"The real voyage of discovery consists not in seeking new landscapes, but in having new eyes." Marcel Proust
D
ESIGN
OF
E
XPERIMENTS
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First printing, January 2000
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Credits and Acknowledgments ... v
Chapter 1 Design of Experiments (DOE) ... 1
DOE Choices ...3
A Simple DOE Example ...6
The DOE Dialog...7
The JMP DOE Data Table ... 11
DOE Utility Commands ... 12
Chapter 2 Introduction to Custom Designs ... 17
Getting Started ... 19
Modify a Design Interactively ... 23
Introducing the Prediction Profiler ... 24
Routine Screening Using Custom Designs ... 29
How the Custom Designer Works ... 32
Chapter 3 Custom Design: Beyond the Textbook ... 33
Custom Situations ... 35
Flexible Block Sizes ... 36
Response Surface Model with Categorical Factors ... 38
Fixed Covariate Factors ... 43
Mixtures with Nonmixture Factors ... 45
Factor Constraints ... 48
Chapter 4 Screening Designs ... 53
Screening Design Types ... 55
A Screening Example ... 58
Loading and Saving Responses and Factors (Optional)... 66
A Simple Effect Screening Analysis ... 67
Chapter 5 Response Surface Designs ... 69
Response Surface Designs ... 71
Taguchi Design Example ... 99
Analyze the Byrne-Taguchi Data ... 103
Chapter 8 Mixture Designs ... 105
The Mixture Design Dialog ... 107
Mixture Designs ... 108
Extreme Vertices Design for Constrained Factors ... 113
Adding Linear Constraints to Mixture Designs... 114
Ternary and Tetrary Plots ... 115
Fitting Mixture Designs... 116
Chemical Mixture Example ... 118
Plotting a Mixture Response Surface ... 119
Chapter 9 Augmented Designs ... 121
The Augment Design Interface ... 123
The Reactor Example Re-visited ... 126
Chapter 10 Prosective Power and Sample Size ... 135
Prospective Power Analysis ... 137
Launch the Sample Size and Power facility ... 137
References ... 145
Credits
JMP was conceived and started by John Sall. Design and development was done by John Sall, Katherine Ng, Michael Hecht, Richard Potter, Brian Corcoran, Annie Dudley, Bradley Jones, Xan Gregg, Eric Wasserman, Charles Soper, and Kevin Hardman. Ann Lehman coordinated product development, production, quality assurance, and documentation. In the SAS Institute Technical Support division, Ryan Gilmore, Maureen Hayes, Craig Devault, Toby Trott, and Peter Ruzza provide technical support and conducted test site administration. Statistical technical support is provided by Duane Hayes, Kathleen Kiernan, and Annette Sanders. Nicole Jones and Jianfeng Ding provide ongoing quality assurance. Additional testing and technical support is done by Kyoko Takenaka and Noriki Inoue from SAS Japan.
Sales and marketing is headed by Colleen Jenkins and includes Dianne Nobles, William Gjertsen, Chris Brown, Carolyn Durst, Mendy Clayton, Bob Hickey, David Sipple, Barbara Droschak, Lisa Rohloff, Bob McCall, Chuck Boiler, Nick Zagone and Bonnie Rigo. Additional support is provided by Kathy Jablonski and Jean Davis.
The JMP manuals were written by Ann Lehman, John Sall, Bradley Jones, and Erin Vang with contributions from Annie Dudley and Brian Corcoran. Editing was done by Lee Bumgarner, Brad Kellam, and Lee Creighton, design and production by Creative Solutions. Lee Creighton
implemented the online help system and online documentation with contribution from Timothy Christensen.
Special thanks to Jim Goodnight for supporting a product outside the usual traditions and to Dave DeLong for valuable ideas and advice on statistical and computational matters.
Thanks also to Robert N. Rodriguez, Ying So, Duane Hayes, Mark Bailey, Donna Woodward, and Mike Stockstill for statistical editorial support and statistical QC advice. Thanks to Georges Guirguis, Warren Sarle, Randall Tobias, Gordon Johnston, Ying So, Wolfgang Hartmann, Russell Wolfinger, and Warren Kuhfeld for statistical R&D support.
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, Al Best, Stan Young, Robert Muenchen, Lenore Herzenberg, Larry Sue, Ramon Leon, Tom Lange, Homer Hegedus, Skip Weed, Michael Emptage, Pat Spagan, John Frei, Paul Wenz, Mike Bowen, Lori Gates, Georgia Morgan, David Coleman, Linda Blazek, Michael Friendly, Joe Hockman, Frank Shen, J.H. Goodman, David Ikle, Lou Valente, Robert Mee, Barry Hembree, Dan Obermiller, Lynn Vanatta, and Kris Ghosh. Also, we thank Dick DeVeaux, Gray McQuarrie, Robert Stein, George Fraction, Al Fulmer, Cary Tuckfield, Ron Thisted, Donna Fulenwider, Nancy McDermott, Veronica Czitrom, Tom Johnson, Avigdor Cahaner, and Andy Mauromoustakos.
We also thank the following individuals for expert advice in their statistical specialties: R. Hocking and P. Spector for advice on effective hypotheses; 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 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; Clay Thompson for advice on contour plotting algorithms.
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.
Past Support
Many people were important in the evolution of JMP. Special thanks Jeffrey Perkinson, Mary Cole, Kristin Nauta, Aaron Walker, Ike Walker, Eric Gjertsen, Dave Tilley, Curt Yeo, Patricia Moell, Patrice Cherry, Mike Pezzoni, Mary Ann Hansen, Ruth Lee, Russell Gardner, and Patsy Poole. SAS Institute quality assurance by Jeanne Martin, Fouad Younan, Jeff Schrilla, Jack Berry, Kari Richardson, Jim Borek, Kay Bydalek, and Frank Lassiter. Additional testing for Versions 3 and 4 was done by Li Yang, Brenda Sun, Katrina Hauser, and Andrea Ritter. Thanks to Walt Martin for Postscript support in documentation production.
