Survey and Analyze Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190
Initial Choice Platform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Find Unit Cost and Trade Off Costs with the Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
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Discrete Choice Designs
Introduction
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Introduction
Most choice experiments involve conducting a market research survey. The survey consists of a series of
questions about attributes of interest about a potential new product or product modification. For example, a
computer manufacturer might be interested in manufacturing a new laptop and wants information about
customer preference before beginning an expensive development process. Computer characteristics change
so rapidly that it is crucial to quickly identify the attributes that help the manufacturers to design and build
a new machine most likely to capture enough market share to be profitable.
Often, the attributes are obvious. For example, the consumer wants a laptop that has a large screen, weighs
almost nothing, costs almost nothing, and lasts forever on a single battery charge. The question, then, is
how much is the customer willing to compromise these desires? How important is each of these attributes,
and which kinds of trade off is the customer most likely to accept and still purchase a new machine?
Assume a simple situation where a computer manufacturer wants to examine preferences for four possible
laptop configurations. Notice that there are no ‘right’ or ‘wrong’ selections. Instead there are just
preferences. A well designed questionnaire and proper analysis of results can tell a manufacturer how to
proceed. The manufacturer wants information about the following four laptop attributes.
•
size of hard drive disk (40 GB or 80 GB)
•
speed of processor (1.5 GHz or 2.0 GHz)
•
battery life (4 Hrs or 6 Hrs)
•
cost of computer ($1000, $1200 or $1500)
If a survey were constructed that offered the possibility of choosing any combination of these attributes, a
respondent would be forced to evaluate 24 possible combinations and make a single response. Instead each
respondent usually evaluates several choice sets and for each choice set, chooses the preferred profile. In the
simplest situation, each respondent chooses between sets of two profiles.
Then, you analyze the choices of multiple respondents. A well designed choice experiment, correctly
analyzed is a efficient way to give the researcher the most information for the least time and expense.
Table 1 shows hypothetical results from a single survey designed to collect information about consumer
preferences about laptop computers.
•
Each column in the survey identifies a laptop attribute.
•
Each line in the survey defines a laptop profile, which is a collection of attribute values.
•
Each choice set consists of two attribute profiles.
•
All of the attribute values are allowed to change across the two profiles in a choice set.
Table 8.1 Hypothetical Choice Survey Results from a single Respondent, Subject ID 2
For each pair, please check the combination of attributes you find most appealing.
Disk Size
1
Speed
Battery Life
Price
Preference
1
40 GB
1.5 GHz
6 hours
$1,000
_X_
2
80 GB
1.5 GHz
4 hours
$1,200
___
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Table 8.1 Hypothetical Choice Survey Results from a single Respondent, Subject ID 2 (Continued)
For each pair, please check the combination of attributes you find most appealing.
Disk Size
2
3
4
5
6
7
8
Speed
Battery Life
Price
Preference
1
40 GB
1.5 GHz
4 hours
$1,500
___
2
80 GB
2.0 GHz
4 hours
$1,200
_X_
1
40 GB
2.0 GHz
4 hours
$1,200
_X_
2
80 GB
2.0 GHz
6 hours
$1,500
___
1
40 GB
2.0 GHz
4 hours
$1,000
_X_
2
80 GB
1.5 GHz
6 hours
$1,200
___
1
40 GB
1.5 GHz
6 hours
$1,000
_X_
2
40 GB
2.0 GHz
4 hours
$1,500
___
1
40 GB
2.0 GHz
6 hours
$1,200
_X_
2
80 GB
1.5 GHz
4 hours
$1,500
___
1
40 GB
2.0 GHz
6 hours
$1,500
___
2
80 GB
1.5 GHz
4 hours
$1,000
_X_
1
40 GB
1.5 GHz
4 hours
$1,200
___
2
80 GB
2.0 GHz
4 hours
$1,000
_X_
The DOE Choice designer can create a survey like that shown in Table 8.1. However, to create an effective
design, the Choice designer needs information about the attributes. For example, most laptop attributes
have values that are intrinsic preferences. That is, a bigger disk size is better, longer battery life is better, and
so forth. The purpose of conducting a choice survey is to find out how the potential laptop purchasers feels
about the advantages of a collection of tractates.
One way to gain prior information about attributes in a survey is to conduct a single example survey, analyze
the results, and use those results as prior information to create the final survey instrument.
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This chapter shows how to create a sample survey and use its results as prior information for a final survey
design
Create an Example Choice Experiment
The Choice design can create a survey like the one in Table 1. As an example,
1. Choose DOE > Choice Design, and complete the initial dialog as shown in Figure 8.1.
Figure 8.1 Choice Design Dialog with Attributes Defined
2. Click Continue. For this example, use the default values in the Model Control panel and in the Design
Generation panel, as shown in Figure 8.2.
Optionally, you can use the DOE Model Controls panel to add interactions to the choice model in
situations where you expect there are interactions and want to generate profile sets that will help detect
them.
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Figure 8.2 Design Generation Panel for the Laptop Experiment
The values in the Design Generation panel describe the laptop survey.
There are four laptop attributes.
Entering 4 as the number of attributes that can change within a choice set means that the Choice
designer can change 4 or fewer attribute values within a single choice set. You can enter fewer than
the total number of attributes to constrain the total number that can be changed within a choice set.
This might be a reasonable thing to do if you had a large number of attributes and you want to make
it easier for the respondents to make a choice. For example, a survey might be interested in 20 or
more attributes of a cell phone, but show and change only 5 or fewer attributes in a choice set.
Number of attributes that can change within a choice set
This example has two profiles per choice set. You can design
choice experiments with more than two profiles in a choice set.
Number of profiles per choice set
Number of choice sets per survey There are eight choice sets in the example survey but often there
are many more.
Number of surveys The example only shows a single survey. Normally you expect multiple
respondents and would request more than one survey.
You might want to give surveys to 10 people, but
use two different surveys. So you enter 2 as the Number of surveys and 5 as the Expected number
of respondents per survey.
Expected number of respondents per survey
Note: Recall that this first example is used to generate prior information, then used to create a more realistic
survey. This example is a single survey given to a single respondent.
3. Click Make Design to see the example survey results in Figure 8.3.
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Figure 8.3 Survey Results based on a Simple Model and Default Prior Information
The Radio buttons beneath the design settings let you choose between having the survey settings in one
JMP table and gathering survey results in a second table, or generating a single table that shows the settings
and has an additional column for the choice response.
You can see that there are eight choice sets, each consisting of two laptop profiles. At this point you can press
Back and modify the design, or click Make Table and generate the JMP table shown in Figure 8.4.
This default design was created with no given prior information. Without prior information, the Choice
designer has no way of knowing which attribute levels are better. That is, the Choice designer cannot know
that a lower price might be more desirable than a higher price, a faster machine is better than a slower
machine, and so forth. As a result, you can see that some choice sets might not convey useful information.
The analysis results are used as prior information in a new Choice design dialog.
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Figure 8.4 JMP Data Table for Preliminary Laptop Choice Survey
Analyze the Example Choice Experiment
Once a survey design is complete, a respondent chooses one profile from each set, entering ‘1’ for the chosen
profile and ‘0’ for the rejected profile. Suppose a respondent completed the example survey as shown in
Figure 8.5. You can now analyze these results using the Choice platform in the Analyze menu (Analyze >
Modeling > Choice).
Figure 8.5 JMP Table with Survey Choice Sets and Responses
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1. Click the Choice script in the Laptop Design data table and select Run Script to analyze that data with
the Choice modeling platform from the Analyze menu.
When you run the Choice model script, the Choice launch dialog shown in Figure 8.6 appears, The Choice
dialog is designed to cover a variety of choice survey results, which can include data saved in multiple data
tables. This example has all data contained in a single table. For details about using the Choice analysis
dialog, see the Choice Modeling chapter in the Modeling and Multivariate Methods book.
Figure 8.6 Choice Model Fitting Dialog
2. Click Run Model on the Choice model fitting dialog.
3. An additional dialog then appears asking if this is a one-table analysis with all the data in the Profile
Table, which is the case in this sample survey. Click Yes in this dialog to continue.
The analysis shows as in Figure 8.7.
To design the final choice survey using prior information, you will need to enter estimates of the mean and
variance of the attribute parameter estimates. The analysis on the left in Figure 8.7 has estimates of the
attribute means, called Estimate, and estimates of the standard deviation of the attributes, called Std Error.
An easy way to see the variance of the attributes is to capture the analysis in a JMP table and compute the
variance:
4. Right click on the Parameter Estimates report and choose Make into Data Table from the menu, as
shown.
5. In the new Untitled data table, create a new column and call it Var.
6. Select Formula from the Cols menu (Cols > Formula), or right-click at the top of the Var column and
select Formula for the menu that shows.
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7. In the Formula Editor, click the Std Error column in the column list and click the exponent button (
on the formula editor panel to compute the variance shown on the right in Figure 8.7.
)
Figure 8.7 Analysis of the sample Laptop Survey
This preliminary survey with its analysis gives you the information needed to design a final survey
appropriate for gathering information from multiple respondents. Keep in mind that in a real situation, you
might have prior information about factor attributes and not need to do a sample design.
Note: Leave the Untitled data table with the mean and variance information open to be used in the next
example.
Design a Choice Experiment Using Prior Information
In some situations, you will know from previous surveys or experience how to give prior information to the
Choice designer about product attributes. This example continues by designing the laptop experiment
again, using the analysis information gained from the sample design.
1. Choose DOE > Choice Design and enter the attributes and values as before.
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2. Click Continue to see the Choice design panels in Figure 8.8.
3. Now enter the values from the JMP table created by the previous analysis into the Prior Mean and Prior
Variance Matrix panels of the Choice Design dialog, as shown in Figure 8.8. You can copy-and-paste to
transfer the values from the data table to the Choice dialog panels.
Figure 8.8 Enter Prior Mean and Variance Information from Preliminary Survey
4. Enter the values into the Design Generation panel, as shown in Figure 8.9.
– Four or fewer attribute levels can change within a choice set.
– There are two profiles per choice set.
– Each survey has eight choice sets.
– The design generates two separate surveys.
– Five respondents are expected to complete each survey (for a total of 10 respondents).
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Figure 8.9 Design Specifications for final Laptop Survey
5. Click Make Design. A partial view of the design is shown in Figure 8.10.
Figure 8.10 Design Runs for Two Choice Surveys
6. Click Make Table. The final data table will have runs for ten survey respondents, giving a total of 160
observations (2 profiles * 8 choice sets * 2 surveys * 5 respondents = 160 observations).
Administer the Survey and Analyze Results
The survey data table with its results is stored in the Sample Data Design Experiment folder installed with
JMP. Figure 8.11 is a partial listing of the survey data table with results. The Choice script created by the
Choice designer and saved with the survey data table can be used to analyze the data. The default data table
created by the Choice designer is named Choice Profiles. Note in Figure 8.11, the data table name is
changed to Laptop Results.
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Initial Choice Platform Analysis
1. To continue the example, open the table called Laptop Results.jmp, saved in the Sample Data folder.
2. To analyze the data, click the Choice script saved with the data and select Run Script from the menu to
see the completed dialog shown in Figure 8.11.
Note that this dialog has three grouping variables (Respondent, Survey, and Choice Set, whereas the dialog
shown Figure 8.6 had only the Choice Set grouping variable because there was a single survey and a single
respondent. This example included multiple surveys and respondents, which must be included in the
analysis.
Figure 8.11 Choice Model Fitting Dialog to Analyze the Laptop Survey
3. Click Run on the Fit Model dialog. The query again appears asking if the analysis is a one-table analysis
with all the data in the profile table. Click Yes to see the initial analysis result shown in Figure 8.12.
The results are clear. While all the effects are significant, the most significant attribute is Speed.
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Figure 8.12 Initial Analysis of the Final Laptop Survey
Find Unit Cost and Trade Off Costs with the Profiler
You would like to know how changing the price, or other characteristics, of a laptop affects the desirability
as perceived by potential buyers. This desirability is called the utility value of the laptop attributes. The
profiler shows the utility value and how it changes as the laptop attributes change.
1. Select Profiler from the menu on the Choice Model title bar to see the Prediction Profiler in
Figure 8.13.
Figure 8.13 Default Prediction Profiler for Laptop Choice Analysis
When each attribute value is set to its lowest value, the Utility value is –0.3406. The first thing you want to
know is the unit utility cost.
2. To find the unit utility cost, move the trace for Price to $1,500 and note how the Utility value changes.
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Compare the Utility values in Figure 8.13 and Figure 8.14. The value of Utility changes from –0.3406 to
–2.3303 when cost is raised from $1,000 to $1,500. That is, raising the price of a laptop $500.00 lowers the
utility (or desirability) approximately 2 units. Thus, you can say that the unit utility cost is roughly
$250.00.
Figure 8.14 Compare Change in Utility Over Price
With this unit utility cost estimate you can now vary the other attributes, note the change in utility, and find
an approximate dollar value associated with that attribute change. For example, the most significant
attribute is speed (see Figure 8.12).
3. In the Prediction Profiler, set Price to its lowest value and change Speed to its higher value.
You can see in Figure 8.15 that the Utility value changes from the original value shown in Figure 8.13 of
–0.3406 to 0.9886, for a total change of 1.3292 units. If the unit utility cost is estimated to be $250.00, as
shown above, then the increase in price for a 2.0 GHz laptop over a 1.5 GHz laptop can be computed to be
1.3292*$250.00 = $332.30. This is the dollar value the Choice survey provides the manufacturer as a basis
for pricing different laptop products. You can make similar calculations for the other attributes.
Figure 8.15 Change Speed in Profiler and Note Utility Value
This simple Choice survey and its analysis shows how this kind if information can be used to help
manufacturers and retailers identify important product attributes and assign values to them.
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The Choice designer allows more complex designs, such as designs with interactions and other terms. The
Choice analysis platform can be used to analyze complex designs, and it can be used to incorporate data
from multiple data sets that include demographic information about the respondents.
Chapter 9
Space-Filling Designs
Space-filling designs are useful in situations where run-to-run variability is of far less concern than the form
of the model. Sensitivity studies of computer simulations is one such situation. For this case, and any
mechanistic or deterministic modeling problem, any variability is small enough to be ignored. For systems
with no variability, randomization and blocking are irrelevant. Replication is undesirable because repeating
the same run yields the same result. In space-filling designs, there are two objectives:
•
Prevent replicate points by spreading the design points out to the maximum distance possible between
any two points.
•
Space the points uniformly.
The following methods are implemented for these types of designs:
•
The Sphere-Packing method emphasizes spread of points.
•
The Latin Hypercube method is a compromise between spread of points and uniform spacing.
•
The Uniform method mimics the uniform probability distribution.
•
The Minimum Potential method minimizes energy designs in a hypersphere.
•
The Maximum Entropy method measures the amount of information contained in the distribution of a
set of data.
•
The Gaussian Process IMSE Optimal method creates a design that minimizes the integrated mean
squared error of the gaussian process over the experimental region.
Figure 9.1 Space-Filling Design
Contents
Introduction to Space-Filling Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Sphere-Packing Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Creating a Sphere-Packing Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Visualizing the Sphere-Packing Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Latin Hypercube Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Creating a Latin Hypercube Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Visualizing the Latin Hypercube Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Uniform Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Comparing Sphere-Packing, Latin Hypercube, and Uniform Methods . . . . . . . . . . . . . . . . . . . . . . . . . 206
Minimum Potential Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Maximum Entropy Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Gaussian Process IMSE Optimal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Borehole Model: A Sphere-Packing Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Create the Sphere-Packing Design for the Borehole Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Guidelines for the Analysis of Deterministic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Results of the Borehole Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
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Introduction to Space-Filling Designs
Space-filling designs are useful for modeling systems that are deterministic or near-deterministic. One
example of a deterministic system is a computer simulation. Such simulations can be very complex
involving many variables with complicated interrelationships. A goal of designed experiments on these
systems is to find a simpler empirical model that adequately predicts the behavior of the system over limited
ranges of the factors.
In experiments on systems where there is substantial random noise, the goal is to minimize the variance of
prediction. In experiments on deterministic systems, there is no variance but there is bias. Bias is the
difference between the approximation model and the true mathematical function. The goal of space-filling
designs is to bound the bias.
There are two schools of thought on how to bound the bias. One approach is to spread the design points out
as far from each other as possible consistent with staying inside the experimental boundaries. The other
approach is to space the points out evenly over the region of interest.
The Space Filling designer supports the following design methods:
Sphere Packing maximizes the minimum distance between pairs of design points.
Latin Hypercube maximizes the minimum distance between design points but requires even spacing
of the levels of each factor. This method produces designs that mimic the uniform distribution. The
Latin Hypercube method is a compromise between the Sphere-Packing method and the Uniform
design method.
minimizes the discrepancy between the design points (which have an empirical uniform
distribution) and a theoretical uniform distribution.