Also thanks to Jenny Kendall, Elizabeth Shaw, and John Hansen, Eddie Routten, David Schlotzhauer, John Boling, and James Mulherin, Thanks to Steve Shack, Greg Weier, and Maura Stokes for testing Version 1. Additional editorial support was given by Marsha Russo, Dea Zullo, and Dee Stribling.
Thanks for support from Morgan Wise, Frederick Dalleska, Stuart Janis, Charles Shipp, Harold Gugel, Jim Winters, Matthew Lay, Tim Rey, Rubin Gabriel, Brian Ruff, William Lisowski, David Morganstein, Tom Esposito, Susan West, Chris Fehily, Dan Chilko, Jim Shook, Bud Martin, Hal Queen, Ken Bodner, Rick Blahunka, Dana C. Aultman, and William Fehlner.
Technology License Notices
JMP software contains portions of the file translation library of MacLinkPlus, a product of DataViz Inc., 55 Corporate Drive, Trumbull, CT 06611, (203) 268-0030.
JMP for the Power Macintosh was compiled and built using the CodeWarrior C compiler from MetroWorks Inc.
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1 JMP DOE
Chapter 1
Design of Experiments (DOE)
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 ofexperiments. A trial-and-error search for the vital few factors that most affect quality is costly and time consuming. Fortunately, researchers in the field of experimental design have invented powerful and elegant ways of making the search process fast and effective. The DOE platform in JMP is a tool for creating designed experiments and saving them in a JMP data table. JMP supports two ways to make a designed experiment.
The first way is to let JMP build a new design that both matches the description of your engineering problem and remains within your budget for time and material. Use the Custom and Augment designers to create these tailor-made designs.
The second way is to choose a pre-formulated design from a list of designs. JMP groups these lists of designs into several types differing by problem type and research goal. For example, the Screening designer provides a list of designs suitable for doing screening experiments. TheResponse Surface, Taguchi, and Mixture designers also involve choosing the design you want from a list.
Each of these two approaches has its advantages. Custom designs are general purpose and flexible. Custom designs are also fine for routine factor screening or response optimization. For problems that are not textbook, custom designs are the only alternative. On the other hand, when you know exactly the design you want, it is convenient to select it from a list. This chapter briefly describes each of the design types, shows how to use the DOE dialog to enter your factors and responses, and points out the special features of a JMP design data table.
Chapter 1
Contents
DOE Choices ...3
Custom Design ...4
Screening Design ...4
Response Surface Design ...4
Full Factorial Design ...5
Taguchi Arrays ...5
Mixture Design ...5
Augment Design ...5
Sample Size and Power ...6
A Simple DOE Example...6
The DOE Dialog ...7
Entering Responses ...8
Entering Factors...9
Select a Design Type ... 10
Modify a Design ... 10
The JMP DOE Data Table ... 11
1 JMP DOE
DOE Choices
TheDOE platform in JMP is an environment for describing the factors, responses and other specifications, creating a designed experiment, and saving it in a JMP table. When you select the DOE tab on the JMP Starter window, you see the list of design command buttons shown on the tab page as in Figure 1.1. Alternatively, you can choose commands from the DOE main menu shown to the right. Figure 1.1 The DOE JMP Starter Tab
Note that the DOE tab in the JMP Starter window tells what each command does. The specific design types are described briefly in the next sections, and covered in detail by the following chapters in this book.
Custom Design
Custom designs give the most flexibility of all design choices. The Custom designer gives you the following options:
❿ continuous factors
❿ categorical factors with arbitrary numbers of levels
❿ mixture ingredients
❿ covariates (factors that already have unchangable values and design around them)
❿ blocking with arbitrary numbers of runs per block
❿ interaction terms and polynomial terms for continuous factors
❿ inequality constraints on the factors
❿ choice of number of experimental runs to do, which can be any number greater than or equal to the number of terms in the model.
After specifying all your requirements, this design solution generates a D-optimal design for those requirements.
Screening Design
As the name suggests, screening experiments “separate the wheat from the chaff.” The
wheat is the group of factors having a significant influence on the response. The chaff is the
rest of the factors. Typically screening experiments involve many factors.
The Screening designer supplies a list of popular screening designs for 2 or more factors. Screening factors can be continuous or categorical with two or three levels. The list of screening designs also includes designs that group the experimental runs into blocks of equal sizes where the size is a power of two.
Response Surface Design
Response Surface Methodology (RSM) is an experimental technique invented to find the optimal response within the 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 find it. In industrial applications, RSM designs involve a small number of factors. This is because the required number of runs increases dramatically
1 JMP DOE
with the number of factors. The Response Surface designer in JMP lists well-known RSM designs for two to eight continuous factors. Some of these designs also allow blocking.
Full Factorial Design
A full factorial design contains all possible combinations of a set of factors. This is the most conservative design approach, but it is also the most costly in experimental resources. The Full Factorial designer supports both continuous factors and categorical factors with arbitrary numbers of levels.
Taguchi Arrays
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 in JMP supports all these features of the Taguchi method. The inner and outer array design lists use the traditional Taguchi orthogonal arrays such as L4, L8, L16, and so forth.
Mixture Design
The Mixture designer lets you define a set of factors that are ingredients in a mixture. You 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.
Augment Design
The Augment designer gives the following four choices for adding new runs to existing design:
❿ add center points
❿ replicate the design a specified number of times
❿ create a foldover design
❿ add runs to the design using a model, which can have more terms than the original model.
The last choice (adding runs to a design) is particularly powerful. You can use this choice to achieve the objectives of response surface methodology by changing a linear model to a full quadratic model and adding the necessary number of runs. For example, 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.
Sample Size and Power
The Sample Size and Power facility computes power, sample size, or the effect size you want to detect, for a given alpha and error standard deviation. You supply two of these values and the Sample Size and Power feature computes the third. If you supply only one of these values, the result is a plot of the other two. This feature is available for the single sample, two sample, and k sample situations.
A Simple DOE Example
The following example demonstrates the interface for choosing designs from a list. It introduces the JMP DOE dialog that lets you
❿ enter factors and responses
❿ choose a design
❿ modify a design
❿ generate a JMP table that contains the design runs.
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:
1 JMP DOE 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
❿ find which factors affect the depth of the weld
❿ quantify those effects
❿ find specific factor settings that predict a weld depth of 0.17 inches. The next sections show how to define this study in JMP with the DOE dialog
The DOE Dialog
When you first select any command from the DOE menu, the DOE dialog appears. It has two basic panels, as illustrated by the dialog shown in Figure 1.2.
❿ The Responses panel has a single default response. You can enter as many responses as you want, and designate response goals as Maximize, Minimize, or Match Target. A response may also have no defined goal. The DOE platform accepts only numeric responses.
❿ The Factors panel requires that you enter one or more factors. The appearance of the Factors panel depends on the DOE command you select. For the 2-level design panel shown in Figure 1.2, enter the number of Continuous,2-Level, or 3-level factors you want and click Add. Factor panels for other types of design are shown in more detail in the following chapters that describe the specific design types.
The results when you click Continue depend on the type of design. There are examples of each design type shown in the chapters that follow. For simplicity, this example uses the Screening designer.
Note that the Responses and Factors panels have disclosure buttons so that you can close them. This lets you simplify the dialog when you are ready to Continue.
Figure 1.2 The DOE Design Experiment Dialog For a Screening Design
Click to see available designs. Enter Factors and
click Add.
Factors Panel
Enter response and edit response names. Define response goal:
Target, Min, Max, or None.
Responses Panel
Edit Factors names.
Entering Responses
By default, The Responses panel in the DOE dialog appears with one response (named Y) that has Maximize as its goal. There are several things you can do in this panel:
❿ Add an additional response with a specific goal type using selections from the Add Response popup menu.
❿ Add N additional responses with the N Responses button. The default goal is maximize.
❿ Specify goals appropriate for each goal type.
To continue with the welding example open the Responses panel if it is not already showing. Note that there is a single default response called Y. Change the default response as follows:
1 JMP DOE
2) The default goal is Maximize, but this process has a target value of 0.17 inches with a lower bound of 0.12 and an upper bound of 0.22. Click on the Goal text edit area and choose Match Target from the popup menu, as shown here. 3) Click the Lower Bound, Upper Bound, areas and enter
0.12 as the target value, 0.22 as a minimum and maximum acceptable values.
Entering Factors
Next enter factors into the Factors panel, which shows beneath the Responses panel. Design factors have different roles that depend on design type. The Factors panel reflects roles appropriate for the design you
choose.
The screening design accepts either continuous or categorical factors. This example has one categorical factor (Operator) and two contin-uous factors (Speed and Current). Enter 1 in the 2-Level Categorical text box and click Add. then click. Enter 2 in the Continuous text box and click Add. These three factors appear with default names (X1, X2, and X3) and the default values shown here.
The factor names and values are editable fields. Double click on these fields to enter new names and values. For this example, use Mary and John as values for the categorical factor called Operator. Name the continuous factors Speed and Current. High and low values for Speed are 3 and 5 rpm. Values
for Current are 150 and 165 amps. After you enter the response, the factors, and edit their values (optional), click Continue.
Select a Design Type
When you click Continue, the next section of the design dialog unfolds. This Choose a Design panel is specific to the Screening designer. Other design types work differently at this stage. Details for each are in the following chapters.
To reproduce the example shown here, click on Full Factorial in the list of designs to select it. The next section discusses addit-ional steps you take in the DOE dialog to give JMP special instructions about details of the design. If necessary you can return (Backup) to the list of
designs and select a different design. After you select a design type, click Continue again and interact with the Display and Modify Design panel to tailor the design. These detail options are different for each type of design.
Modify a Design
Special features for screening designs include the ability to list the Aliasing of Effects, Change Generating Rules for aliasing, and view the Coded Design. A standard feature for all designs lets you specify the Run Orderwith selections from the run order popup menu. These features are used in examples and discussed in detail in the following chapters.
When the design details are complete, click Make Table to create a JMP table that contains the specified design.
1 JMP DOE
Note: All dialogs have a Backup button that returns you to the previous stage of the design
generation, where you can change the design type selection.
The JMP DOE Data Table
The example in the discussion above is for a factorial design with one 2-level categorical and two continuous factors. When you click Make Table, the JMP table in Figure 1.3 appears. The table uses the names for responses, factors, and levels assigned in the DOE dialog panels. The Pattern variable shows the coded design runs.
This data table is called DOE Example 1.jmp in the Design Experiment folder in the sample data.
Figure 1.3 The Generated DOE JMP Data Table
The table panels show table properties automatically created by the DOE platform:
❿ The name of the table is the design type that generated it.
❿ A table variable called Design also shows the design type. You can edit this table variable to further document the table, or you can create new table variables.
❿ A script to generate the analysis model is saved with the table. The icon labeled Model is a Table Property that runs a script that generates a Model Specification dialog with the analysis specification for the design type you picked. In this example the Model Specification dialog shows a single response, Depth (In.), three main effects, Operator,
Figure 1.4 The Model Specification dialog Generated by the DOE Dialog
DOE Utility Commands
The DOE dialog has a number of efficiency features accessible using the popup menu on the Design Experiment title bar. Most of these features are for saving and loading information about variables. This is handy when you plan several
experiments using the same factors and responses.
There are examples of each feature in the list below. Many of the DOE case studies later in this manual also show how to benefit from these utilities.
Save Responses
1 JMP DOE
completed DOE dialog. The table has a row for each response with a column called
Response Name that iden-tifies them. Four additional columns identify response goals to the DOE facility: Lower Limit, Upper Limit, Response Goal, and an Importance weight. This example shows a DOE dialog for four responses with a variety
of response goals, and the JMP table that contains the response information. Load Responses
If the responses and response goals are in a JMP table as described previously, you can use that table to complete the DOE dialog for an experiment. When the responses table you want is open and is the current table, the Load Responses command copies the response names and goals into the DOE dialog. If there is no response table open, Load Responses displays the Open File dialog for you to open the table you want to use. Save Factors
If an experiment has many factors, it can take time to enter the names and values for each factor. After you finish you can use the Save Factors command to save your work, so you only have to do this job once. The Save Factors command creates a JMP data table that contains the information in a completed factor list. The table has a column for each factor and a row for each factor level.