Uniform
Minimum Potential
spreads points out inside a sphere around the center.
Maximum Entropy
measures the amount of information contained in the distribution of a set of
data.
Gaussian Process IMSE Optimal creates a design that minimizes the integrated mean squared error
of the Gaussian process over the experimental region.
Sphere-Packing Designs
The Sphere-Packing design method maximizes the minimum distance between pairs of design points. The
effect of this maximization is to spread the points out as much as possible inside the design region.
Creating a Sphere-Packing Design
1. Select DOE > Space Filling Design.
2. Enter responses and factors. (See “Enter Responses and Factors into the Custom Designer,” p. 65.)
3. Alter the factor level values, if necessary. For example, Figure 9.2 shows the two existing factors, X1 and
X2, with values that range from 0 to 1 (instead of the default –1 to 1).
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Figure 9.2 Space-Filling Dialog for Two Factors
4. Click Continue.
5. In the design specification dialog, specify a sample size (Number of Runs). Figure 9.3 shows a sample
size of eight.
Figure 9.3 Space-Filling Design Dialog
6. Click Sphere Packing.
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JMP creates the design and displays the design runs and the design diagnostics. Figure 9.4 shows the
Design Diagnostics panel open with 0.518 as the Minimum Distance. Your results might differ slightly
from the ones below, but the minimum distance will be the same.
Figure 9.4 Sphere-Packing Design Diagnostics
7. Click Make Table. Use this table to complete the visualization example, described next.
Visualizing the Sphere-Packing Design
To visualize the nature of the Sphere-Packing technique, create an overlay plot, adjust the plot’s frame size,
and add circles using the minimum distance from the diagnostic report shown in Figure 9.4 as the radius for
the circles. Using the table you just created:
1. Select Graph > Overlay Plot.
2. Specify X1 as X and X2 as Y, then click OK.
3. Adjust the frame size so that the frame is square by right-clicking the plot and selecting Size/Scale >
Size to Isometric.
4. Right-click the plot and select Customize. When the Customize panel appears, click the plus sign to see
a text edit area and enter the following script:
For Each Row(Circle({:X1, :X2}, 0.518/2))
where 0.518 is the minimum distance number you noted in the Design Diagnostics panel. This script
draws a circle centered at each design point with radius 0.259 (half the diameter, 0.518), as shown on
the left in Figure 9.5. This plot shows the efficient way JMP packs the design points.
5. Now repeat the procedure exactly as described in the previous section, but with a sample size of 10
instead of eight.
Remember to change 0.518 in the graphics script to the minimum distance produced by 10 runs.
When the plot appears, again set the frame size and create a graphics script using the minimum distance
from the diagnostic report as the diameter for the circle. You should see a graph similar to the one on the
right in Figure 9.5. Note the irregular nature of the sphere packing. In fact, you can repeat the process a
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third time to get a slightly different picture because the arrangement is dependent on the random
starting point.
Figure 9.5 Sphere-Packing Example with Eight Runs (left) and 10 Runs (right)
Latin Hypercube Designs
In a Latin Hypercube, each factor has as many levels as there are runs in the design. The levels are spaced
evenly from the lower bound to the upper bound of the factor. Like the sphere-packing method, the Latin
Hypercube method chooses points to maximize the minimum distance between design points, but with a
constraint. The constraint maintains the even spacing between factor levels.
Creating a Latin Hypercube Design
To use the Latin Hypercube method:
1. Select DOE > Space Filling Design.
2. Enter responses, if necessary, and factors. (See “Enter Responses and Factors into the Custom Designer,”
p. 65.)
3. Alter the factor level values, if necessary. For example, Figure 9.6 shows adding two factors to the two
existing factors and changing their values to 1 and 8 instead of the default –1 and 1.
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Figure 9.6 Space-Filling Dialog for Four Factors
4. Click Continue.
5. In the design specification dialog, specify a sample size (Number of Runs). This example uses a sample
size of eight.
6. Click Latin Hypercube (see Figure 9.3). Factor settings and design diagnostics results appear similar to
those in Figure 9.7, which shows the Latin Hypercube design with four factors and eight runs.
Note: The purpose of this example is to show that each column (factor) is assigned each level only once,
and each column is a different permutation of the levels.
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Figure 9.7 Latin Hypercube Design for Four Factors and Eight Runs with Eight Levels
Visualizing the Latin Hypercube Design
To visualize the nature of the Latin Hypercube technique, create an overlay plot, adjust the plot’s frame size,
and add circles using the minimum distance from the diagnostic report as the radius for the circle.
1. First, create another Latin Hypercube design using the default X1 and X2 factors.
2. Be sure to change the factor values so they are 0 and 1 instead of the default –1 and 1.
3. Click Continue.
4. Specify a sample size of eight (Number of Runs).
5. Click Latin Hypercube. Factor settings and design diagnostics are shown in Figure 9.8.
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Figure 9.8 Latin Hypercube Design with two Factors and Eight Runs
The minimum
distance
6. Click Make Table.
7. Select Graph > Overlay Plot.
8. Specify X1 as X and X2 as Y, then click OK.
9. Right-click the plot and select Size/Scale > Size to Isometric to adjust the frame size so that the frame
is square.
10. Right-click the plot, select Customize from the menu. In the Customize panel, click the large plus sign
to see a text edit area, and enter the following script:
For Each Row(Circle({:X1, :X2}, 0.404/2))
where 0.404 is the minimum distance number you noted in the Design Diagnostics panel (Figure 9.8).
This script draws a circle centered at each design point with radius 0.202 (half the diameter, 0.404), as
shown on the left in Figure 9.9. This plot shows the efficient way JMP packs the design points.
11. Repeat the above procedure exactly, but with 10 runs instead of eight (step 5). Remember to change
0.404 in the graphics script to the minimum distance produced by 10 runs.
You should see a graph similar to the one on the right in Figure 9.9. Note the irregular nature of the sphere
packing. In fact, you can repeat the process to get a slightly different picture because the arrangement is
dependent on the random starting point.
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Figure 9.9 Comparison of Latin Hypercube Designs with Eight Runs (left) and 10 Runs (right)
Note that the minimum distance between each pair of points in the Latin Hypercube design is smaller than
that for the Sphere-Packing design. This is because the Latin Hypercube design constrains the levels of each
factor to be evenly spaced. The Sphere-Packing design maximizes the minimum distance without any
constraints.
Uniform Designs
The Uniform design minimizes the discrepancy between the design points (empirical uniform distribution)
and a theoretical uniform distribution.
Note: These designs are most useful for getting a simple and precise estimate of the integral of an unknown
function. The estimate is the average of the observed responses from the experiment.
1. Select DOE > Space Filling Design.
2. Enter responses, if necessary, and factors. (See “Enter Responses and Factors into the Custom Designer,”
p. 65.)
3. Alter the factor level values to 0 and 1.
4. Click Continue.
5. In the design specification dialog, specify a sample size. This example uses a sample size of eight
(Number of Runs).
6. Click the Uniform button. JMP creates this design and displays the design runs and the design
diagnostics as shown in Figure 9.10.
Note: The emphasis of the Uniform design method is not to spread out the points. The minimum
distances in Figure 9.10 vary substantially.
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Figure 9.10 Factor Settings and Diagnostics for Uniform Space-Filling Designs with Eight Runs
7. Click Make Table.
A Uniform design does not guarantee even spacing of the factor levels. However, increasing the number of
runs and running a distribution on each factor (use Analyze > Distribution) shows flat histograms.
Figure 9.11 Histograms are Flat for each Factor when Number of Runs is Increased to 20
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Comparing Sphere-Packing, Latin Hypercube, and Uniform
Methods
To compare space-filling design methods, create the Sphere Packing, Latin Hypercube, and Uniform
designs, as shown in the previous examples. The Design Diagnostics tables show the values for the factors
scaled from zero to one. The minimum distance is based on these scaled values and is the minimum distance
from each point to its closest neighbor. The discrepancy value is the integrated difference between the
design points and the uniform distribution.
Figure 9.12 shows a comparison of the design diagnostics for three eight-run space-filling designs. Note that
the discrepancy for the Uniform design is the smallest (best). The discrepancy for the Sphere-Packing design
is the largest (worst). The discrepancy for the Latin Hypercube takes an intermediate value that is closer to
the optimal value.
Also note that the minimum distance between pairs of points is largest (best) for the Sphere-Packing
method. The Uniform design has pairs of points that are only about half as far apart. The Latin Hypercube
design behaves more like the Sphere-Packing design in spreading the points out.
For both spread and discrepancy, the Latin Hypercube design represents a healthy compromise solution.
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Minimum Potential Designs
207
Figure 9.12 Comparison of Diagnostics for Three Eight-Run Space-Filling Methods
SpherePacking
Latin
Hypercube
Uniform
Another point of comparison is the time it takes to compute a design. The Uniform design method requires
the most time to compute. Also, the time to compute the design increases rapidly with the number of runs.
For comparable problems, all the space-filling design methods take longer to compute than the D-optimal
designs in the Custom Designer.
Minimum Potential Designs
The Minimum Potential design spreads points out inside a sphere. To understand how this design is created,
imagine the points as electrons with springs attached to every other point, as illustrated to the right. The
coulomb force pushes the points apart, but the springs pull them together. The design is the spacing of
points that minimizes the potential energy of the system.
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Figure 9.13 Minimum Potential Design
Minimum Potential designs:
•
have spherical symmetry
•
are nearly orthogonal
•
have uniform spacing
To see a Minimum Potential example:
1. Select DOE > Space Filling Design.
2. Add 1 continuous factor. (See “Enter Responses and Factors into the Custom Designer,” p. 65.)
3. Alter the factor level values to 0 and 1, if necessary.
4. Click Continue.
5. In the design specification dialog (shown on the left in Figure 9.14), enter a sample size (Number of
Runs). This example uses a sample size of 12.
6. Click the Minimum Potential button. JMP creates this design and displays the design runs (shown on
the right in Figure 9.14) and the design diagnostics.
Figure 9.14 Space-Filling Methods and Design Diagnostics for Minimum Potential Design
7. Click Make Table.
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209
You can see the spherical symmetry of the Minimum Potential design using the Scatterplot 3D graphics
platform.
1. After you make the JMP design table, choose the Graph > Scatterplot 3D command.
2. In the Scatterplot 3D launch dialog, select X1, X2, and X3 as Y, Columns and click OK to see the initial
three dimensional scatterplot of the design points.
3. To see the results similar to those in Figure 9.15, select the Normal Contour Ellipsoids option from the
menu in the Scatterplot 3D title bar, and make the points larger by right-clicking on the plot and
selecting Settings, then increasing the Marker Size slider.
Now it is easy to see the points spread evenly on the surface of the ellipsoid.
Figure 9.15 Minimum Potential Design Points on Sphere
Maximum Entropy Designs
The Latin Hypercube design is currently the most popular design assuming you are going to analyze the
data using a Gaussian-Process model. Computer simulation experts like to use the Latin Hypercube design
because all projections onto the coordinate axes are uniform.
However, as the example at the top in Figure 9.16 shows, the Latin Hypercube design does not necessarily
do a great job of space filling. This is a two-factor Latin Hypercube with 16 runs, with the factor level
settings set between -1 and 1. Note that this design appears to leave a hole in the bottom right of the overlay
plot.
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Chapter 9
Figure 9.16 Two-factor Latin Hypercube Design
Note the empty area in the
Latin Hypercube design
The Maximum Entropy design is a competitor to the Latin Hypercube design for computer experiments
because it optimizes a measure of the amount of information contained in an experiment. See the technical
note below. With the factor levels set between -1 and 1, the two-factor Maximum Entropy design shown in
Figure 9.17 covers the region better than the Latin hypercube design in Figure 9.16. The space-filling
property generally improves as the number of runs increases without bound.
Figure 9.17 Two-Factor Maximum Entropy Design
Technical Note: Maximum Entropy designs maximize the Shannon information (Shewry and Wynn
(1987)) of an experiment, assuming that the data come from a normal (m, s2 R) distribution, where
⎛
2⎞
R ij = exp ⎜ – ∑ θ k ( x ik – x jk ) ⎟
⎝ k
⎠
Chapter 9
Space-Filling Designs
Gaussian Process IMSE Optimal Designs
211
is the correlation of response values at two different design points, xi and xj. Computationally, these designs
maximize |R|, the determinant of the correlation matrix of the sample. When xi and xj are far apart, then Rij
approaches zero. When xi and xj are close together, then Rij is near one.
Gaussian Process IMSE Optimal Designs
The Gaussian process IMSE optimal design is also a competitor to the Latin Hypercube design because it
minimizes the integrated mean squared error of the Gaussian process model over the experimental region.
You can compare the IMSE optimal design to the Latin Hypercube (shown previously in Figure 9.16). The
table and overlay plot in Figure 9.18 show a Gaussian IMSE optimal design. You can see that the design
provides uniform coverage of the factor region.
Figure 9.18 Comparison of Two-factor Latin Hypercube and Gaussian IMSE Optimal Designs
Note: Both the Maximum Entropy design and the Gaussian Process IMSE Optimal design were created
using 100 random starts.
Borehole Model: A Sphere-Packing Example
Worley (1987) presented a model of the flow of water through a borehole that is drilled from the ground
surface through two aquifers. The response variable y is the flow rate through the borehole in m3/year and is
determined by the following equation:
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2πT u ( H u – H l )
y = -----------------------------------------------------------------------------------------Tu
2LT u
ln ( r ⁄ r w ) 1 + --------------------------------------- + -----2
Tl
ln ( r ⁄ r w ) r w K w
There are eight inputs to this model:
rw = radius of borehole, 0.05 to 0.15 m
r = radius of influence, 100 to 50,000 m
Tu = transmissivity of upper aquifier, 63,070 to 115,600 m2/year
Hu = potentiometric head of upper aquifier, 990 to 1100 m
Tl = transmissivity of lower aquifier, 63.1 to 116 m2/year
Hl = potentiometric head of lower aquifier, 700 to 820 m
L = length of borehole, 1120 to 1680 m
Kw = hydraulic conductivity of borehole, 9855 to 12,045 m/year
This example is atypical of most computer experiments because the response can be expressed as a simple,
explicit function of the input variables. However, this simplicity is useful for explaining the design methods.
Create the Sphere-Packing Design for the Borehole Data
To create a Sphere-Packing design for the borehole problem:
1. Select DOE > Space Filling Design.
2. Click the red triangle icon on the Space Filling Design title bar and select Load Factors.
3. Open the Sample Data folder installed with JMP. In the Design Experiment folder, open Borehole
Factors.jmp from the Design Experiment folder to load the factors (Figure 9.19).
Figure 9.19 Factors Panel with Factor Values Loaded for Borehole Example
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213
Note: The logarithm of r and rw are used in the following discussion.
4. Click Continue.
5. Specify a sample size (Number of Runs) of 32 as shown in Figure 9.20.
Figure 9.20 Space-Filling Design Method Panel Showing 32 Runs
6. Click the Sphere Packing button to produce the design.
7. Click Make Table to make a table showing the design settings for the experiment. The factor settings in
the example table might not have the same ones you see when generating the design because the designs
are generated from a random seed.
8. To see a completed data table for this example, open Borehole Sphere Packing.jmp from the Design
Experiment Sample Data folder installed with JMP. This table also has a table variable that contains a
script to analyze the data. The results of the analysis are saved as columns in the table.
Guidelines for the Analysis of Deterministic Data
It is important to remember that deterministic data have no random component. As a result, p-values from
fitted statistical models do not have their usual meanings. A large F statistic (low p-value) is an indication of
an effect due to a model term. However, you cannot make valid confidence intervals about the size of the
effects or about predictions made using the model.
Residuals from any model fit to deterministic data are not a measure of noise. Instead, a residual shows the
model bias for the current model at the current point. Distinct patterns in the residuals indicate new terms
to add to the model to reduce model bias.
Results of the Borehole Experiment
The example described in the previous sections produced the following results:
•
A stepwise regression of the response, log y, versus the full quadratic model in the eight factors, led to the
prediction formula column.
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•
The prediction bias column is the difference between the true model column and the prediction
formula column.
•
The prediction bias is relatively small for each of the experimental points. This indicates that the model
fits the data well.
In real world examples, the true model is generally not available in a simple analytical form. As a result, it is
impossible to know the prediction bias at points other than the observed data without doing additional
runs.
In this case, the true model column contains a formula that allows profiling the prediction bias to find its
value anywhere in the region of the data. To understand the prediction bias in this example:
1. Select Graph > Profiler.
2. Highlight the prediction bias column and click the Y, Prediction Formula button.
3. Check the Expand Intermediate Formulas box, as shown at the bottom on the Profiler dialog in
Figure 9.21, because the prediction bias formula is a function of columns that are also created by
formulas.