As an example, suppose you entered the informa-tion showing in the dialog to the right. Save Factors produces the data table shown below. The columns of this table have a Column
Property called Design Role, that identifies them as DOE factors to the DOE facility, and tells what kind of factors they are (continuous, categorical, blocking, and so on.).
You can also create a factors table by keying data into an empty table, but you have to assign each column its factor type. Use the New Property menu in the Column Info dialog and select Design Role. Then choose the appropriate design role from the popup menu on the design role column property tab page.
Load Factors
If the factors and levels for an experiment are in a JMP table as described previously, you can use that table to complete the DOE dialog for an experiment. If the factors table you want is open and is the current table, the Load Factors command copies the factor names, values, and factor types into the DOE dialog. If there is no factor table open, Load Factors displays the Open File dialog for you to open the factors table you want to use.
Save Constraints
Entering constraints on continuous factors is another example of work you only want to do once. In the next example, there are three variables, X1, X2, and X3, with three linear constraints. The Save Constraints
command creates a JMP table that contains the information you enter into a constraints panel like the one shown here. There is a columns for each constraint with a column property called Constraint State that identifies them as constraints (< or >) to the DOE facility. There is a row for each variable and an additional row that has the inequality condition for each variable. Load Constraints
If the responses and response goals are in a JMP table as described previously, you can use that table to complete the DOE dialog for an experiment. When the responses table you want is open and is the current table, the Load Constraints command copies the
1 JMP DOE
response names and goals into the DOE dialog. If there is no response table open, Load Responses displays the Open File dialog for you to open the table you want to use. Set Random Seed
The Custom designer begins the design process with a random number. After a design
is complete the Set Random Number command displays a dialog that shows the generating seed for that design. On this dialog you can set that design to run again, or continue with a new random number.
Simulate Responses
When you check Simulate Response, that item shows as checked for the current design only. It adds simulated response values to the JMP design data table for custom and augmented designs.
2 Customized I
Chapter 2
Introduction to Custom Designs
The DOE platform in JMP has thefollowing two approaches for building an experimental design:
❿ You can let JMP build a design for your specific problem that is consistent with your resource budget.
❿ You can choose a predefined design from one of the design catalogs, which are grouped by problem type.
choose from catalogues of listed designs create design to solve a problem modify any design
The Custom designer supports the first of these approaches. You can use it for routine factor screening, response optimization, and mixture problems. Also, the custom designer can find designs for special conditions not covered in the lists of predefined designs. This chapter introduces you to the Custom designer. It shows how to use the Custom Design interface to build a design using this easy step-by-step approach:
Use model to find best factor settings for on-target responses and minimum variability. Identify factors and responses. Compute design for maximum infromation from runs.
Use design to set factors; measure responses for each run.
Compute best fit of mathematical model to data from test runs.
Key mathematical steps: appropriate computer-based tools are empowering. Key engineering steps: process knowledge and engineering judgement are important.
Predict Fit
Collect Describe Design
Chapter 3, “Custom Design: Beyond the Textbook," uses a case study approach to introduce the advanced capabilities of the Custom Design personality.
Chapter 2
Contents
Getting Started ... 19 Define Factors in the Factors Panel ... 19 Describe the Model in the Model Panel ... 20 The Design Generation Panel ... 20 The Design Panel and Output Options ... 21 Make Table ... 22 Modify a Design Interactively ... 23 Introducing the Prediction Variance Profiler ... 24 A Quadratic Model ... 24 A Cubic Model ... 26 Routine Screening Using Custom Designs ... 28 Main Effects Only ... 28 All Two-Factor Interactions Involving Only One Factor... 30 All Two-Factor Interactions ... 31 How the Custom Designer Works ... 322 Customized I
Getting Started
The purpose of this chapter is to guide you through the interface of the Custom Design personality. You interact with this facility to describe your experimental situation, and JMP creates a design that fits your requirements.
The Custom Design interface has these key steps:
1) Enter and name one or more responses, if needed. The DOE dialog always begins with a single response, called Y, and the Response panel is closed by default.
2) Use the Factors panel to name and describe the types of factors you have. 3) Enter factor constraints, if there are any.
4) Choose a model.
5) Modify the sample size alternatives. 6) Choose the run order.
7) Optionally, add center points and replicates.
You can use the custom design dialog to enter main effects, then add interactions, and specify center points and replicates.
Define Factors in the Factors Panel
When you select Custom Design from the DOE menu, or from the DOE tab on the JMP Starter, the dialog on the right in Figure 2.1 appears. One way to enter factors is to click
Add N Factors text edit box and enter the number of continuous factors you want. If you want other kinds of factors click Add Factor and select a factor type: Continuous, Categorical, Blocking, Covariate, Mixture, or Constant.
When you finish defining factors, Click Continue in the Factors panel to proceed to the next step.
Figure 2.1 Select Custom Design and Enter Factors
Describe the Model in the Model Panel
When you click Continue, the Model panel initially appears with only the main effects corresponding to the factors you entered. Next, you might want to enter additional effects to estimate. That is, if you do not want to limit your model to main effects, you can add factor interactions or powers of
continuous factors to the model. This simple example has two continuous factors, X1 and X2. When you click Continue, the current Model panel appears with only those factors, as shown here. The Model panel has buttons for you to add specific factor types to the model. For example, when you select 2nd from the Interaction popup menu, the
X1*X2 interaction term is added to the model effects.
The Design Generation Panel
As you add effects to the model, the Design Generation panel shows the minimum number of runs needed to perform the experiment. It also shows alternate numbers of runs, or lets
2 Customized I
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.
The Design Generation panel has the following radio buttons:
Minimum is the number of terms in the design model. The resulting design is saturated (no degrees of freedom for error). This is the most risky choice. Use it only when the cost of extra runs is prohibitive.