4. Click OK.
The profile plots at the bottom in Figure 9.21 show the prediction bias at the center of the design region. If
there were no bias, the profile traces would be constant between the value ranges of each factor. In this
example, the variables Hu and Hl show nonlinear effects.
Figure 9.21 Profiler Dialog and Profile of the Prediction Bias in the Borehole Sphere-Packing Data
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215
The range of the prediction bias on the data is smaller than the range of the prediction bias over the entire
domain of interest. To see this, look at the distribution analysis (Analyze > Distribution) of the prediction
bias in Figure 9.22. Note that the maximum bias is 1.826 and the minimum is –0.684 (the range is 2.51).
Figure 9.22 Distribution of the Prediction Bias
The top plot in Figure 9.23 shows the maximum bias (2.91) over the entire domain of the factors. The plot
at the bottom shows the comparable minimum bias (–4.84). This gives a range of 7.75. This is more than
three times the size of the range over the observed data.
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Figure 9.23 Prediction Plots showing Maximum and Minimum Bias Over Factor Domains
Keep in mind that, in this example, the true model is known. In any meaningful application, the response at
any factor setting is unknown. The prediction bias over the experimental data underestimates the bias
throughout the design domain.
There are two ways to assess the extent of this underestimation:
•
Cross-validation refits the data to the model while holding back a subset of the points and looks at the
error in estimating those points.
•
Verification runs (new runs performed) at different settings to assess the lack of fit of the empirical
model.
Chapter 10
Accelerated Life Test Designs
Designing Experiments for Accelerated Life Tests
The Accelerated Life Test Design platform can be used to design experimental plans for accelerated life
testing. You can design initial experiments, or augment existing experiments.
Figure 10.1 Accelerated Life Test Design
Contents
Overview of Accelerated Life Test Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Using the ALT Design Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Platform Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224
Chapter 10
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Overview of Accelerated Life Test Designs
219
Overview of Accelerated Life Test Designs
Often in reliability studies, the product reliability at use conditions is so high that the time required to test
the product until it fails is prohibitive. As an alternative, you can test the product in conditions that are
more extreme than normal use conditions. The extreme conditions enable the product to degrade and fail
sooner, making a reliability study possible. Results are used to predict product reliability at normal use
conditions.
The Accelerated Life Testing Design platform can be used to design experimental plans for accelerated life
testing (ALT) experiments. The ALT Design platform can be used to design initial experiments, or augment
existing experiments. Augmenting designs is useful if you want to obtain more data, so that you can decrease
the variance associated with predicting product reliability.
The ALT Design platform can create designs for situations involving one or two accelerating factors. For
two accelerating factors, you can choose to include the interaction. You can use D-optimality or two types of
I-optimality criterion.
The process requires estimates of acceleration model parameters. Since those parameters might not be
known in advance, you can specify prior distributions to account for the uncertainty. Designs can be created
for either Lognormal or Weibull life distributions.
Using the ALT Design Platform
1. To launch the ALT Design platform, select DOE > Accelerated Life Test Design.
Figure 10.2 Initial ALT Design Window
2. Select one or two accelerating factors.
For two factors, you can choose to include the interaction between the factors.
3. Click Continue.
A window appears for specifying details of the accelerating factor or factors.
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Figure 10.3 Accelerating Factor Details Window
4. Fill in these options:
Factor Name Enter a name for the accelerating factor.
Number of Levels Enter the number of levels of the factor to include in the experiment.
Select a transformation for the factor. The options are Arrhenius Celsius,
Reciprocal, Log, Square Root, and Linear.
Factor Transformation
Low Usage Condition Enter a value for the low usage conditions.
High Usage Conditions
Enter a value for the high usage conditions.
5. Click Continue.
A window appears for specifying additional information about the assumed distribution and desired
experimental conditions.
Chapter 10
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Using the ALT Design Platform
221
Figure 10.4 Distribution Details
6. Fill in these options:
Factors Enter the values of the accelerating factor levels.
Distribution Choice
Select the failure distribution, either Weibull or LogNormal.
Enter the acceleration model parameters. These values can be either a best guess or the
estimates of the parameters from an analysis of previously obtained data.
Prior Mean
Prior Variance Matrix Enter variances and covariances for the acceleration model parameters. These
values can be either a best guess or the estimated variance matrix from an analysis of previously
obtained data.
Ignore prior variance Select this option to ignore variances and covariances for model parameters.
When the variances and covariances are ignored, the design is created for the specific fixed
parameters entered under Prior Mean. The resulting design is said to be a locally optimal design.
This design will be good if the prior mean parameters are close to the true values. However, the
design will not be robust to mis-specified parameters. When the variances and covariances are used,
a multivariate normal prior distribution is assumed for the acceleration model parameters. This is
useful when the values entered under Prior Mean are estimates or guesses, and you want the design
to reflect the associated uncertainty.
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Diagnostic Choices Enter values for the following:
Time range of interest are values for which you want an estimated probability of failure. For
example, if you are interested in the probability of failure by 100,000 hours, then enter 100,000 for
both the lower and upper ranges.
Probability of interest is the value for which you want an estimated time of failure. For example, if
you are most interested in obtaining the time until 10% of units fail, then enter 0.1.
Design Choices Enter values for the following:
Length of Test is the length of time to run the experiment.
Number of Units Under Test is the number of units in the experiment. If augmenting a previous
experiment, enter the number of units from the previous experiment plus the number of units that
you want to run for the next experiment. If designing an initial experiment, enter the number of
units that you want to run.
7. Click Continue.
New outline nodes appear as shown in Figure 10.5.
Figure 10.5 Additional Outline Nodes
Enter the minimum and maximum number of runs allowed at each level of the
acceleration factor. If augmenting a previous experiment, enter the number of units already run at
each level for the Minimum Units.
Candidate Runs
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Using the ALT Design Platform
223
Parameter Variance for Balanced Design Gives the expected variances and covariances for the
acceleration model parameters after running and analyzing the balanced design (same number of
runs at each level of the acceleration factor). These values are valid under the assumption that the
values for the prior mean and variance are correct.
These values can be compared to the ones entered under Prior Variance Matrix, to see whether the
balanced design can improve the variances of the parameter estimates. They can also be compared to
the final parameter variances for the optimal design, after clicking Make Design.
Use the profiler to visualize the probability that a unit will fail at different values
of the acceleration factor and time.
Distribution Profiler
8. Click Update Profiler to update the profiler if changes are made to the distribution choice, means,
variances, design choices, or candidate runs.
9. Click Make Design to create the optimal design and display the results.
Figure 10.6 Design Results
The information below describes the results you get after clicking Make Design.
The Design report gives the expected number of failures for each level of the acceleration factor. Also given
is the probability that none of the units at this setting will fail.
The Parameter Variance for Optimal Design report gives the variances and covariances for the acceleration
model parameters for the optimal design. These values are valid under the assumption that the values for the
prior mean and variance are correct. These values can be compared to those under Parameter Variance for
Balanced Design to determine whether the optimal design is able to reduce the parameter variances more
than the balanced design.
The Optimality Criteria report gives the values of the optimality criterion for the optimal design. For more
information about the optimality criterion, see “Platform Options,” p. 224.
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Chapter 10
The Make Design button updates the optimal design if any changes are made to the distribution choice,
prior means or variances, design choices, or candidate runs.
The Make Test Plan button creates a data table with the acceleration factor levels and the number of units
to include in the experiment for each level.
The Make Table button creates a table that can be used for data collection during the experiment.
Platform Options
The red triangle menu for Accelerated Life Test Plan has the following options:
Simulate Responses
Adds simulated responses to the table when you click Make Table.
ALT Optimality Criterion
Gives three choices for design optimality:
Make D-Optimal Design creates a design that minimizes the variance of the model coefficients.
Make Time I-Optimal Design creates a design that minimizes the prediction variance when
predicting the time to failure for the probability given in “Diagnostic Choices,” p. 222.
Make Probability I-Optimal Design creates a design that minimizes the prediction variance when
predicting the failure probability for the times given in “Diagnostic Choices,” p. 222.
Advanced Options Gives the N Monte Carlo Spheres option, which affects the speed and accuracy of
numerical integration. For more information, see “Advanced Options for the Nonlinear Designer,”
p. 245 in the “Nonlinear Designs” chapter.
Example
This example shows how to use the Accelerated Life Test Design to augment an existing design.
An accelerated life test was performed, and the results are in the Capacitor ALT.jmp sample data table (in the
Design Experiment folder). Fifty units were tested at each of three temperatures (85o, 105o, and 125o
Celsius) for 1500 hours. The resulting model is used to predict the probability of failure at 100,000 hours at
normal use conditions of 25o.
1. Open the Capacitor ALT.jmp data table in the Design Experiment folder.
2. Run the Fit Life by X table script.
3. In the Distribution Profiler, enter 25 for Temperature and 100,000 for hours.
The profiler is shown in Figure 10.7.
Chapter 10
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Example
225
Figure 10.7 Distribution Profiler for Capacitor Model
The predicted probability of failure at 100,000 hours at 25o is 0.00358, with a confidence interval of
0.00056 to 0.0227. The analyst wants to decrease the width of the confidence interval. To do so, the
experiment needs to be augmented with additional data.
To augment the design in the optimal way, use the Accelerated Life Test Design platform. Follow the steps
below to use the platform:
1. Select DOE > Accelerated Life Test Design.
2. Select Design for one accelerating factor and click Continue.
3. Enter Temperature for Factor Name.
4. Enter 5 for Number of Levels.
5. Enter 25 for both Low Usage Condition and High Usage Condition.
6. Click Continue.
7. Enter 85, 95, 105, 115, and 125 for the Temperature Level Values.
8. Select Weibull for Distribution Choice.
9. Under Prior Mean, enter the acceleration model parameters from the Fit Life by X Estimates report. See
Figure 10.8.
– Enter –35.200 for Intercept.
– Enter 1.389 for Temperature.
– Enter 1.305 for scale.
Figure 10.8 Fit Life by X Estimates
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Chapter 10
In the Fit Life by X output, under the Estimates report, is the Covariance Matrix report. Note the variances
for the acceleration model parameters are 22, 0.025, and 0.013. These values will be compared to our final
results at the end.
10. Enter 100,000 for both boxes for Time range of interest.
11. Enter 1500 for Length of Test.
12. Enter 300 for Number of Units Under Test. The previous experiment used 150 units, and the next
experiment uses 150 units, for a total of 300.
The completed window is shown in Figure 10.9.
Figure 10.9 Completed Window
13. Click Continue.
14. To account for the units in the previous experiment, enter the following under Candidate Runs.
– Enter 50 for Minimum Units for 85o.
– Enter 50 for Minimum Units for 105o.
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227
– Enter 50 for Minimum Units for 125o.
15. From the red triangle menu for Accelerated Life Test Plan, select ALT Optimality Criterion > Make
Probability I-Optimal Design.
16. Click Make Design.
The optimal experimental design is returned, along with other results. See Figure 10.10.
Figure 10.10 Optimal Design
The optimal design is computed based on the Number of Units Under Test, Candidate Runs, and other
information that you specified earlier. The optimal design consists of the following number of units at each
temperature level:
•
184 units at 85o. Since the previous experiment used 50 units, 134 additional units are needed.
•
0 units at 95o. The next experiment will not utilize any units at this level.
•
50 units at 105o. Since the previous experiment already used 50 units, no additional units are needed.
•
0 units at 115o. The next experiment will not utilize any units at this level.
•
66 units at 125o. Since the previous experiment used 50 units, 16 additional units are needed.
As we entered earlier, a total of 134+16=150 units are used for the new experiment.
An estimate of the acceleration model parameters variances is given. Note that, due to the additional data,
all three variances are smaller than before from the original Fit Life by X report.
In the Profiler, enter 25 for Temperature and 100,000 for Time. The estimated probability of failure is
0.00357, with an estimated confidence interval of 0.00106 to 0.01201. This interval is narrower than the
one from the previous experiment, as a result of the additional units to be tested.
To decrease the interval further, try entering more than 300 units to be tested.
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Chapter 10
Chapter 11
Nonlinear Designs
Design of experiments with models that are nonlinear in their parameters is available using either the DOE
menu or the JMP Starter DOE category.
Nonlinear designs offer both advantages and disadvantages compared to designs for linear models.
On the positive side, predictions using a well chosen model are likely to be good over a wider range of factor
settings. It is also possible to model response surfaces with more curvature and with asymptotic behavior.
On the negative side, the researcher needs a greater understanding of both the system and of the nonlinear
design tool.
Contents
Examples of Nonlinear Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Using Nonlinear Fit to Find Prior Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Creating a Nonlinear Design with No Prior Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237
Creating a Nonlinear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Identify the Response and Factor Column with Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Set Up Factors and Parameters in the Nonlinear Design Dialog. . . . . . . . . . . . . . . . . . . . . . . . . . . . .242
Enter the Number of Runs and Preview the Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243
Make Table or Augment the Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Advanced Options for the Nonlinear Designer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245
Chapter 11
Nonlinear Designs
Examples of Nonlinear Designs
231
Examples of Nonlinear Designs
The Nonlinear Designer allows scientists to generate optimal designs and optimally augment data for fitting
models that are nonlinear in their parameters. Such models, when they are descriptive of the underlying
process, can yield more accurate prediction of process behavior than is possible with the standard
polynomial models.
To use the Nonlinear Designer, you first need a data table that has
•
one column for each factor
•
one column for the response
•
a column that contains a formula showing the functional relationship between the factor(s) and the
response.
This is the same format for a table you would supply to the nonlinear platform for modeling.
The first example in this section describes how to approach creating a nonlinear design when there is prior
data. The second example describes how to approach creating the design without data, but with reasonable
starting values for the parameter estimates.
Using Nonlinear Fit to Find Prior Parameter Estimates
Suppose you have already collected experimental data and placed it in a JMP data table. That table can be
used to create a nonlinear design for improving the estimates of the model’s parameters.
To follow along with this example, open Chemical Kinetics.jmp from the Nonlinear Examples folder found
in the sample data installed with JMP.
Chemical Kinetics.jmp (Figure 11.1) contains a column (Model (x))whose values are formed by a formula
with a poor guess of the parameter values.
Figure 11.1 Chemical Kinetics.jmp
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Examples of Nonlinear Designs
Chapter 11
First, fit the data to the model using nonlinear least squares to get better parameter values.
1. Select Analyze > Modeling > Nonlinear.
2. Select Velocity (y) and click Y, Response on the nonlinear launch dialog.
3. Select Model (x) and click X, Predictor Formula (see Figure 11.2). Note that the formula given by
Model (X) shows in the launch dialog.
Figure 11.2 Initial Nonlinear Analysis Launch Dialog
4. Click OK on the launch dialog to see the Nonlinear control panel.
5. Click the Go button on the Nonlinear control panel to see the results (Figure 11.3).
The Save Estimates button adds the new fitted parameter values in the Model (x) column in the
Chemical Kinetics.jmp data table.
The Confidence Limits button computes confidence intervals used to create a nonlinear design.
The ranges for LowerCL and UpperCL are the intervals for Vmax and k. They are asymptotically
normal. Use these limits to create a nonlinear design in JMP.
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Examples of Nonlinear Designs
233
Figure 11.3 Nonlinear Fit Results
6. Click the Confidence Limits button to produce confidence intervals.
7. Click Save Estimates to add the new fitted parameter values in the Model (x) column in the Chemical
Kinetics.jmp data table, which contains the formula:
Figure 11.4 Formula Produced by Save Estimates
Note: Leave the nonlinear analysis report open because these results are needed in the DOE nonlinear
design dialog described next.
Now create a design for fitting the model’s nonlinear parameters.
1. With the Chemical Kinetics.jmp data table open, select DOE > Nonlinear Design.
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2. Complete the launch dialog the same way as the Nonlinear Analysis launch dialog shown previously.
That is, Select Velocity (y) and click Y, Response. Select Model (x) and click the X, Predictor Formula.
Figure 11.5 shows the completed dialog.
Figure 11.5 Initial Nonlinear Design Launch Dialog
3. Click OK to see the completed Design panels for factors and parameters, as shown in Figure 11.6.
Figure 11.6 Nonlinear Design Panels for Factors and Parameters
Note that in Chemical Kinetics.jmp (Figure 11.1), the range of data for Concentration goes from 0.417 to
6.25. Therefore, these values initially appear as the high and low values in the Factors control panel as
follows:
4. Change the factor range for Concentration to a broader interval—from 0.1 to 7 (Figure 11.7).
Note that the a priori distribution of the parameters Vmax and k is Normal, which is correct for this
example. Change the current level of uncertainty in the two parameters using the analysis results.
Chapter 11
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235
5. Look back at the analysis report inFigure 11.3 and locate the upper and lower confidence limits for
Vmax and k in the Solution table. Change the values for Vmax and k to correspond to those limits, as
shown in Figure 11.7.
Now you have described the current level of uncertainty of the two parameters.