Default is a custom design suggestion for the number of runs. This value is based on heuristics for creating balanced designs with a minimum of additional runs above the minimum.
Compromise is a second suggestion that is more conservative than the Default. Its value is generally between Default and Grid.
Grid, in most cases, shows the number of points in a full-factorial design. Exceptions are for mixture and blocking designs. Generally Grid is unnecessarily large and is included as an options for reference and comparison.
UserSpecified highlights the Number of Runs text box. You key in a number of runs that is at least the minimum.
When the Design Generation panel is the way you want it, click Make Design to see the factor design layout, the Design panel, appended to the Model panel in the DOE dialog.
The Design Panel and Output Options
Before you create a JMP data table of design runs you can use the Run Order option to designate the order you want the runs to appear in the JMP data table when it is created. If you select Keep the Same, the rows (runs) in the JMP table appear as they show in the Design panel. Alternatively, you can sort the table columns or randomize the runs.
There are edit boxes to request additional runs at the center points be added, and to request rows that replicate the design (including any additional center points).
Note: You can double-click any title bar to change its text. It can be helpful to give your design dialog a meaningful name in the title bar labeled Custom Design
by default.
Make Table
When the Design panel shows the layout you want, click Make Table. This creates the JMP data table whose rows are the runs you defined. Make Table also updates the runs in the Design panel to match the JMP data table.
The table to the right is the initial two-factor design shown above, which has four additional center points, and is replicated once as specified above. initial design 4 added center points replicate initial design replicate 4 added center points
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Modify a Design Interactively
There is a Backup button at several stages in the design dialog that allows you to change your mind and go back to a previous step and modify the design. For example, you can modify the previous design by adding quadratic terms to the model, by removing the center points and the replicate. Figure 2.2 shows the steps to modify a design interactively. When you click Continue the Design panel shows with 8 runs as default. If you choose the
Grid option, the design that results has 9 runs. Figure 2.2 Back up to Interactively Modify a Design
Introducing the Prediction Variance Profiler
All of the listed designs in the other design types require at least two factors. The following examples have a single continuous factor and compare designs for quadratic and cubic models. The purpose of these examples is to introduce the prediction variance profile plot.
A Quadratic Model
You can follow the steps in Figure 2.3 to create a simple quadratic model with a single continuous factor.
1) Add one continuous factor and click Continue.
2) Select 2nd from the Powers popup menu in the Model panel to create a quadratic term. 3) Use the default number of runs, 6, and click Make Design.
Figure 2.3 Use One Continuous Factor and Create a Quadratic Model
When the design appears, open the Prediction Variance Profile (as shown next). 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 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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The Y axis is the relative variance of prediction of the expected value of the response. The prediction variance is relative to the error variance. When the prediction variance is 1, the absolute variance is equal to the error variance of the regression model.
What you are deciding when you choose a sample size is how much variance in the expected response you are willing to tolerate. As the number of runs increases, the prediction curve (prediction variance) decreases.
To compare profile plots, Backup and choose Minimum in the Design Generation panel, which gives a sample size of 3. This produces a curve that has the same shape as the previous plot, but the maxima are at 1 instead of 0.5. Figure 2.4 compares plots for sample size 6 and sample size 3 for this quadratic model example. You can see the prediction variance increase as the sample size decreases.
Figure 2.4 Comparison of Prediction Variance Profiles.
These profiles are for middle variance and
lowest variance, for
sample sizes 6 (top charts) and sample size 3 (bottom charts).
.
Note: You can CONTROL-click (COMMAND-click on the Mac) on the factor to set a factor level precisely
For a final look at the Prediction Variance Profile for the quadratic model, Backup and enter a sample size of 4 in the Design Generation panel and click Make Design.
The sample size of 4 adds a point at –1 (Figure 2.5). Therefore, the variance of prediction at –1 is lower (half the value) than the other sample points. The symmetry of the plot is related to the balance of the factor settings. When the design points are balanced, the plot is symmetric, like those in Figure 2.4; when the design is unbalanced, the prediction plot is not symmetric, as shown below.
Figure 2.5 Sample Size of Four for the One-Factor Quadratic Model
A Cubic Model
The runs in the quadratic model are equally spaced. This is not true for the single-factor cubic model shown in this section. To create a one-factor cubic model, follow the same steps as shown previously in Figure 2.3. In addition, add a cubic term to the model with the Powers popup menu. Use the Default number of runs in the Design Generation panel.
Click Make Design to continue. Then open the Prediction Variance Profile Plot to see the Prediction Variance Profile and its associated design shown in Figure 2.6. The cubic model has a variance profile that is a 6th degree polynomial. Note that the points are not equally spaced in X. It is interestingly non-intuitive that this design has a better
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You can reproduce the plots in Figure 2.6 with JSL code. The following JSL code shows graphically that the design with unequally spaced points has a better prediction variance than the equally spaced design. Open the file called Cubic Model.jsl, found in the Scripts folder in the Sample Data, and select Submit Scriptfrom the Editmenu. When the plot appears, move the free values from the equally spaced points to the optimal points to see that the maximum variance on the interval decreases by more that 10%.
// DOE for fitting a cubic model. n = 4; // number of points //Start with equally spaced points. u = [-0.333 0.333];
x = {-1,u[1],u[2],1}; y = j(2,1,.2);
cubicx = function({x1},
rr=j(4,1,1);for(i=1,i<=3,i++,rr[i+1]=x1^i); rr;);
NewWindow("DOE - Variance Function of a Cubic Polynomial", Graph(FrameSize(500,300),XScale(-1.0,1.0),yScale(0,1.2), Double Buffer, M = j(n,1,1); for(i=1,i<=3,i++, M = M||(x^i)); V = M`*M; C = inverse(V); yFunction(xi=cubicx(x);sqrt(xi`*C*xi),x); detV = det(V); text({-0.3,1.1},"Determinant = ",char(detV,6,99)); DragMarker(u,y); for(i=1,i<=2,i++,Text({u[i],.25},char(u[i],6,99)));)); show(n,d,u);
// Drag the middle points to -0.445 and 0.445 for a D-Optimal design.
Figure 2.6 Comparison of Prediction Variance Profiles For Cubic Design with Unequally Spaced Points and Augmented to Have Equally Spaced Points
Routine Screening Using Custom Designs
You can usethe Screening designer to create screening designs, but it is not necessary. The straightforward screening examples described next show that ‘custom’ is not equivalent to ‘exotic.’ The Custom designer is a general purpose design environment. As such, it can create screening designs.
The first example shows the steps to generate a main-effects-only screening design, an easy design to create and analyze. This is also easy using the Screening designer.
Main Effects Only
First, enter the number of factors you want into the Factors panel and click Continue as shown in Figure 2.7. This example uses 6 factors. Because there are no complex terms in the model no further action is needed in the Model panel. The default number of runs (8) is correct for the main-effects-only model.
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Note to DOE experts:
The result is a resolution 3 screening design. All main effects are estimable but are confounded with two factor interactions.
Click Make Design to see the Factor Design table in Figure 2.7. Figure 2.7 A Main Effects Only Screening Design
The Prediction Variance Profile in Figure 2.8 shows a variance of 0.125 (1/8) at the center of the design, which are the settings that show when you open the Prediction Variance Profile. If you did all of your runs at this point, you would have the same prediction variance. But, then you could not make predictions for any other row of factor settings. The prediction variance profile for each factor is a parabola centered at the midrange of each factor. The maximum prediction variance is at each design point and is equal to p/n, where p is the number of parameters and n is the number of runs.
Figure 2.8 A Main Effects Only Screening Design
All Two-Factor Interactions Involving Only One Factor
Sometimes there is reason to believe that some two-factor interactions may be important. The following example illustrates adding all the two-factor interactions involving one factor. The example has 5 continuous factors.
Note to DOE experts:
This design is a resolution 4 design equivalent to folding over on the factor for which all two factor interactions are estimable.
To get a specific set of crossed factors (rather than all interactions or response surface terms) Select the factor to cross (X1, for example) in the Factors table. Select the other factors in the Model Table and click Cross to see the interactions in the model table, as shown in Figure 2.9 .
The default sample size for designs with only two-level factors is the smallest power of two that is larger than the number of terms in the design model. For example, in Figure 2.9 there are 9 terms in the model, so 24=16 is the smallest power of two that is greater than 9.
2 Customized I Figure 2.9 Two-factor Interactions that Involve Only One of the Factors
All Two-Factor Interactions
In situations where there are few factors and experimental runs are cheap, you can run screening experiments that allow for estimating all the two-factor interactions.
Note to DOE experts:
The result is a resolution 5 screening design. Two-factor interactions are estimable but are confounded with three-factor interactions.
The custom design interface makes this simpl e (see Figure 2.10.). Enter the number of factors. Then click Continue and choose 2nd from the Interactions popup in the Model outline, then click Make Design. Figure 2.10 shows a partial listing of the two-factor design with all interactions. The default design has the minimum power of two sample size consistent with fitting the model.
Figure 2.10 All Two-Factor Interactions
How the Custom Designer Works
The Custom designer starts with a random design with each point inside the range of each factor. The computational method is an iterative algorithm called coordinate exchange. 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. Iteration continues until no replacement occurs in an entire iterate.
To avoid converging to a local optimum, the whole process is repeated several times using a different random start. The designer displays the best of these designs.
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 will generate different (but equivalent) designs if you press the Backup and Make Design buttons repeatedly.
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Chapter 3
Custom Design: Beyond the Textbook
No list of defined designs has an exact match for every industrial process. To use a pre-fabricated design you usually have to modify the process description to suit the design or make ad hoc modifications to the design so that it does a better job of modeling the process. Using the Custom designer, you first describe process variables and constraints, then JMP tailors a design that fits. This approach is general and requires less experience and expertise in statistical design of experiments.The ability to mix factor roles as required by the engineering situation is what makes the Custom Design facility so flexible.
The Add Factorpopup menu shows the list of roles factors can take. Here is a sample of what you can do.
❿ You can add factors with any role in any experiment.
❿ Categorical factors can have as many levels as you need.
❿" You can specify any number of runs per block.
❿ Any design can have continuous or categorical covariate factors—factors whose values are fixed in advance of the experiment.
❿ You can have non-mixture factors in a mixture experiment.
❿ You can disallow certain regions of the factor space by defining linear inequality constraints.
Once you generate a design, you can use the Prediction Variance Profiler as a diagnostic tool to assess the quality of the design. You can use this tool to compare many candidate designs and choose the one that best meets your needs.
This chapter presents several examples with aspects that are common in industry but which make them beyond the scope of any design catalog. It introduces various features of the Custom designer in the context of solving real-world problems.
Chapter 3
Contents
Custom Situations ... 35 Flexible Block Sizes ... 36 Response Surface Model with Categorical Factors ... 38 Fixed Covariate Factors ... 43 Mixtures with Nonmixture Factors... 45 Factor Constraints ... 483 Customized II
Custom Situations
When your design situation does not fit a standard design, the Custom designer gives you the flexibility to tailor a design to specific circumstances. Here are some examples.
❿ The listed designs in the Screening designer allow only 2-level or 3-level factors. Moreover, the designs that allow blocking limit the block sizes to powers of two. Suppose you are able to do a total of 12 runs, and want to complete one block per day. With a block size of two the experiment takes six days. If you could do three runs a day, it would take only four days instead of six.
❿ The Response Surfacedesigner allows only continuous factors. Suppose you wanted to model the behavior of three kinds of epoxy under varying temperatures and pressures in a lamination process. Repeating a complete response surface design for each type of epoxy requires more runs than a single response surface design arranged over the epoxy levels.
❿ Preformulated designs rely on the assumption that the experimenter controls all the factors. It is common to have quantitative measurements (a covariate) on the experimental units before the experiment begins. If these measures affect the experimental response, the covariate should be a design factor. The preformulated design that allows only a few discrete values is too restrictive.