Figure 11.7 Change Values for Factor and Parameters
6. If necessary, type the desired number of runs (17 is the default value for this example) into the text box.
Use commands from the menu on the Nonlinear Design title bar to get the best possible design:
7. Select Number of Starts from the menu on the title bar and enter 100 in the text box.
8. Select Advanced Options > Number of Monte Carlo Samples and enter 2 in the text box.
9. Click Make Design to preview the design (Figure 11.8). Your results might differ from those shown for
the additional runs.
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Figure 11.8 Selecting the Number of Runs
10. Click Make Table.
This creates a new JMP design table (Figure 11.9) whose rows are the runs defined by the nonlinear
design.
Note: This example creates a new table to avoid altering the sample data table Chemical Kinetics.jmp. In
most cases, however, you can augment the original table using the Augment Table option in the Nonlinear
Designer instead of making a new table. This option adds the new runs shown in the Design to the existing
data table.
Figure 11.9 Making a Table with the Nonlinear Designer
Chapter 11
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Examples of Nonlinear Designs
237
The new runs use the wider interval of allowed concentration, which leads to more precise estimates of k
and Vmax.
Creating a Nonlinear Design with No Prior Data
This next example describes how to create a design when you have not yet collected data, but have a guess
for the unknown parameters.
To follow along with this example, open Reaction Kinetics Start.jmp, found in the Design Experiment
folder in the sample data installed with JMP. Notice that the table is a template. That is, the table has
columns with properties and formulas, but there are no observations in the table. The design has not yet
been created and data have not been collected.
This table is used to supply the formula in the yield model column to the Nonlinear DOE platform. The
formula is used to create a nonlinear design for fitting the model’s nonlinear parameters.
Figure 11.10 Yield Model Formula
This model is from Box and Draper (1987). The formula arises from the fractional yield of the intermediate
product in a consecutive chemical reaction. It is written as a function of time and temperature.
1. With the Reaction Kinetics Start.jmp data table open, select DOE > Nonlinear Design to see the initial
launch dialog.
2. Select observed yield and click Y, Response.
3. Select yield model (the column with the formula) and click X, Predictor Formula.
The completed dialog should look like the one in Figure 11.11.
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Figure 11.11 Nonlinear Design launch Dialog
4. Click OK to see the nonlinear design Factors and Parameters panels in Figure 11.12.
5. Change the two factors’ values to be a reasonable range of values. (In your experiment, these values
might have to be an educated guess.) For this example, use the values 510 and 540 for Reaction
Temperature. Use the values 0.1 and 0.3 for Reaction Time.
6. Change the values of the parameter t1 to 25 and 50, and t3 to 30 and 35.
7. Click on the Distribution of each parameter and select Uniform from the menu to change the
distribution from the default Normal (see Figure 11.12).
8. Change the number of runs to 12 in the Design Generation panel.
Figure 11.12 Change Factor Values, Parameter Distributions, and Number of Runs
9. Click Make Design, then Make Table. Your results should look similar to those in Figure 11.13.
Chapter 11
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239
Figure 11.13 Design Table
10. To analyze data that contains values for the response, observed yield, open Reaction Kinetics.jmp from
the Design Experiment folder in the sample data installed with JMP (Figure 11.14).
Figure 11.14 Reaction Kinetics.jmp
First, examine the design region with an overlay plot.
11. Select Graph > Overlay Plot.
12. Remove all existing column assignments.
13. Select Reaction Temperature and click Y
14. Select Reaction Time and click X as shown in the Overlay Plot launch dialog in Figure 11.15.
15. Click OK to see the overlay plot in Figure 11.15.
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Figure 11.15 Create an Overlay Plot
Notice that the points are not at the corners of the design region. In particular, there are no points at low
temperature and high time—the lower right corner of the graph.
16. Select Analyze > Modeling > Nonlinear.
17. Remove all existing column assignments.
18. Select observed yield and click Y, Response.
19. Select yield model and click the X, Predictor Formula, then click OK.
20. Click Go on the Nonlinear control panel.
21. Now, choose Profilers > Profiler from the red triangle menu on the Nonlinear Fit title bar.
22. To maximize the yield, choose Maximize Desirability from the red triangle menu on the Prediction
Profiler title bar.
The maximum yield is approximately 63.5% at a reaction temperature of 540 degrees Kelvin and a reaction
time of 0.1945 minutes.
Chapter 11
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Creating a Nonlinear Design
241
Figure 11.16 Time and Temperature Settings for Maximum Yield
Creating a Nonlinear Design
To begin, open a data table that has a column whose values are formed by a formula (for details about
formulas, see the Using JMP). This formula must have parameters.
Select DOE >Nonlinear Design, or click the Nonlinear Design button on the JMP Starter DOE page.
Then, follow the steps below:
•
“Identify the Response and Factor Column with Formula,” p. 241
•
“Set Up Factors and Parameters in the Nonlinear Design Dialog,” p. 242
•
“Enter the Number of Runs and Preview the Design,” p. 243
•
“Make Table or Augment the Table,” p. 244
Identify the Response and Factor Column with Formula
1. Open a data table that contains a column whose values are formed by a formula that has parameters.
This example uses Corn.jmp from the Nonlinear Examples folder in the sample data installed with JMP.
2. Select DOE > Nonlinear Design to see the initial launch dialog.
3. Select yield and click Y, Response. The response column cannot have missing values.
4. Select quad and click X, Predictor Formula. The quad variable has a formula that includes nitrate and
three parameters (Figure 11.17).
5. Click OK on the launch dialog to see the Nonlinear Design DOE panels.
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Figure 11.17 Identify Response (Y) and the Column with the Nonlinear Formula (X)
Set Up Factors and Parameters in the Nonlinear Design Dialog
First, look at the formula for quad, shown in Figure 11.18, and notice there are three parameters. These
parameters show in the Parameters panel of the Nonlinear design dialog, with initial parameter values.
Figure 11.18 Formula for quad has Parameters a, b, and c
Use Figure 11.19 to understand how to set up factor and parameter names and values.
•
The initial values for the factor and the parameters are reasonable and do not need to be changed.
•
If necessary, change the Distribution of the parameters to Uniform, as shown in Figure 11.19.
Chapter 11
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Creating a Nonlinear Design
243
Figure 11.19 Example of Setting Up Factors and Parameters
Double-click to edit the factor or
parameter name.
Click to enter or change factor values.
Click to edit the distribution: Uniform, Normal,
Lognormal, or Exponential.
Enter the Number of Runs and Preview the Design
1. The Design Generation panel shows 150 as the default number of runs. This number includes
observations in the current data. 147 runs are desired. Since there are originally 144 rows, 3 additional
runs need to be added. Enter 147 in the Number of Runs edit box.
2. Click Make Design before creating the data table to preview the design. Figure 11.20 shows a partial
listing of the design.
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Figure 11.20 Example Preview Design
Make Table or Augment the Table
3. The last step is to click either Make Table or Augment Table. The Make Table command creates a new
table (Figure 11.21) with all runs included. The Augment Table command adds the new runs to the
existing table.
Figure 11.21 Partial Listing of an Example Nonlinear Design Table
Chapter 11
Nonlinear Designs
Advanced Options for the Nonlinear Designer
245
Advanced Options for the Nonlinear Designer
For advanced users, the Nonlinear Designer has the two additional options, as shown in Figure 11.22.
These advanced options are included because finding nonlinear DOE solutions involves minimizing the
integral of the log of the determinant of the Fisher information matrix with respect to the prior distribution
of the parameters. These integrals are complicated and have to be calculated numerically.
The way the integration is done for Normal distribution priors uses a numerical integration technique
where the integral is reparameterized into a radial direction, and the number of parameters minus one
angular directions. The radial part of the integral is handled using Radau-Gauss-Laguerre quadrature with
an evaluation at radius=0. A randomized Mysovskikh quadrature is used to calculate the integral over the
spherical part, which is equivalent to integrating over the surface of a hypersphere.
Note: If some of the prior distributions are not Normal, then the integral is reparameterized so that the new
parameters have normal distribution, and then the radial-spherical integration method is applied. However,
if the prior distribution set for the parameters does not lend itself to a solution, sometimes the process fails
and gives the message that the Fisher information is singular in a region of the parameter space, and advises
changing the prior distribution or the ranges of the parameters.
Figure 11.22 Advanced Options for the Nonlinear Designer
The following is a technical description for these two advanced options:
Number of Monte Carlo Samples sets the number of octahedra per sphere. Because each
octahedron is a fixed unit, this option can be thought of as setting the number of octahedra per
sphere.
are the number of nonzero radius values used. The default is two spheres
and one center point. Each radial value requires integration over the angular dimensions. This is
done by constructing a certain number of hyperoctahedra (the generalization of an octagon in
arbitrary dimensions), and randomly rotating each of them.
N Monte Carlo Spheres
Technical Note: The reason for the approach given by these advanced options is to get good integral
approximations much faster than using standard methods. For instance, with six parameters, using two radii
and one sample per sphere, these methods give a generalized five- point rule that needs only 113
observations to get a good approximation. Using the most common approach (Simpson’s rule) would need
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Chapter 11
56 = 15,625 evaluations. The straight Monte Carlo approach also requires thousands of function evaluations
to get the same level of quality in the answer. For more details, see Gotwalt, Jones, and Steinberg (2009).
Keep in mind that if the number of radii is set to zero, then just the center point is used, which gives a local
design that is optimal for a particular value of the parameters. For some people this is good enough for their
purposes. These designs are created much faster than if the integration is performed.
Chapter 12
Taguchi Designs
Quality was the watchword of 1980s, and Genichi Taguchi was a leader in the growth of quality
consciousness. One of Taguchi’s technical contributions to the field of quality control was a new approach
to industrial experimentation. The purpose of the Taguchi method was to develop products that worked well
in spite of natural variation in materials, operators, suppliers, and environmental change. This is robust
engineering.
Much of the Taguchi method is traditional. His orthogonal arrays are two-level, three-level, and mixed-level
fractional factorial designs. The unique aspects of his approach are the use of signal and noise factors, inner
and outer arrays, and signal-to-noise ratios.
The goal of the Taguchi method is to find control factor settings that generate acceptable responses despite
natural environmental and process variability. In each experiment, Taguchi’s design approach employs two
designs called the inner and outer array. The Taguchi experiment is the cross product of these two arrays.
The control factors, used to tweak the process, form the inner array. The noise factors, associated with process
or environmental variability, form the outer array. Taguchi’s signal-to-noise ratios are functions of the
observed responses over an outer array. The Taguchi designer supports all these features of the Taguchi
method. You choose from inner and outer array designs, which use the traditional Taguchi orthogonal
arrays, such as L4, L8, and L16.
Dividing system variables according to their signal and noise factors is a key ingredient in robust
engineering. Signal factors are system control inputs. Noise factors are variables that are typically difficult or
expensive to control.
The inner array is a design in the signal factors and the outer array is a design in the noise factors. A
signal-to-noise ratio is a statistic calculated over an entire outer array. Its formula depends on whether the
experimental goal is to maximize, minimize or match a target value of the quality characteristic of interest.
A Taguchi experiment repeats the outer array design for each run of the inner array. The response variable in
the data analysis is not the raw response or quality characteristic; it is the signal-to-noise ratio.
The Taguchi designer in JMP supports signal and noise factors, inner and outer arrays, and signal-to-noise
ratios as Taguchi specifies.
Contents
The Taguchi Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Taguchi Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Analyze the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Creating a Taguchi Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254
Detail the Response and Add Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254
Choose Inner and Outer Array Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Display Coded Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256
Make the Design Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257
Chapter 12
Taguchi Designs
The Taguchi Design Approach
249
The Taguchi Design Approach
The Taguchi method defines two types of factors: control factors and noise factors. An inner design
constructed over the control factors finds optimum settings. An outer design over the noise factors looks at
how the response behaves for a wide range of noise conditions. The experiment is performed on all
combinations of the inner and outer design runs. A performance statistic is calculated across the outer runs
for each inner run. This becomes the response for a fit across the inner design runs. Table 12.1 lists the
recommended performance statistics.
Table 12.1 Recommended Performance Statistics
Goal
S/N Ratio Formula
nominal is best
⎛ 2⎞
S--------⎟
= 10 log ⎜ Y
N
⎝ s2 ⎠
larger-is-better (maximize)
⎛
S1⎞
--= – 10 log ⎜ 1--- ∑ ------2⎟
N
n
⎝ i Y ⎠
i
⎛
⎞
S--= – 10 log ⎜ 1--- ∑Y 2 ⎟
N
⎝n i i ⎠
smaller-is-better (minimize)
Taguchi Design Example
The following example is an experiment done at Baylock Manufacturing Corporation and described by
Byrne and Taguchi (1986). The objective of the experiment is to find settings of predetermined control
factors that simultaneously maximize the adhesiveness (pull-off force) and minimize the assembly costs of
nylon tubing.
To follow along with this example, open the Byrne Taguchi Data.jmp table found in the Design Experiment
folder of the Sample Data installed with JMP. Or, generate the original design table on your own using
DOE > Taguchi Arrays.
Table 12.2 shows the signal and noise factors in the Byrne Taguchi Data for this example.
Table 12.2 Signal and Noise Factors
Factor Name
Type
Levels
Comment
Interfer
control
3
tubing and connector interference
Wall
control
3
the wall thickness of the connector
Depth
control
3
insertion depth of the tubing into the connector
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Chapter 12
Table 12.2 Signal and Noise Factors (Continued)
Factor Name
Type
Levels
Comment
Adhesive
control
3
percent adhesive
Time
noise
2
the conditioning time
Temperature
noise
2
temperature
Humidity
noise
2
the relative humidity
To start this example:
1. Select DOE > Taguchi Arrays.
2. Click the red triangle icon on the Taguchi Design title bar and select Load Factors.
3. When the Open File dialog appears, open the factors table, Byrne Taguchi Factors.jmp found in the
Design Experiment Sample Data folder installed with JMP.
The factors panel then shows the four three-level control (signal) factors and three noise factors, as shown in
Figure 12.1.
Figure 12.1 Response, and Signal and Noise Factors for the Byrne-Taguchi Example
4. Highlight L9-Taguchi to give the L9 orthogonal array for the inner design.
5. Highlight L8 to give an eight-run outer array design.
6. Click Continue.
Chapter 12
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Taguchi Design Example
251
The outer design has three two-level factors. A full factorial in eight runs is generated. However, it is only
used as a guide to identify a new set of eight columns in the final JMP data table—one for each
combination of levels in the outer design.
7. Click Make Table to create the design table shown in Figure 12.2.
Figure 12.2 Taguchi Design Before Data Entry
Now, suppose the pull-off adhesive force measures are collected and entered into the columns containing
missing data, as shown in Figure 12.3. The missing data column names are appended with the letter Y
before the levels (+ or –) of the noise factors for that run. For example, Y--- is the column of measurements
taken with the three noise factors set at their low levels.
8. To see the completed experiment, open the data table, Byrne Taguchi Data.jmp found in the Design
Experiment Sample Data folder installed with JMP. Figure 12.3 shows the completed design.
Figure 12.3 Complete Taguchi Design Table (Byrne Taguchi Data.jmp)
The column named SN Ratio Y is the performance statistic computed with the formula shown below. In
this case, it is the “larger-the-better” (LTB) formula, which is –10 times the common logarithm of the
average squared reciprocal:
252
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Taguchi Design Example
–10Log10
Chapter 12
[Mean [(
1
Y– – –)2
,
1
1
1
1
,
,
,
,
(Y– – +)2 (Y– + –)2 (Y– ++)2 (Y+ – –)2
1
1
1
,
,
(Y+ – +)2 (Y+ + –)2 (Y+++)2
]]
This expression is large when all of the individual y values are large.
Analyze the Data
The data in Byrne Taguchi Data.jmp are now ready to analyze. The goal of the analysis is to find factor
settings that maximize both the mean and the signal-to-noise ratio.
1. Click the red triangle icon next to Model on the upper left of the data table and select Run Script. The
Model script produces the Fit Model dialog shown in Figure 12.4.
The Fit Model dialog that appears automatically has the appropriate effects. It includes the main effects of
the four signal factors. The two responses are the mean (Mean Y) and signal-to-noise ratio (SN Ratio Y)
over the outer array.
Figure 12.4 Fit Model Dialog for Taguchi Data
2. Click Run on the Fit Model dialog.
The prediction profiler is a quick way to find settings that give the highest signal-to-noise ratio for this
experiment.
Chapter 12
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Taguchi Design Example
253
3. To open the Prediction Profiler, click the red triangle on the Response Mean Y title bar and select Factor
Profiling > Profiler.
The profile traces (Figure 12.5) indicate that different settings of the first three factors would increase SN
Ratio Y.
Figure 12.5 The Prediction Profiler
4. To find optimal settings, click the red triangle on the Prediction Profiler title bar and select Desirability
Functions. This adds the row of traces and a column of function settings to the profiler, as shown in
Figure 12.6. The default desirability functions are set to larger-is-better, which is what you want in this
experiment. See Modeling and Multivariate Methods for more details about the prediction profiler.
5. Again click the red triangle on the Prediction Profiler title bar and select Maximize Desirability to
automatically set the prediction traces that give the best results according to the desirability functions.
In this example, the settings for Interfer and Wall changed from 1 to 2. (See Figure 12.5 and Figure 12.6).
The Depth setting changed from 1 to 3. The settings for Adhesive did not change. These new settings
increased the signal-to-noise ratio from 24.0253 to 26.9075.
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Chapter 12
Figure 12.6 Best Factor Settings for Byrne Taguchi Data
Creating a Taguchi Design
To start a Taguchi design, select DOE >Taguchi Arrays, or click the Taguchi Arrays button on the JMP
Starter DOE page. Then, follow the steps below:
•
“Detail the Response and Add Factors,” p. 254
•
“Choose Inner and Outer Array Designs,” p. 255
•
“Display Coded Design,” p. 256
•
“Make the Design Table,” p. 257
Detail the Response and Add Factors
The Responses panel has a single default response. The steps for setting up the details of this response are
outlined in Figure 12.7. For information on importance weights and lower and upper limits, see
“Understanding Response Importance Weights,” p. 66.
1. Double-click to edit the response name.
2. Click to change the response goal: Larger Is Better, Nominal is Best, Smaller is Better, or None.
3. Click to enter lower and upper limits and importance weights.
Chapter 12
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Creating a Taguchi Design
255
Figure 12.7 Setting Up the Response
3
1
2
The steps for setting up the factors are outlined in Figure 12.8.
1. Click to add a signal, then select a signal type: 2 Level, or 3 Level.
Or click to add a noise.
2. Double-click to edit the factor name.
3. To change the value of a signal or noise, click and then type the new value.
Figure 12.8 Entering Factors
1
3
2
When you finish adding factors, click Continue.
Choose Inner and Outer Array Designs
Your choice for inner and outer arrays depends on the number and type of factors you have. Figure 12.9
shows the available inner array choices when you have eight signal factors. If you also have noise factors,
choices include designs for the outer array. To follow along, enter eight two-level Signal factors and click
Continue. Then highlight the design you want and again click Continue. This example uses the L12 design.
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Chapter 12
Figure 12.9 Selecting a Design for Eight Signal Factors
If you did not specify a noise factor, after you click Continue, a dialog appears that asks you to specify how
many times you want to perform each inner array run. Specify two (2) for this example.
Display Coded Design
After you select a design type, the Coded Design (Figure 12.10) is shown below the Factors panel.
Figure 12.10 Coding for Eight Factor L12 Design
The Coded Design shows the pattern of high and low values for the factors in each run. For more details on
the coded design, see “Understanding Design Codes,” p. 118.
Chapter 12
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Creating a Taguchi Design
257
Make the Design Table
When you click Make Table, a table similar to that shown in Figure 12.11 appears. In the data table, each
row represents a run. In the values for the Pattern variable, plus signs designate high levels and minus signs
represent low levels.
Figure 12.11 Taguchi Design Table for Eight Factor L12 Design
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Chapter 13
Augmented Designs
If you treat experimentation as an iterative process, you can master the temptation to assume that one
successful screening experiment has optimized your process. You can also avoid disappointment if a
screening experiment leaves behind some ambiguities. The augment designer helps facilitate
experimentation as an iterative process.
The augment designer modifies an existing design data table, supporting your iterative process. It gives the
following five choices:
•
replicate the design a specified number of times
•
add center points
•
create a foldover design
•
add axial points together with center points to transform a screening design to a response surface design
•
add runs to the design using a model that can have more terms than the original model
This chapter provides an overview of the augment designer. It also presents a case study of design
augmentation.
Contents
A D-Optimal Augmentation of the Reactor Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Analyze the Augmented Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Creating an Augmented Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Replicate a Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272
Add Center Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
Creating a Foldover Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Adding Axial Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Adding New Runs and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .278
Special Augment Design Commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Save the Design (X) Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .282
Modify the Design Criterion (D- or I- Optimality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .282
Select the Number of Random Starts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Specify the Sphere Radius Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Disallow Factor Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284
Chapter 13
Augmented Designs
A D-Optimal Augmentation of the Reactor Example
261
A D-Optimal Augmentation of the Reactor Example
This example, adapted from Meyer, et al. (1996), demonstrates how to use the augment designer in JMP to
resolve ambiguities left by a screening design. In this study, a chemical engineer investigates the effects of
five factors on the percent reaction of a chemical process.
To begin, open Reactor 8 Runs.jmp found in the Design Experiment Sample Data folder installed with
JMP. Then select Augment Design from the DOE menu. When the initial launch dialog appears:
1. Select Percent Reacted and click Y, Response.
2. Select all other variables except Pattern and click X, Factor.
3. Click OK on the launch dialog to see the Augment Design dialog in Figure 13.1.
Note: You can check Group new runs into separate block to add a blocking factor to any design.
However, the purpose of this example is to estimate all two-factor interactions in 16 runs, which can’t be
done when there is the additional blocking factor in the model.
Figure 13.1 Augment Design Dialog for the Reactor Example
4. Now click Augment on the Augment Design dialog to see the display in Figure 13.2.
This model shown in Figure 13.2 is the result of the model stored with the data table when it was created by
the Custom designer. However, the augmented design is to have 16 runs in order to estimate all two-factor
interactions.
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Figure 13.2 Initial Augmented Model
To continue with the augmented reactor design:
5. Choose 2nd from the Interactions menu as shown in Figure 13.3. This adds all the two-factor
interactions to the model. The Minimum number of runs given for the specified model is 16, as shown
in the Design Generation text edit box.
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Figure 13.3 Augmented Model with All Two-Factor Interactions
6. Click Make Design.
JMP now computes D-optimally augmented factor settings, similar to the design shown in Figure 13.4.
Figure 13.4 D-Optimally Augmented Factor Settings
Note: The resulting design is a function of an initial random number seed. To reproduce the exact factor
settings table in Figure 13.4, (or the most recent design you generated), choose Set Random Seed from the
popup menu on the Augment Design title bar. A dialog shows the most recently used random number.
Click OK to use that number again, or Cancel to generate a design based on a new random number. The
dialog in Figure 13.5 shows the random number (12834729) used to generate the runs in Figure 13.4.
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Figure 13.5 Specifying a Random Number
7. Click Make Table to generate the JMP table with D-Optimally augmented runs.
Analyze the Augmented Design
Suppose you have already run the experiment on the augmented data and recorded results in the Percent
Reacted column of the data table.
1. To see these results, open Reactor Augment Data.jmp found in the Design Experiment Sample Data
folder installed with JMP.
It is desirable to maximize Percent Reacted, however its column in this sample data table has a response
limits column property set to Minimize.
2. Click the asterisk next to the Percent Reacted column name in the Columns panel of the data table and
select Response Limits, as shown on the left in Figure 13.6.
3. In the Column Info dialog that appears, change the response limit to Maximize, as shown on the right
in Figure 13.6.
Figure 13.6 Change the Response Limits Column Property for the Percent Reacted Column
You are now ready to run the analysis.
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4. To start the analysis, click the red triangle for Model in the upper left of the data table and select Run
Script from the menu, as shown in Figure 13.7.
Figure 13.7 Completed Augmented Experiment (Reactor Augment Data.jmp)
The Model script, stored as a table property with the data, contains the JSL commands that display the Fit
Model dialog with all main effects and two-factor interactions as effects.
5. Change the fitting personality on the Fit Model dialog from Standard Least Squares to Stepwise, as
shown in Figure 13.8.
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Figure 13.8 Fit Model Dialog for Stepwise Regression on Generated Model
6. When you click Run, the stepwise regression control panel appears. Click the check boxes for all the
main effect terms.
Important: Choose P-value Threshold from the Stopping Rule menu, Mixed from the Direction menu,
and make sure Prob to Enter is 0.050 and Prob to Leave is 0.100. These are not the default values. You
should see the dialog shown in Figure 13.9.
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Figure 13.9 Initial Stepwise Model
7. Click Go to start the stepwise regression and watch it continue until all terms are entered into the model
that meet the Prob to Enter and Prob to Leave criteria in the Stepwise Regression Control panel.
Figure 13.10, shows the result of this example analysis. Note that Feed Rate is out of the model while the
Catalyst*Temperature, Stir Rate*Temperature, and the Temperature*Concentration interactions have
entered the model.
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Figure 13.10 Completed Stepwise Model
8. After Stepwise is finished, click Make Model on the Stepwise control panel to generate this reduced
model, as shown in Figure 13.11.
9. Click Run and fit the reduced model to do additional diagnostic work, make predictions, and find the
optimal factor settings.
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Figure 13.11 New Prediction Model Dialog
The Analysis of Variance and Lack of Fit Tests in Figure 13.12, indicate a highly significant regression
model with no evidence of Lack of Fit.
Figure 13.12 Prediction Model Analysis of Variance and Lack of Fit Tests
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The Sorted Parameter Estimates table in Figure 13.13 shows that Catalyst has the largest main effect.
However, the significance of the two-factor interactions are of the same order of magnitude as the main
effects. This is the reason that the initial screening experiment, shown in the chapter “Screening Designs,”
p. 101, had ambiguous results.
Figure 13.13 Prediction Model Estimates Plot
10. Choose Maximize Desirability from the menu on the Prediction Profiler title bar.
The prediction profile plot at the bottom in Figure 13.14 shows that maximum occurs at the high levels of
Catalyst, Stir Rate, and Temperature and the low level of Concentration. At these extreme settings, the
estimate of Percent Reacted increases from 65.17 to 98.38.
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Figure 13.14 Maximum Percent Reacted
To summarize, compare the analysis of 16 runs with the analyses of reactor data from previous chapters:
•
“Screening Designs,” p. 101, the analysis of a screening design with only 8 runs produced a model with
the five main effects and two interaction effects with confounding. None of the factors effects were
significant, although the Catalyst factor was large enough to encourage collecting data for further runs.
•
“Full Factorial Designs,” p. 143, a full factorial of the five two-level reactor factors, 32 runs, was first
subjected to a stepwise regression. This approach identified three main effects (Catalyst, Temperature,
and Concentration) and two interactions (Temperature*Catalyst, Contentration*Temperature) as
significant effects.
•
By using a D-optimal augmentation of 8 runs to produce 8 additional runs, a stepwise analysis returned
the same results as the analysis of 32 runs. The bottom line is that only half as many runs yielded the
same information. Thus, using an iterative approach to DOE can save time and money.
Creating an Augmented Design
The augment designer modifies an existing design data table. It gives the following five choices:
Replicate replicates the design a specified number of times. See “Replicate a Design,” p. 272.
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Add Centerpoints adds center points. See “Add Center Points,” p. 275.
Fold Over
creates a foldover design. See “Creating a Foldover Design,” p. 276.
Add Axial adds axial points together with center points to transform a screening design to a response
surface design. See “Adding Axial Points,” p. 277.
adds runs to the design (augment) using a model, which can have more terms than the
original model. See “Adding New Runs and Terms,” p. 278.
Augment
Replicate a Design
Replication provides a direct check on the assumption that the error variance is constant. It also reduces the
variability of the regression coefficients in the presence of large process or measurement variability.
To replicate the design a specified number of times:
1. Open a data table that contains a design you want to augment. This example uses Reactor 8 Runs.jmp
from the Design Experiment Sample Data folder installed with JMP.
2. Select DOE > Augment Design to see the initial dialog for specifying factors and responses.
3. Select Percent Reacted and click Y, Response.
4. Select all other variables (except Pattern) and click X, Factor to identify the factors you want to use for
the augmented design (Figure 13.15).
Figure 13.15 Identify Response and Factors
5. Click OK to see the Augment Design panel shown in Figure 13.16.
6. If you want the original runs and the resulting augmented runs to be identified by a blocking factor,
check the box beside Group New Runs into Separate Block on the Augment Design panel.
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Figure 13.16 Choose an Augmentation Type
7. Click the Replicate button to see the dialog shown on the left in Figure 13.17. Enter the number of
times you want JMP to perform each run, then click OK.
Note: Entering 2 specifies that you want each run to appear twice in the resulting design. This is the same
as one replicate (Figure 13.17).
8. View the design, shown on the right in Figure 13.17.
Figure 13.17 Reactor Data Design Augmented With Two Replicates
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9. Click the disclosure icons next to Prediction Variance Profile and Prediction Variance Surface to see the
profile and surface plots shown in Figure 13.18.
Figure 13.18 Prediction Profiler and Surface Plot
10. Click Make Table to produce the design table shown in Figure 13.19.
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Figure 13.19 The Replicated Design
Add Center Points
Adding center points is useful to check for curvature and reduce the prediction error in the center of the
factor region. Center points are usually replicated points that allow for an independent estimate of pure
error, which can be used in a lack-of-fit test.
To add center points:
1. Open a data table that contains a design you want to augment. This example uses Reactor 8 Runs.jmp
found in the Design Experiment Sample Data folder installed with JMP.
2. Select DOE > Augment Design.
3. In the initial Augment Design dialog, identify the response and factors you want to use for the
augmented design (see Figure 13.15) and click OK.
4. If you want the original runs and the resulting augmented runs to be identified by a blocking factor,
check the box beside Group new runs into separate block. (Figure 13.16 shows the check box location
directly under the Factors panel.)
5. Click the Add Centerpoints button and type the number of center points you want to add. For this
example, add two center points, and click OK.
6. Click Make Table to see the data table in Figure 13.20.
The table shows two center points appended to the end of the design.
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Figure 13.20 Design with Two Center Points Added
Creating a Foldover Design
A foldover design removes the confounding of two-factor interactions and main effects. This is especially
useful as a follow-up to saturated or near-saturated fractional factorial or Plackett-Burman designs.
To create a foldover design:
1. Open a data table that contains a design you want to augment. This example uses Reactor 8 Runs.jmp,
found in the Design Experiment Sample Data folder installed with JMP.
2. Select DOE > Augment Design.
3. In the initial Augment Design dialog, identify the response and factors you want to use for the
augmented design (see Figure 13.15) and click OK.
4. Check the box to the left of Group new runs into separate block. (Figure 13.16 shows the check box
location directly under the Factors panel.) This identifies the original runs and the resulting augmented
runs with a blocking factor.
5. Click the Fold Over button. A dialog appears that lists all the design factors.
6. Choose (select) which factors to fold. The default, if you choose no factors, is to fold on all design
factors. If you choose a subset of factors to fold over, the remaining factors are replicates of the original
runs. The example in Figure 13.21 folds on all five factors and includes a blocking factor.
7. Click Make Table. The design data table that results lists the original set of runs as block 1 and the new
(foldover) runs are block 2.
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Figure 13.21 Listing of a Foldover Design On All Factors
Adding Axial Points
You can add axial points together with center points, which transforms a screening design to a response
surface design. To do this:
1. Open a data table that contains a design you want to augment. This example uses Reactor 8 Runs.jmp,
from the Design Experiment Sample Data folder installed with JMP.
2. Select DOE > Augment Design.
3. In the initial Augment Design dialog, identify the response and factors you want to use for the
augmented design (see Figure 13.15) and click OK.
4. If you want the original runs and the resulting augmented runs to be identified by a blocking factor,
check the box beside Group New Runs into Separate Block (Figure 13.16).
5. Click Add Axial.
6. Enter the axial values in units of the factors scaled from –1 to +1, then enter the number of center points
you want. When you click OK, the augmented design includes the number of center points specified
and constructs two axial points for each variable in the original design.
Figure 13.22 Entering Axial Values
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7. Click Make Table. The design table appears. Figure 13.23 shows a table augmented with two center
points and two axial points for five variables.
Figure 13.23 Design Augmented With Two Center and Ten Axial Points
center points
axial points
Adding New Runs and Terms
A powerful use of the augment designer is to add runs using a model that can have more terms than the
original model. For example, you can achieve the objectives of response surface methodology by changing a
linear model to a full quadratic model and adding the necessary number of runs. Suppose you start with a
two-factor, two-level, four-run design. If you add quadratic terms to the model and five new points, JMP
generates the 3 by 3 full factorial as the optimal augmented design.
D-optimal augmentation is a powerful tool for sequential design. Using this feature you can add terms to
the original model and find optimal new test runs with respect to this expanded model. You can also group
the two sets of experimental runs into separate blocks, which optimally blocks the second set with respect to
the first.
To add new runs and terms to the original model:
1. Open a data table that contains a design you want to augment. This example uses Reactor Augment
Data.jmp, from the Design Experiment Sample Data folder installed with JMP.
2. Select DOE > Augment Design.
3. In the initial Augment Design dialog, identify the response and factors you want to use for the
augmented design (see Figure 13.15) and click OK.