❿ The Mixture designer requires all factors to be mixture components. It seems natural to vary the process settings along with the percentages of the mixture ingredients. After all, the optimal formulation could change depending on the operating environment.
❿ Screening and RSM designs assume it is possible to vary all the factors independently over their experimental ranges. The experimenter might know in advance that running a process at certain specified settings has an undesirable result. Leaving these runs out of an available listed design type destroys the mathematical properties of the design. The Custom designer can supply a reasonable design for all these examples. Instead of a list of tables, the Custom designer creates a design table from scratch according to your specifications. Instead of forcing you to modify your problem to conform to the restrictions of a tabled design, it tailors a design to fit your needs.
Flexible Block Sizes
When you create a design using the Screening designer, the available block sizes for the listed designs are a power of 2. Custom designs can have blocks of any size. The blocking shown below is flexible because there are 3 runs per block, instead of a power of 2. When you first enter the factors, the blocking factor
shows only one level because the sample size is unknown at this point. When you complete the design, the number of blocks is the sample size divided by the number of runs per block.
Click Continue to see the Design Generation panel shown on the right in Figure 3.1. The
choice of three runs per block leads to a default sample size of six runs. This sample size requires 2 blocks, which now shows in the Factors panel. If you chose the Grid option with 24 runs, the Factors panel changes to show 24/3 = 8 blocks.
Figure 3.1 Examples of Blocking Factor Levels
If you add the two-factor interactions of X1-X3 to the design, as shown by the Model panel and Design Generation panel in Figure 3.2, the default number of runs changes to 12. The blocking factor then has 4 levels. The table in the example results from the Randomize within Blocks option in the Run Order popup menu on the Display and Modify Design panel..
3 Customized II Figure 3.2 Model Design Table For Blocking Factor With Four Levels
The initial Prediction Variance Profile for this design (Figure 3.3) shows that at the center of the design, the block-to-block variance is a constant. This results from the fact that each block has three runs.
Figure 3.3 Constant Block-to-Block Variance at Design Center
If you drag the vertical reference lines in the plots of X1 through X3 to their high value of 1, you see the top plot in Figure 3.4. The bottom plot results from dragging the vertical reference line for X4 to block 4. At this vertex the prediction variance is not constant over the blocks. This is due to an unavoidable lack of balance resulting from the fact that there are three runs in each block, but only two values for each continuous variable.
Figure 3.4 Block 1 and Block 4 Prediction Variance at Point (1,1,1)
The main question here is whether the size of the prediction variance over the possible factor settings is acceptably small. If not, adding more runs (up to 15 or 18) will lower the prediction variance traces.
Response Surface Model with Categorical Factors
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 only deals with quantitative factors. The only way to handle a RSM design with a qualitative factor is to replicate the design over each level of the factor, which can be unnecessarily time consuming and expensive.
The following example shows how easy it is to build these designs using the Custom designer.
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First, define two continuous factors (X1 and X2). Click Continue and then click the RSM button in the Model panel. You should see the
panels as they are shown here.
Now, use the Add Factor popup above the Factors panel to create a 3-level categorical factor (X3). As soon as you add the categorical factor, the model updates to show the main effect of the categorical factor in the Model panel. Ignoring the categorical factor, it seems natural to use a 32 factorial design to fit
an RSM model for two continuous factors, which gives the design illustrated to the right. The traditional approach would be to repeat this design three times (once for each level of the categorical variable), giving a sample size of 27. This is overkill. In fact, its not strictly necessary to add any runs to accommodate the categorical factor. When you click Continue for this example, the Design Generation panel shows the default number of runs to be 12, but the Minimum
option is 8.
Note: The minimum number of runs needed for this example is eight because the RSM model for two continuous factors has six parameters (constant, two linear terms, interaction, and two quadratic terms). The main effect of the 3-level categorical factor adds two more parameters, giving a total of eight parameters.
The rest of this example compares the results of 8 runs, 9 runs and the 9-run design with 3 center points added. To see these designs:
❿ Make a design with the Minimum runs (8).
❿ Make a second design by typing “9” in the Design Generation Panel Number of Runs text box.
❿ For the third design, add three center points to the previously 9-run design and make the design again.
Figure 3.5 shows these three designs after making JMP tables for them, sorted right to left. Figure 3.5 8 runs (Left) 9 runs (Middle) 9 runs with 3 Center Points Added (Right)
Figure 3.6 gives a geometric view of the designs generated by this example. These plots were generated for the runs in each JMP table with the Overlay command in the Graph menu, using the block factor as the Group By variable.
3 Customized II Figure 3.6 Geometric View of RSM Designs
8 runs 9 runs 9 runs with 3 center points
The Prediction Variance Profilers for each of these designs are shown in Figures 3.7-3.9. Figure 3.7 shows the variance traces for the minimum design. Note that at the center of the design the prediction variance is larger than the error variance. If the error variance is small relative to the size of the effect that is important, this should not concern you. If the process variability is sizeable, then adding runs will help reduce the noise in the parameter
estimates.
Figure 3.7 Prediction Variance Profile For Minimum Design
The prediction variance trace in Figure 3.8 shows that adding just one more run to the minimum (saturated) design reduces the prediction variance at the center of the design by nearly 40%. If extra runs are not prohibitively expensive, this is a desirable choice.
Figure 3.8 Prediction Variance Profile For 9 Run Design
Figure 3.9 shows the prediction trace after adding three center points to the 9-Run design. The additional center points give the prediction trace a bowl shape which is desirable if you are confident that you have already bracketed the optimum response. There is a further 40% drop in the prediction variance at the center of the design, but this is at the cost of three extra runs instead of one.
Figure 3.9 Prediction Variance Profile For 12-Run Design
Any of the designs described in this section could be acceptable, depending on your research objectives and budget. The Prediction Variance Profile is a tool for assessing the trade-off between improved prediction and extra cost.