4. If you want the original runs and the resulting augmented runs to be identified by a blocking factor,
check the box beside Group New Runs into Separate Block (not used in this example).
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5. Click the Augment button. The original number of runs (Figure 13.24) appear in the Factor Design
panel.
Figure 13.24 Viewing the Existing Design
6. In the Design Generation panel, enter the number of total runs you want this design to contain. The
number you enter is the original number of runs plus the number of additional runs you want.
7. Click the Make Design button. The new number of runs (Figure 13.25) appear in the Design panel.
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Figure 13.25 24 Total Runs
8. If desired, view the prediction variance profile and the prediction variance surface.
9. Click Make Table to create the augmented design JMP table (Figure 13.26) with the additional runs.
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Figure 13.26 The Augmented Design Table with New Runs
additional
runs
Technical Note: The Augment designer does not support designs that have terms whose estimability has
been set to If Possible instead of Necessary, as is done in some screening designs that have fewer runs than
terms.
Special Augment Design Commands
After you select DOE > Augment Design and identify factors and responses, the window in Figure 13.27
appears. Click the red triangle icon on the Augment Design title bar to see a list of commands. Most of these
commands are for saving and loading information about variables; they are available in all designs and more
information is in “Special Custom Design Commands,” p. 83. The following sections describe commands
found in this menu that are specific to augment designs.
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Figure 13.27 Click the Red Triangle Icon to Reveal Commands
Save the Design (X) Matrix
To create a script and save it as a table property in the JMP design data table, click the red triangle icon in
the Augment Design title bar (Figure 13.27) and select Save X Matrix. Two or three scripts are saved to the
table. Moments Matrix and Design Matrix scripts are always saved. If the design is a split plot design, an
additional V Inverse script is also saved. When you run the Moments Matrix script, JMP creates a matrix
called Moments and displays its number of rows in the log. When you run the Design Matrix script, JMP
creates a matrix called X and displays its number of rows in the log. When you run the V Inverse script,
JMP creates the inverse of the variance matrix of the responses, and displays its number of rows in the log. If
you do not have the log visible, select View > Log or Window > Log on the Macintosh.
Modify the Design Criterion (D- or I- Optimality)
To modify the design optimality criterion, click the red triangle icon in the Augment Design title bar
(Figure 13.28) and select Optimality Criterion, then choose Make D-Optimal Design or Make I-Optimal
Design. The default criterion for Recommended is D-optimal for all design types unless you have used the
RSM button in the Model panel to add effects that make the model quadratic.
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Figure 13.28 Change the Optimality Criterion
Select the Number of Random Starts
To override the default number of random starts, click the red triangle icon in the Augment Design title bar
(Figure 13.28) and select Number of Starts. The window in Figure 13.29 appears with an edit box for you
to enter the number of random starts for the design you want to build. The number you enter overrides the
default number of starts, which varies depending on the design.
Figure 13.29 Changing the Number of Starts
For additional information on the number of starts, see “Why Change the Number of Starts?,” p. 89.
Specify the Sphere Radius Value
Augment designs can be constrained to a hypersphere. To edit the sphere radius for the design in units of the
coded factors (–1, 1), click the red triangle icon in the Augment Design title bar (Figure 13.27) and select
Sphere Radius. Enter the appropriate value and click OK.
Or, use JSL and submit the following command before you build a custom design:
DOE Sphere Radius = 1.0;
In this statement you can replace 1.0 with any positive number.
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Disallow Factor Combinations
In addition to linear inequality constraints on continuous factors and constraining a design to a
hypersphere, you can define general factor constraints on the factors. You can disallow any combination of
levels of categorical factors if you have not already defined linear inequality constraints.
For information on how to do this, see “Disallowed Combinations: Accounting for Factor Level
Restrictions,” p. 90.
Chapter 14
Prospective Sample Size and Power
Use the DOE > Sample Size and Power command to answer the question “How many runs do I need?”
The important quantities are sample size, power, and the magnitude of the effect. These depend on the
significance level, alpha, of the hypothesis test for the effect and the standard deviation of the noise in the
response. You can supply either one or two of the three values. If you supply only one of these values, the
result is a plot of the other two. If you supply two values, the third value is computed.
The Sample Size and Power platform can answer the question, “Will I detect the group differences I am
looking for, given my proposed sample size, estimate of within-group variance, and alpha level?” In this type
of analysis, you must approximate the group means and sample sizes in a data table as well as approximate
the within-group standard deviation (σ).
The sample size and power computations determine the sample size necessary for yielding a significant
result, given that the true effect size is at least a certain size. It requires that you enter two out of three
possible quantities; difference to detect, sample size, and power. The third quantity is computed for the
following cases:
•
difference between a one sample mean and a hypothesized value
•
difference between two sample means
•
differences in the means among k samples
•
difference between a standard deviation and a hypothesized value
•
difference between a one sample proportion and a hypothesized value
•
difference between two sample proportions
•
difference between counts per unit in a Poisson-distributed sample and a hypothesized value.
The calculations assume that there are equal numbers of units in each group. You can apply this platform to
more general experimental designs, if they are balanced and an adjustment for the number-of-parameters is
specified.
You can also compute the required sample sizes needed for reliability studies and demonstrations.
Contents
Launching the Sample Size and Power Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287
One-Sample and Two-Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287
Single-Sample Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .289
Sample Size and Power Animation for One Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .292
Two-Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293
k-Sample Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
One Sample Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
One Sample Standard Deviation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
One-Sample and Two-Sample Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .298
One Sample Proportion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .298
Two Sample Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Counts per Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Counts per Unit Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304
Sigma Quality Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Sigma Quality Level Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Number of Defects Computation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Reliability Test Plan and Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Reliability Test Plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Reliability Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .309
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Launching the Sample Size and Power Platform
The Sample Size and Power platform helps you plan your study for a single mean or proportion
comparison, a two sample mean or proportion comparison, a one-sample standard deviation comparison, a
k sample means comparison, or a counts per unit comparison. Depending upon your experimental
situation, you supply one or two quantities to obtain a third quantity. These quantities include:
•
required sample size
•
expected power
•
expected effect size
When you select DOE > Sample Size and Power, the panel in Figure 14.1 appears with button selections
for experimental situations. The following sections describe each of these selections and explain how to
enter the quantities and obtain the desired computation.
Figure 14.1 Sample Size and Power Choices
One-Sample and Two-Sample Means
After you click either One Sample Mean, or Two Sample Means in the initial Sample Size selection list
(Figure 14.1), a Sample Size and Power window appears. (See Figure 14.2.)
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Figure 14.2 Initial Sample Size and Power Windows for Single Mean (left) and Two Means (right)
The windows are the same except that the One Mean window has a button at the bottom that accesses an
animation script.
The initial Sample Size and Power window requires values for Alpha, Std Dev (the error standard
deviation), and one or two of the other three values: Difference to detect, Sample Size, and Power. The
Sample Size and Power platform calculates the missing item. If there are two unspecified fields, a plot is
constructed, showing the relationship between these two values:
•
power as a function of sample size, given specific effect size
•
power as a function of effect size, given a sample size
•
effect size as a function of sample size, for a given power.
The Sample Size and Power window asks for these values:
Alpha s the probability of a type I error, which is the probability of rejecting the null hypothesis when
it is true. It is commonly referred to as the significance level of the test. The default alpha level is
0.05. This implies a willingness to accept (if the true difference between groups is zero) that, 5%
(alpha) of the time, a significant difference is incorrectly declared.
is the error standard deviation. It is a measure of the unexplained random variation around
the mean. Even though the true error is not known, the power calculations are an exercise in
probability that calculates what might happen if the true value is the one you specify. An estimate of
the error standard deviation could be the root mean square error (RMSE) from a previous model fit.
Std Dev
Extra Parameters is only for multi-factor designs. Leave this field zero in simple cases. In a
multi-factor balanced design, in addition to fitting the means described in the situation, there are
other factors with extra parameters that can be specified here. For example, in a three-factor
two-level design with all three two-factor interactions, the number of extra parameters is five. (This
includes two parameters for the extra main effects, and three parameters for the interactions.) In
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practice, the particular values entered are not that important, unless the experimental range has very
few degrees of freedom for error.
is the smallest detectable difference (how small a difference you want to be able
to declare statistically significant) to test against. For single sample problems this is the difference
between the hypothesized value and the true value.
Difference to Detect
Sample Size is the total number of observations (runs, experimental units, or samples) in your
experiment. Sample size is not the number per group, but the total over all groups.
Power is the probability of rejecting the null hypothesis when it is false. A large power value is better,
but the cost is a higher sample size.
Continue evaluates at the entered values.
Back returns to the previous Sample Size and Power window so that you can either redo an analysis or
start a new analysis.
Animation Script runs a JSL script that displays an interactive plot showing power or sample size. See
the section, “Sample Size and Power Animation for One Mean,” p. 292, for an illustration of the
animation script.
Single-Sample Mean
Using the Sample Size and Power window, you can test if one mean is different from the hypothesized value.
For the one sample mean, the hypothesis supported is
H0 : μ = μ0
and the two-sided alternative is
Ha : μ ≠ μ0
where μ is the population mean and μ0 is the null mean to test against or is the difference to detect. It is
assumed that the population of interest is normally distributed and the true mean is zero. Note that the
power for this setting is the same as for the power when the null hypothesis is H0: μ=0 and the true mean is
μ0.
Suppose you are interested in testing the flammability of a new fabric being developed by your company.
Previous testing indicates that the standard deviation for burn times of this fabric is 2 seconds. The goal is to
detect a difference of 1.5 seconds when alpha is equal to 0.05, the sample size is 20, and the standard
deviation is 2 seconds. For this example, μ0 is equal to 1.5. To calculate the power:
1. Select DOE > Sample Size and Power.
2. Click the One Sample Mean button in the Sample Size and Power Window.
3. Leave Alpha as 0.05.
4. Leave Extra Parameters as 0.
5. Enter 2 for Std Dev.
6. Enter 1.5 as Difference to detect.
7. Enter 20 for Sample Size.
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8. Leave Power blank. (See the left window in Figure 14.3.)
9. Click Continue.
The power is calculated as 0.8888478174 and is rounded to 0.89. (See right window in Figure 14.3.)
The conclusion is that your experiment has an 89% chance of detecting a significant difference in the
burn time, given that your significance level is 0.05, the difference to detect is 1.5 seconds, and the
sample size is 20.
Figure 14.3 A One-Sample Example
Power versus Sample Size Plot
To see a plot of the relationship of Sample Size and Power, leave both Sample Size and Power empty in
the window and click Continue.
The plots in Figure 14.4, show a range of sample sizes for which the power varies from about 0.1 to about
0.95. The plot on the right in Figure 14.4 shows using the crosshair tool to illustrate the example in
Figure 14.3.
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Figure 14.4 A One-Sample Example Plot
Power versus Difference Plot
When only Sample Size is specified (Figure 14.5) and Difference to Detect and Power are empty, a plot
of Power by Difference appears, after clicking Continue.
Figure 14.5 Plot of Power by Difference to Detect for a Given Sample Size
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Sample Size and Power Animation for One Mean
Clicking the Animation Script button on the Sample Size and Power window for one mean shows an
interactive plot. This plot illustrates the effect that changing the sample size has on power. As an example of
using the Animation Script:
1. Select DOE > Sample Size and Power.
2. Click the One Sample Mean button in the Sample Size and Power Window.
3. Enter 2 for Std Dev.
4. Enter 1.5 as Difference to detect.
5. Enter 20 for Sample Size.
6. Leave Power blank.
The Sample Size and Power window appears as shown on the left of Figure 14.6.
7. Click Animation Script.
The initial animation plot shows two t-density curves. The blue curve shows the t-distribution when the
true mean is zero. The red curve shows the t-distribution when the true mean is 1.5, which is the
difference to be detected. The probability of committing a type II error (not detecting a difference when
there is a difference) is shaded blue on this plot. (This probability is often represented as β in the
literature.) Similarly, the probability of committing a type I error (deciding that the difference to detect
is significant when there is no difference) is shaded as the red areas under the red curve. (The red-shaded
areas under the curve are represented as α in the literature.)
Select and drag the square handles to see the changes in statistics based on the positions of the curves. To
change the values of Sample Size and Alpha, click on their values beneath the plot.
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Figure 14.6 Example of Animation Script to Illustrate Power
Two-Sample Means
The Sample Size and Power windows work similarly for one and two sample means; the Difference to
Detect is the difference between two means. The comparison is between two random samples instead of
one sample and a hypothesized mean.
For testing the difference between two means, the hypothesis supported is
H0 : μ – μ2 = D0
1
and the two-sided alternative is
Ha : μ – μ2 ≠ D0
1
where μ1 and μ2 are the two population means and D0 is the difference in the two means or the difference
to detect. It is assumed that the populations of interest are normally distributed and the true difference is
zero. Note that the power for this setting is the same as for the power when the null hypothesis is
H 0 : μ – μ 2 = 0 and the true difference is D0.
1
Suppose the standard deviation is 2 (as before) for both groups, the desired detectable difference between
the two means is 1.5, and the sample size is 30 (15 per group). To estimate the power for this example:
1. Select DOE > Sample Size and Power.
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2. Click the Two Sample Means button in the Sample Size and Power Window.
3. Leave Alpha as 0.05.
4. Enter 2 for Std Dev.
5. Leave Extra Parameters as 0.
6. Enter 1.5 as Difference to detect.
7. Enter 30 for Sample Size.
8. Leave Power blank.
9. Click Continue.
The Power is calculated as 0.51. (See the left window in Figure 14.7.) This means that you have a 51%
chance of detecting a significant difference between the two sample means when your significance level
is 0.05, the difference to detect is 1.5, and each sample size is 15.
Plot of Power by Sample Size
To have a greater power requires a larger sample. To find out how large, leave both Sample Size and
Power blank for this same example and click Continue. Figure 14.7 shows the resulting plot, with the
crosshair tool estimating that a sample size of about 78 is needed to obtain a power of 0.9.
Figure 14.7 Plot of Power by Sample Size to Detect for a Given Difference
k-Sample Means
Using the k-Sample Means option, you can compare up to 10 means. Consider a situation where 4 levels
of means are expected to be in the range of 10 to 13, the standard deviation is 0.9, and your sample size is
16.
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The hypothesis to be tested is:
H0: μ1=μ2=μ3=μ4 versus Ha: at least one mean is different
To determine the power:
1. Select DOE > Sample Size and Power.
2. Click the k Sample Means button in the Sample Size and Power Window.
3. Leave Alpha as 0.05.
4. Enter 0.9 for Std Dev.
5. Leave Extra Parameters as 0.
6. Enter 10, 11, 12, and 13 as the four levels of means.
7. Enter 16 for Sample Size.
8. Leave Power blank.
9. Click Continue.
The Power is calculated as 0.95. (See the left of Figure 14.8.) This means that there is a 95% chance of
detecting that at least one of the means is different when the significance level is 0.05, the population
means are 10, 11, 12, and 13, and the total sample size is 16.
If both Sample Size and Power are left blank for this example, the sample size and power calculations
produce the Power versus Sample Size curve. (See the right of Figure 14.8.) This confirms that a
sample size of 16 looks acceptable.
Notice that the difference in means is 2.236, calculated as square root of the sum of squared deviations
from the grand mean. In this case it is the square root of (–1.5)2+ (–0.5)2+(0.5)2+(1.5)2, which is the
square root of 5.
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Figure 14.8 Prospective Power for k-Means and Plot of Power by Sample Size
One Sample Standard Deviation
Use the One-Sample Standard Deviation option on the Sample Size and Power window (Figure 14.1) to
determine the sample size needed for detecting a change in the standard deviation of your data. The usual
purpose of this option is to compute a large enough sample size to guarantee that the risk of a type II error,
β, is small. (This is the probability of failing to reject the null hypothesis when it is false.)
In the Sample Size and Power window, specify:
Alpha is the significance level, usually 0.05. This implies a willingness to accept (if the true difference
between standard deviation and the hypothesized standard deviation is zero) that a significant
difference is incorrectly declared 5% of the time.
Hypothesized Standard Deviation is the hypothesized or baseline standard deviation to which the
sample standard deviation is compared.
can select Larger or Smaller from the menu to indicate the
direction of the change you want to detect.
Alternative Standard Deviation
is the smallest detectable difference (how small a difference you want to be able
to declare statistically significant). For single sample problems this is the difference between the
hypothesized value and the true value.
Difference to Detect
Sample Size is how many experimental units (runs, or samples) are involved in the experiment.
Power is the probability of declaring a significant result. It is the probability of rejecting the null
hypothesis when it is false.
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In the lower part of the window you enter two of the items and the Sample Size and Power calculation
determines the third.
Some examples in this chapter use engineering examples from the online manual of The National Institute
of Standards and Technology (NIST). You can access the NIST manual examples at http://www.itl.nist.gov/
div898/handbook.
One Sample Standard Deviation Example
One example from the NIST manual states a problem in terms of the variance and difference to detect. The
variance for resistivity measurements on a lot of silicon wafers is claimed to be 100 ohm-cm. The buyer is
unwilling to accept a shipment if the variance is greater than 155 ohm-cm for a particular lot (55 ohm-cm
above the baseline of 100 ohm-cm).
In the Sample Size and Power window, the One Sample Standard Deviation computations use the standard
deviation instead of the variance. The hypothesis to be tested is:
H0: σ = σ0, where σ0 is the hypothesized standard deviation. The true standard deviation is σ0 plus the
difference to detect.
In this example the hypothesized standard deviation, σ0, is 10 (the square root of 100) and σ is 12.4499
(the square root of 100 + 55 = 155). The difference to detect is 12.4499 – 10 = 2.4499.
You want to detect an increase in the standard deviation of 2.4499 for a standard deviation of 10, with an
alpha of 0.05 and a power of 0.99. To determine the necessary sample size:
1. Select DOE > Sample Size and Power.
2. Click the One Sample Standard Deviation button in the Sample Size and Power Window.
3. Leave Alpha as 0.05.
4. Enter 10 for Hypothesized Standard Deviation.
5. Select Larger for Alternate Standard Deviation.
6. Enter 2.4499 as Difference to Detect.
7. Enter 0.99 for Power.
8. Leave Sample Size blank. (See the left of Figure 14.9.)
9. Click Continue.
The Sample Size is calculated as 171. (See the right of Figure 14.9.) This result is the sample size
rounded up to the next whole number.
Note: Sometimes you want to detect a change to a smaller standard deviation. If you select Smaller from
the Alternative Standard Deviation menu, enter a negative amount in the Difference to Detect field.
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Figure 14.9 Window To Compare Single-Direction One-Sample Standard Deviation
One-Sample and Two-Sample Proportions
The Sample Size windows and computations to test sample sizes and power for proportions are similar to
those for testing means. You enter a true Proportion and choose an Alpha level. Then, for the one-sample
proportion case, enter the Sample Size and Null Proportion to obtain the Power. Or, enter the Power and
Null Proportion to obtain the Sample Size. Similarly, to obtain a value for Null Proportion, enter values
for Sample Size and Power. For the two-sample proportion case, either the two sample sizes or the desired
Power must be entered. (See Figure 14.10 and Figure 14.11.)
The sampling distribution for proportions is normal, but the computations to determine sample size and
test proportions use exact methods based on the binomial distribution. Exact methods are more reliable
since using the normal approximation to the binomial can provide erroneous results when small samples or
proportions are used. Exact power calculations are used in conjunction with a modified Wald test statistic
described in Agresti and Coull (1998).
One Sample Proportion
Clicking the One Sample Proportion option on the Sample Size and Power window yields a One
Proportion window. In this window, you can specify the alpha level and the true proportion. The sample
size, power, or the hypothesized proportion is calculated. If you supply two of these quantities, the third is
computed, or if you enter any one of the quantities, you see a plot of the other two.
For example, if you have a hypothesized proportion of defects, you can use the One Sample Proportion
window to estimate a large enough sample size to guarantee that the risk of accepting a false hypothesis (β)
is small. That is, you want to detect, with reasonable certainty, a difference in the proportion of defects.
For the one sample proportion, the hypothesis supported is
H0 : p = p0
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and the two-sided alternative is
Ha : p ≠ p0
where p is the population proportion and p0 is the null proportion to test against. Note that if you are
interested in testing whether the population proportion is greater than or less than the null proportion, you
use a one-sided test. The one-sided alternative is either
Ha : p < p0
or
Ha : p > p0
One-Sample Proportion Window Specifications
In the top portion of the Sample Size window, you can specify or enter values for:
Alpha
is the significance level of your test. The default value is 0.05.
Proportion
is the true proportion, which could be known or hypothesized. The default value is 0.1.
One-Sided or Two-Sided Specify either a one-sided or a two-sided test. The default setting is the
two-sided test.
In the bottom portion of the window, enter two of the following quantities to see the third, or a single
quantity to see a plot of the other two.
Null Proportion is the proportion to test against (p0) or is left blank for computation. The default
value is 0.2.
Sample Size is the sample size, or is left blank for computation. If Sample Size is left blank, then
values for Proportion and Null Proportion must be different.
Power is the desired power, or is left blank for computation.
One-Sample Proportion Example
As an example, suppose that an assembly line has a historical proportion of defects equal to 0.1, and you
want to know the power to detect that the proportion is different from 0.2, given an alpha level of 0.05
and a sample size of 100.
1. Select DOE > Sample Size and Power.
2. Click One Sample Proportion.
3. Leave Alpha as 0.05.
4. Leave 0.1 as the value for Proportion.
5. Accept the default option of Two-Sided. (A one-sided test is selected if you are interested in testing if
the proportion is either greater than or less than the Null Proportion.)
6. Leave 0.2 as the value for Null Proportion.
7. Enter 100 as the Sample Size.
8. Click Continue.
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The Power is calculated and is shown as approximately 0.7. See Figure 14.10.
Figure 14.10 Power and Sample Window for One-Sample Proportions
Two Sample Proportions
The Two Sample Proportions option computes the power or sample sizes needed to detect the difference
between two proportions, p1 and p2.
For the two sample proportion, the hypothesis supported is
H0 : p1 – p2 = D0
and the two-sided alternative is
Ha : p1 – p2 ≠ D0
where p1 and p2 are the population proportions from two populations, and D0 is the hypothesized
difference in proportions.
The one-sided alternative is either
Ha : ( p1 – p2 ) < D0
or
Ha : ( p1 – p2 ) > D0
Two Sample Proportion Window Specifications
Specifications for the Two Sample Proportions window include:
Alpha
is the significance level of your test. The default value is 0.05.
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Proportion 1 is the proportion for population 1, which could be known or hypothesized. The default
value is 0.5.
Proportion 2 is the proportion for population 2, which could be known or hypothesized. The default
value is 0.1.
One-Sided or Two-Sided Specify either a one-sided or a two-sided test. The default setting is the
two-sided test.
Null Difference in Proportion is the proportion difference (D0) to test against, or is left blank for
computation. The default value is 0.2.
Sample Size 1 is the sample size for population 1, or is left blank for computation.
Sample Size 2 is the sample size for population 2, or is left blank for computation.
Power is the desired power, or is left blank for computation.
If you enter any two of the following three quantities, the third quantity is computed:
•
Null Difference in Proportion
•
Sample Size 1 and Sample Size 2
•
Power
Example of Determining Sample Sizes with a Two-Sided Test
As an example, suppose you are responsible for two silicon wafer assembly lines. Based on the knowledge
from many runs, one of the assembly lines has a defect rate of 8%; the other line has a defect rate of 6%. You
want to know the sample size necessary to have 80% power to reject the null hypothesis of equal
proportions of defects for each line.
To estimate the necessary sample sizes for this example:
1. Select DOE > Sample Size and Power.
2. Click Two Sample Proportions.
3. Accept the default value of Alpha as 0.05.
4. Enter 0.08 for Proportion 1.
5. Enter 0.06 for Proportion 2.
6. Accept the default option of Two-Sided.
7. Enter 0.0 for Null Difference in Proportion.
8. Enter 0.8 for Power.
9. Leave Sample Size 1 and Sample Size 2 blank.
The completed window appears as the left window in Figure 14.11.
10. Click Continue.
The Sample Size window shows sample sizes of 2554. (See the figure on the right in Figure 14.11.)
Testing for a one-sided test is conducted similarly. Simply select the One-Sided option.
Note: The computations for finding two sample sizes can take a little time so be patient.
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Figure 14.11 Difference Between Two Proportions for a Two-Sided Test
Example of Determining Power with Two Sample Proportions Using a One-Sided Test
Suppose you want to compare the effectiveness of two chemical additives. The standard additive is known to
be 50% effective in preventing cracking in the final product. The new additive is assumed to be 60%
effective. You plan on conducting a study, randomly assigning parts to the two groups. You have 800 parts
available to use for the study (400 parts for each additive). Your objective is to determine the power of your
test, given a null difference in proportions of 0.01 and an alpha level of 0.05. Because you are interested in
testing that the difference in proportions is greater than 0.01, you use a one-sided test.
1. Select DOE > Sample Size and Power.
2. Click Two Sample Proportions.
3. Accept the default value of Alpha as 0.05.
4. Enter 0.6 for Proportion 1.
5. Enter 0.5 for Proportion 2.
6. Select One-Sided.
7. Enter 0.01 as the Null Difference in Proportion.
8. Enter 400 for Sample Size 1.
9. Enter 400 for Sample Size 2.
10. Leave Power blank.
Figure 14.12 (left side) shows the completed Two Proportions window.
11. Click Continue.
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Figure 14.12 (right side) shows the Two Proportions windows with the estimated Power calculation of
0.82.
Figure 14.12 Difference Between Two Proportions for a One-Sided Test
You conclude that there is about an 82% chance of rejecting the null hypothesis at the 0.05 level of
significance, given that the sample sizes for the two groups are each 400.
Counts per Unit
You can use the Counts per Unit option from the Sample Size and Power window (Figure 14.1) to calculate
the sample size needed when you measure more than one defect per unit. A unit can be an area and the
counts can be fractions or large numbers.
Although the number of defects observed in an area of a given size is often assumed to have a Poisson
distribution, it is understood that the area and count are large enough to support a normal approximation.
Questions of interest are:
•
Is the defect density within prescribed limits?
•
Is the defect density greater than or less than a prescribed limit?
In the Counts per Unit window, options include:
Alpha
is the significance level of your test. The default value is 0.05.
Baseline Count per Unit is the number of targeted defects per unit. The default value is 0.1.
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Difference to detect is the smallest detectable difference to test against and is specified in defects per
unit, or is left blank for computation.
Sample Size is the sample size, or is left blank for computation.
Power is the desired power, or is left blank for computation.
In the Counts per Unit window, enter Alpha and the Baseline Count per Unit. Then enter two of the
remaining fields to see the calculation of the third. The test is for a one-sided (one-tailed) change. Enter the
Difference to Detect in terms of the Baseline Count per Unit (defects per unit). The computed sample
size is expressed as the number of units, rounded to the next whole number.
Counts per Unit Example
As an example, consider a wafer manufacturing process with a target of 4 defects per wafer. You want to
verify that a new process meets that target within a difference of 1 defect per wafer with a significance level
of 0.05. In the Counts per Unit window:
1. Leave Alpha as 0.05 (the chance of failing the test if the new process is as good as the target).
2. Enter 4 as the Baseline Counts per Unit, indicating the target of 4 defects per wafer.
3. Enter 1 as the Difference to detect.
4. Enter a power of 0.9, which is the chance of detecting a change larger than 1 (5 defects per wafer). In
this type of situation, alpha is sometimes called the producer’s risk and beta is called the consumer’s risk.
5. Click Continue to see the results in Figure 14.13, showing a computed sample size of 38 (rounded to
the next whole number).
The process meets the target if there are less than 190 defects (5 defects per wafer in a sample of 38 wafers).
Figure 14.13 Window For Counts Per Unit Example
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Sigma Quality Level
The Sigma Quality Level feature is a simple statistic that puts a given defect rate on a “six-sigma” scale. For
example, on a scale of one million opportunities, 3.397 defects result in a six-sigma process. The
computation that gives the Sigma Quality Level statistic is
Sigma Quality Level = NormalQuantile(1 – defects/opportunities) + 1.5
Two of three quantities can be entered to determine the Sigma Quality Level statistic in the Sample Size and
Power window:
•
Number of Defects
•
Number of Opportunities
•
Sigma Quality Level
When you click Continue, the sigma quality calculator computes the missing quantity.
Sigma Quality Level Example
As an example, use the Sample Size and Power feature to compute the Sigma Quality Level for 50 defects in
1,000,000 opportunities:
1. Select DOE > Sample Size and Power.
2. Click the Sigma Quality Level button.
3. Enter 50 for the Number of Defects.
4. Enter 1000000 as the Number of Opportunities. (See window to the left in Figure 14.14.)
5. Click Continue.
The results are a Sigma Quality Level of 5.39 defects in 1,000,000 opportunities. (See right window in
Figure 14.14.)
Figure 14.14 Sigma Quality Level Example 1
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Number of Defects Computation Example
If you want to know how many defects reduce the Sigma Quality Level to “six-sigma” for 1,000,000
opportunities:
1. Select DOE > Sample Size and Power.
2. Click the Sigma Quality Level button.
3. Enter 6 as Sigma Quality Level.
4. Enter 1000000 as the Number of Opportunities. (See left window in Figure 14.14.)
5. Leave Number of Defects blank.
6. Click Continue.
The computation shows that the Number of Defects cannot be more than approximately 3.4. (See
right window in Figure 14.15.)
Figure 14.15 Sigma Quality Level Example 2
Reliability Test Plan and Demonstration
You can compute required sample sizes for reliability tests and reliability demonstrations using the
Reliability Test Plan and Reliability Demonstration features.
Reliability Test Plan
The Reliability Test Plan feature computes required sample sizes, censor times, or precision, for estimating
failure times and failure probabilities.
To launch the Reliability Test Plan calculator, select DOE > Sample Size and Power, and then select
Reliability Test Plan. Figure 14.16 shows the Reliability Test Plan window.
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Figure 14.16 Reliability Test Plan Window
The Reliability Test Plan window has the following options:
Alpha is the significance level. It is also 1 minus the confidence level.
Distribution is the assumed failure distribution, with the associated parameters.
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Precision Measure is the precision measure. In the following definitions, U and L correspond to the
upper and lower confidence limits of the quantity being estimated (either a time or failure
probability), and T corresponds to the true time or probability for the specified distribution.
Interval Ratio is sqrt(U/L), the square root of the ratio of the upper and lower limits.
Two-sided Interval Absolute Width is U-L, the difference of the upper and lower limits.
Lower One-sided Interval Absolute Width is T-L, the true value minus the lower limit.
Two-sided Interval Relative Width is (U-L)/T, the difference between the upper and lower limits,
divided by the true value.
Lower One-sided Interval Relative Width is (T-L)/T, the difference between the true value and the
lower limit, divided by the true value.
Objective is the objective of the study. The objective can be one of the following two:
•
estimate the time associated with a specific probability of failure.
•
estimate the probability of failure at a specific time.
is a plot of the CDF of the specified distribution. When estimating a time, the true time
associated with the specified probability is written on the plot. When estimating a failure probability,
the true probability associated with the specified time is written on the plot.
CDF Plot
Sample Size is the required number of units to include in the reliability test.
Censor Time
is the amount of time to run the reliability test.
Precision is the level of precision. This value corresponds to the Precision Measure chosen above.
gives the approximate variances and covariance for
the location and scale parameters of the distribution.
Large-sample approximate covariance matrix
Continue click here to make the calculations.
Back click here to go back to the Power and Sample Size window.
After the Continue button is clicked, two additional statistics are shown:
Expected number of failures is the expected number of failures for the specified reliability test.
Probability of fewer than 3 failures is the probability that the specified reliability test will result in
fewer than three failures. This is important because a minimum of three failures is required to
reliably estimate the parameters of the failure distribution. With only one or two failures, the
estimates are unstable. If this probability is large, you risk not being able to achieve enough failures
to reliably estimate the distribution parameters, and you should consider changing the test plan.
Increasing the sample size or censor time are two ways of lowering the probability of fewer than three
failures.
Example
A company has developed a new product and wants to know the required sample size to estimate the time
till 20% of units fail, with a two-sided absolute precision of 200 hours. In other words, when a confidence
interval is created for the estimated time, the difference between the upper and lower limits needs to be
approximately 200 hours. The company can run the experiment for 2500 hours. Additionally, from studies
done on similar products, they believe the failure distribution to be approximately Weibull (2000, 3).
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To compute the required sample size, do the following steps:
1. Select DOE > Sample Size and Power.
2. Select Reliability Test Plan.
3. Select Weibull from the Distribution list.
4. Enter 2000 for the Weibull α parameter.
5. Enter 3 for the Weibull β parameter.
6. Select Two-sided Interval Absolute Width from the Precision Measure list.
7. Select Estimate time associated with specified failure probability.
8. Enter 0.2 for p.
9. Enter 2500 for Censor Time.
10. Enter 200 for Precision.
11. Click Continue. Figure 14.17 shows the results.
Figure 14.17 Reliability Test Plan Results
The required sample size is 217 units if the company wants to estimate the time till 20% failures with a
precision of 200 hours. The probability of fewer than 3 failures is small, so the experiment will likely result
in enough failures to reliably estimate the distribution parameters.
Reliability Demonstration
A reliability demonstration consists of testing a specified number of units for a specified period of time. If
fewer than k units fail, you pass the demonstration, and conclude that the product reliability meets or
exceeds a reliability standard.
The Reliability Demonstration feature computes required sample sizes and experimental run-times for
demonstrating that a product meets or exceeds a specified reliability standard.
To launch the Reliability Demonstration calculator, select DOE > Sample Size and Power, and then select
Reliability Demonstration. Figure 14.18 shows the Reliability Demonstration window.
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Figure 14.18 Reliability Demonstration Window
The Reliability Demonstration window has the following options:
Alpha is the alpha level.
Distribution is the assumed failure distribution. After selecting a distribution, specify the associated
scale parameter in the text field under the Distribution menu.
is the maximum number of failures you want to allow during the
demonstration. If we observe this many failures or fewer, then we say we passed the demonstration.
Max Failures Tolerated
Time is the time component of the reliability standard you want to meet.
Probability of Surviving is the probability component of the reliability standard you want to meet.
Time of Demonstration is the required time for the demonstration.
Number of Units Tested is the required number of units for the demonstration.
Continue click here to make the calculations.
Back click here to go back to the Power and Sample Size window.
After the Continue button is clicked, a plot appears (see Figure 14.19).
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Figure 14.19 Reliability Demonstration Plot
The true probability of a unit surviving to the specified time is unknown. The Y axis of the plot gives the
probability of passing the demonstration (concluding the true reliability meets or exceeds the standard) as a
function of the true probability of a unit surviving to the standard time. Notice the line is increasing,
meaning that the further the truth is above the standard, the more likely you are to detect the difference.
Example
A company wants to get the required sample size for assessing the reliability of a new product against an
historical reliability standard of 90% survival after 1000 hours. From prior studies on similar products, it is
believed that the failure distribution is Weibull, with a β parameter of 3. The company can afford to run the
demonstration for 800 hours, and wants the experiment to result in no more than 2 failures.
To compute the required sample size, do the following steps:
1. Select DOE > Sample Size and Power.
2. Select Reliability Demonstration.
3. Select Weibull from the Distribution list.
4. Enter 3 for the Weibull β.
5. Enter 2 for Max Failures Tolerated.
6. Enter 1000 for Time.
7. Enter 0.9 for Probability of Surviving.
8. Enter 800 for Time of Demonstration.
9. Click Continue. Figure 14.20 shows the results.
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Chapter 14
Figure 14.20 Reliability Demonstration Results
The company needs to run 118 units in the demonstration. Furthermore, if they observe 2 or fewer failures
by 800 hours, we can conclude that the new product reliability is at least as reliable as the standard.
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Index
Design of Experiments
A
B
ABCD designs 168
acceptable values See lower limits and upper limits
Actual-by-Predicted plots 125
adding
center points in augment designs 275, 277
factors 40, 67
linear constraints 164
runs and terms in augment designs 279
additional runs 78, 120, 140, 152
advanced options (nonlinear designer) 245
algorithms, coordinate exchange 99
alias matrix 76
explanation 18
showing 18
aliasing effects 115
Alias-optimal designs 60, 88
Alpha 288
animation scripts 289
Anova reports 171
assigning importance (of responses) 65, 109, 254
assigning importance of responses 84
attribute, discrete choice designs 181
augment nonlinear design table 244
Augmented Designs 259–284
add centerpoints 272, 275
add new runs and terms 272, 278
adding axial points 272, 277
disallow factor combinations 284
D-optimal 278
extending experiments 271
foldover 272, 276
modify D- or I- optimality criterion 282
replicate 271–272
saved design (X) matrix 282
select number of random starts 283
special commands 281
specify sphere radius value 283
axial
points 127
scaling, central composite designs 138
balanced designs 71
Bayesian D-optimal designs 31, 60–61
Bayesian I-optimal designs 60, 62
Big Class.jmp 51
block sizes 27
Borehole Factors.jmp 212
Borehole Sphere Packing.jmp 213
Bounce Data.jmp 129
Bounce Factors.jmp 129–130
Box-Behnken designs 127, 129, 138
See also Response Surface designs
Byrne Taguchi Factors.jmp 250–251
C
categorical factors 68, 111
CCD See central composite designs
center points
augment designs 275, 277
central composite designs 127
response surface designs 127
simplex centroid designs 158
central composite designs 127, 138
See also response surface designs
centroid points 163
Chakravarty 114
Chemical Kinetics.jmp 231
chemical mixture, examples 171
choice designs
example 179
purpose 179
choice set, discrete choice designs 181
coding, column property 94
coefficients, relative variance 75
column properties
coding 94
constrained state 85
design roles 84, 97
factors for mixture experiments 95
high and low values 94
318
response limits 96
combinations, disallowing 90, 283
CONAEV method 167
confounding 116
resolution numbers 112
confounding pattern 76, 116
constraints
adding 164
disallowing combinations 90, 283
entering 85
linear 166
loading 85
saving 85
continuous factors 67, 111
control factors 247, 249
CONVRT method 167
coordinate exchange algorithms 99
Corn.jmp 241
Cotter designs 114
counts per unit (power and sample size) 303
covariate factors 51
creating factors tables 85
criterion, optimality 88, 282
custom design commands 83
custom designs
data tables 79
Design Generation panel 7, 71
factors, defining 65
flexible block size 27
how they work 99
models, describing 69
random block design 79
screening 17
special commands 83
split plot 80
split-split plot 81
strip plot 82
D
defaults
number of random starts 88
defects 303
D-efficiencies 43
describing models 69, 158
design
matrix table properties 87
Moments Matrix 87
resolutions 112
roles 84
Index
Save X Matrix 87
design diagnostics table 206
Design of Experiments. See DOE
design roles 97
designs
ABCD 168
aberration 112
augment 271
balanced 71
Bayesian D-optimal 31, 61
Bayesian I-optimal 62
Box-Behnken 127, 129, 138
central composite 127
discrete choice 179
foldover 276
fractional factorial 112
full factorial 101, 143, 151
full factorials 112
Gaussian process IMSE optimal space filling 211
I-optimal 61
Latin hypercube space filling design 200–204
maximum entropy space filling 209
minimum aberration 112
minimum potential space filling 207
mixed-level 113
mixture 155, 171
nonlinear 231
orthogonal
screening designs 112
screening experiments 155
surface designs 138
orthogonal arrays 113
Plackett-Burman 113
random block 79
replicating 272–273
response surface 127
saturated 71
screening 101
simplex centroid 158
simplex centroids 158
simplex lattice 161
space filling 197–216
sphere packing space filling 197
split plot 80
split-split plot 81
strip plot 82
uniform precision 138
uniform space filling 204
desirability
functions 133, 253
319
Index
maximizing 133
values 66, 110
desirability traces 133
desirability values, as column properties 96
determinant formula 89
diagnostics for custom designs 86, 282
Difference to Detect option 288–289, 293, 296
disallowing combinations 90, 283
discrete choice designs
analysis 186, 190
attribute 181
attributes panel 183
choice set 181
definition 179
design generation panel 184
dialog 183
example 181, 183–184
prior information 188
profile 181
survey 181
DOE (Design of Experiments)
coding 94
simple examples 103
DOE Sphere Radius 283
Donev Mixture Factors.jmp 44
D-optimal
augmentation 278
designs 31, 60–61
optimality criteria 88, 282
E
effect
aliasing 116
eigenvalue 132
eigenvector 132
nonestimable 112
orthogonal 139, 247
sparsity 101, 113–114
efficiencies
D, G, and A 77
eigenvalue of effect 132
eigenvector of effect 132
equivalent solutions 99
error standard deviation 288
error variance 36
excluding factor combinations 90
extra parameters 288
extreme vertices 163
algorithms 167
finding optimal subsets 164
range constraints 164
F
Factor Changes 84
factor combinations, disallowing 90
Factor Profiling option 132, 134, 150, 175, 253
factorial designs
fractional 112
full 101, 112, 143, 151
three level 113
factors
adding 40, 67
categorical 68, 111
column 98
continuous 67, 111
control factors 247, 249
covariate 51
for mixture experiments 95
key factors 101
loading 85
nonmixture 44
saving 84
tables, creating 85
false negatives 114
finding optimal subsets (extreme vertices) 164
fitting mixture designs 170
fixed covariate factors 51
fixed covariates example 51
folding the calculation 171
foldover designs 276
fraction of design space plot 74
fractional factorial designs 112
Full Factorial Designs 143–153
design generation panel 71
examples 145
full factorial designs 112, 143
functions, desirability 133, 253
G
Gaussian process IMSE optimal design 195, 211
global optimum 89
goal types 65, 109
matching targets 66, 110
minimizing and maximizing 66, 110
Group New Runs into Separates Block option 275
320
Index
H
M
hidden intercepts 171
hyperspheres 90, 283
matching target goals 65–66, 96, 109–110
matrix, alias 76
matrix, design 87
maximize responses 65, 109
maximizing
desirability 133
goals 66, 110
maximum entropy space filling design 195, 209
means, one- and two-sample (power and sample
size) 287
methods
CONAEV 167
minimize responses 65, 109
minimizing goals 66, 110
minimum aberration designs 112
minimum potential design 207
minimum potential space filling design 195
mixed-level designs 113
Mixture Designs 155–179
ABCD design 168
compared to screening designs 155
definition 155
examples 171–174
extreme vertices 163
fitting 170
linear constraints 166
optimal 157
optimal subsets 164
response surfaces 175
simplex centroids 158
simplex lattice 161
ternary plots 168
with nonmixture factors 44
mixture designs
examples 171
mixture experiments (column property) 95
mixture, column properties 95
Model script
Model Specification dialog 123, 130
models
custom design 69
describing 69, 158
I
identifying key factors 101
importance of responses 65, 84, 109, 254
entering 96
industrial experimentation 179
inner arrays, inner designs 247, 249
Inscribe option 139
interactions 114
high-order 112
intercepts, hidden 171
I-optimal
designs 60–62
optimality criteria 88, 282
I-optimal designs 61
J
JSL (JMP Scripting Language)
animation scripts 289
augmented designs 265
sphere radius 283
K
Keep the Same 78
k-Sample Means (power and sample size) 294
L
L18 Chakravarty 114
L18 Hunter 114
L18 John 113
L36 114
L9-Taguchi 250
Label column 120, 140
larger–the–better formulas (LTB) 251
Latin hypercube space filling design 200–204
limits, responses 96
linear constraints 164, 166
loading
constraints 85
factors 85
responses 85
local optimum 89
N
N factors, adding 67
N Monte Carlo Spheres (nonlinear design
option) 245
N Responses 65
321
Index
N Responses button 109
no-intercept model 95
noise factors 247, 249
nonestimable effect 112
nonlinear design
advanced options 245
augment table 236, 244
create using prior parameter estimates 233
creating 241
creating with no prior data 237
distribution of parameters 238
fitting to find prior parameter estimates 231
introduction 229
launch dialog 234, 241–242
number of Monte Carlo samples 235
number of starts 235
overlay plot of design points 239
using prior parameter estimates 233
Nonlinear Designs 229–246
advanced options 245
creating 241
examples 231
nonmixture factors in mixture designs 44
Number of Monte Carlo Samples (nonlinear design
option) 245
number of runs 15, 71
screening designs 112
number of starts 88, 283
O
On Face option 139
one-sample and two-sample means (power and sample
size) 287
one-sample and two-sample proportion (power and
sample size) 298
one-sample variance (power and sample size) 296
optimal determinant formula 89
optimal subsets (mixture designs) 164
Optimality Criterion 88, 282
optimality criterion 88
order of runs 78, 120, 140, 152, 159, 164, 168
orthogonal array designs 113, 247
orthogonal designs
screening designs 112
screening experiments 155
surface designs 138
Orthogonal option 139
outer arrays, outer designs 247, 249
P
parameters, extra 288
Pattern column 105, 120, 140, 153
patterns, confounding 116
performance statistics 249
Placket-Burman design
creating 120
Plackett-Burman designs 113
Plastifactors.jmp 171
plots
Actual-by-Predicted 125
Scatterplot 3D 136
ternary 155, 168
points
axial 127
center See center points
centroid 163
potential terms (DOE) 31
power and sample size calculations 285–306
analyses 287
animation 292
counts per unit 303
in statistical tests on means 296
k-sample means 294
one-sample and two sample proportions 298
one-sample mean 288–289
one-sample variance 296, 298, 300
Sigma quality level 305
two-sample means 293
prediction
profilers 35, 38
traces 132
variances 36, 138
prediction formulas, saving 174
prediction profiler 132
prediction variance profile plot 35
prediction variance surface plot 75
primary terms (Bayesian D-optimal design) 31
profile, discrete choice designs 181
profilers
mixture response surface 175
prediction profilers 38, 132
prediction variance plot 35
prediction Variance profiler 35
properties, columns 93
proportions (power and sample size) 298
prospective power and sample size 285
prospective power and sample size calculations (see
power and sample size) 306
pseudocomponent (mixture column property) 95
322
pure error 275
Q
quadratic model 34–76
R
radius, sphere 90, 283
random block designs 79
random seed, setting 85
random starts 88
Randomize within Blocks 78
randomizing runs 78
range constraints 163
Reaction Kinetics Start.jmp 237
Reaction Kinetics.jmp 239
Reactor 8 Runs.jmp 261
Reactor Factors.jmp 145–146
Reactor Response.jmp 145
regressor columns 114
relative proportions See mixture designs
relative variance of coefficients 75
relative variance of coefficients and power table 30
replicating
designs 273
replicating designs 272
reponses, loading 85
requesting additional runs 78, 120, 140, 152
rescaling designs 139
resolution numbers 112
resolutions of designs 112
response limits, column property 96
Response Surface Designs 127–141
examples 129–136
introduction 136
purpose 127
reports 131
with blocking factors 40
with categorical factors 40
response surface effects 171
Response Surface Methodology (RSM) 136
response surfaces
mixture designs 175
responses
custom designs 65, 109, 137
desirability values 66, 110
goals 65, 109
goals, desirability functions 133
lower limits 66, 110
saving 84
Index
upper limits 66, 110
responses, multiple 96
RMSE 125, 149
robust engineering 247
roles, design 97
Rotatable option 139
RSM (Response Surface Methodology) 136
runs
additional 78, 120, 140, 152
order they appear in table 78, 120, 140, 152, 159,
164, 168
requesting additional 78, 120, 140, 152
screening designs 112
S
sample means (power and sample size) 287
sample sizes
example comparing single-direction one-sample
variances 296, 298
example with counts per unit 303
one and two sample means 289
prospective power analysis 296
screening designs 143
saturated designs 71
saving
constraints 85
factors 84
prediction formulas 174
responses 84
X Matrix 87, 282
scaling
axial 138
designs 139
Scatterplot 3D
Box-Behnken designs 136
Scheffé polynomial 170
screening designs 101
custom designs 17
design types 111
dialogs 106, 120
examples 105–140
scripts
animation 289
generating the analysis model 123, 130
Model script See Model table property
scripting See JSL
Set Random Seed command 85
sigma quality level (power and sample size) 305
signal factors 247
323
Index
signal-to-noise ratios 247
simplex 155
centroid designs 158
lattice designs 161
single-sample means (power and sample size) 289
solutions, equivalent 99
Sort Left to Right options 78
Sort Right to Left option 78
Space Filling Designs 197–216
borehole problem, sphere packing example 212
Gaussian process IMSE optimal 195, 211
Latin hypercube 195, 200–204
maximum entropy 195, 209
minimum potential 195, 207
sphere packing 195, 197
uniform 195, 204
sparsity, effect 101, 113–114
sphere packing design 195, 197
sphere radius 90, 283
split plot designs 68, 80
split-split plot designs 81
star points 127
starts, number of 88, 283
starts, random 88
statistics, performance 249
strip plot designs 82
subsets, finding optimal 164
supersaturated designs 23
surface designs See response surface designs
survey, discrete choice designs 181
T
tables
factors table, creating 85
making in custom designs 79
Taguchi designs 247–254
description 247
methods 247
target values 66, 96, 110
ternary plots 155, 168
three-dimensional scatterplot
Box-Behnken designs 136
traces, desirability 133
trade-off in screening designs 112
tutorial examples
augment designs 261–271
custom design 1–13
DOE 103
flexible block size 27
full factorial designs 145
mixture designs 171–174
response surface designs 129, 134, 136
screening designs 105
two-level categorical 104
two-level fractional factorials 112
two-level full factorials 112
two-sample and one-sample means 293
two-sample means (power and sample size) 287
two-sample proportion (power and sample size) 298
U
Uniform (space filling design) 195
uniform (space filling design) 204
uniform precision designs 138
User Defined option 139
V
values
high and low for columns (in DOE) 94
target 66, 110
variance of prediction 139
variance, error and prediction 36
vertices
extreme 163
extreme, finding optimal subsets 164
W-Z
weight, importance of response 65, 84, 97, 109, 254
whole model test 171
X Matrix, saving 87, 282
XVERT method 167
324
Index
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