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Fixed Covariate Factors
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 (X1) and the calories for breakfast (X2) and ask each student 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, open the Big Class.jmp sample data table.
Build the custom design as follows:
❿ Add 2 continuous variables to the model, as shown in previous examples.
❿ Click Continue and add the interaction to the model.
❿ Then select Covariate from the Add Factors popup menu as shown here.
The Covariate selection displays a variable list of the variables in the current data table.
Note: If you have more than one data table open, be sure the table that contains the covariate you want is the active, or current data table.
The covariate, weight, shows in the Factors panel with its minimum and maximum as levels, and is a term in the model. The data table in Figure 3.10 shows the Factors panel and the resulting JMP data table.
Figure 3.10 Design With Fixed Covariate Factor
You can see that weight is nearly independent of the X1 and X2 factors by running the model with the two-factor interaction as in the Model Specification dialog in Figure 3.11. The leverage plots are nearly horizontal, and the analysis of variance table (not shown) shows that the model sum of squares is near zero compared to the residuals.
Figure 3.11 Analysis To Check That weight is Independent of X1 and X2
You can save the prediction equation from by this analysis and use it to generate a set of predicted weight values over a grid of X1 and X2 values, and append them to the column of observed weight values in the experimental design JMP table. Then use the Spinning Plot platform to generate a plot of X1, X2, and weight. This is a way to illustrate that the X1
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Three-dimensional Spinning Plot of Two Design Factors, Observed Covariate Values and Predicted Covariate Grid
Mixtures with Nonmixture Factors
This example taken 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.
❿ The response is the electromagnetic damping of an acrylonitrile powder.
❿ The three mixture ingredients are copper sulphate, sodium thiosulphate, and glyoxal.
❿ 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 treat it as a categorical factor with three levels.
The Responses panel in Figure 3.13 shows Damping as the response. The authors do not mention how much damping is desirable so the response goal is None.
The Factors panel shows the three mixture ingredients and the categorical factor,
Wavelength. The mixture ingredients have range constraints that arise from the mechanism of the chemical reaction. To load these factors choose Load Factors from the popup menu on the Factors panel title bar. When the open file dialog appears, open the file Donev Mixture factors.JMP in the DOE folder in the Sample Data.
Figure 3.13 Mixture Experiment Response Panel and Factors Panel
The model in Figure 3.14 is a response surface model in the mixture ingredients along with the additive effect of the wavelength. There are several reasonable choices for sample size. The grid option in the Design Generation Panel (Figure 3.14) corresponds to repeating a 6-run mixture design in the mixture ingredients once for each level of the categorical factor. The resulting data table is on the right.
Figure 3.14 Mixture Experiment Design Generation Panel and Data Table
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Atkinson and Donev provide the response values shown in Figure 3.14. They also discuss the design where the number of runs is limited to 10. In this case it is not possible to run a complete mixture response surface design for every wavelength.
Typing "10" in the Number of Runs edit box in the Design Generation panel (Figure 3.15) sets the run choice to User Specified. The Design table to the right in Figure 3.15 shows the factor settings for 10 runs.
Figure 3.15 Ten-Run Mixture Response Surface Design.
Note that there are unequal numbers of runs for each wavelength. Because of this lack of balance it a good idea to look at the prediction variance plot Figure 3.16.
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. Figure 3.17 shows the result of increasing the copper sulphate percentage from 0.38462 to 0.61476. The other two ingredients both drop, keeping their ratio constant. The ratio of Na2S2O3 to Glyoxal is 5:3 in both plots.
Figure 3.17 Increasing the Copper Sulphate Percentage.
Factor Constraints
Sometimes it is impossible to vary all the factors independently over their experimental ranges. The experimenter might know in advance that running a process at certain specified settings has an undesirable result. Leaving these runs out of an available listed design type destroys the mathematical properties of the design, which is unacceptable. The solution is to support factor constraints as an integral part of the design requirements.
For this example, define two factors. Suppose that it is impossible or dangerous to perform an experimental run where both factors are at either extreme. That is, none of the corners of the factor region are acceptable points.
Figure 3.18 shows a set of four constraints that cut off the corner points. The figure on the right in Figure 3.18 shows the geometric view of the constrains. The allowable region is inside the diamond defined by the four constraints.
If you want to avoid entering these constraints yourself, choose Load Constraints from the Design Experiments title bar. Open the sample data file Diamond Constraints.jmp in the DOE folder.
3 Customized II Figure 3.18 Factor Constraints
Y X Y = –X + 1 X + Y < 1 Y = X + 1 Y = X – 1 Y = –X – 1 –X + Y > –1 X + Y > –1 –X + Y < 1
Next, click the RSM button in the Model panel to include the two-factor interaction term and both quadratic effects in the model. This is a second order empirical approximation to the true functional relationship between the factors and the response.
Suppose the complexity of this relationship required third order terms for an adequate approximation. Figure 3.19 shows how to create a higher order cross product term. First select one or more factors from the Factors panel and one or more terms from the Model panel. Then click the Cross button to add the cross product terms.
Figure 3.19 Creating a Cross-Product Term
Similarly, you can add the X1*X2*X2 cross product term. To complete the full third order model, select both factors and choose 3rdfrom the Powers popup menu in the Model panel.
There are 10 terms in the design model. A 4 by 4 grid design would be 16 runs. Choosing an intermediate value of 12 runs yields a design similar to the one in Figure 3.20. The geometric view shows many design points at or near the constraint boundary.
Figure 3.20 Factor Settings and Geometric View
Figure 3.21 shows the prediction variance as a function of the factor settings at the center of the design and at the upper right constraint boundary. The variance of prediction at the center of the design is 0.602301, nearly the same as it is at the boundary, 0.739579.
Figure 3.21 Prediction Variance at the Center of the Design and at a Boundary.
In many situations it is preferable to have lower prediction variance at the center of the design. You can accomplish this by adding centerpoints to the design. Figure 3.22 shows
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the result of adding two center points after having generated the 12 run design shown in Figure 3.20.
Snee (1985) calls this exercising the boss option. It is practical to add centerpoints to a design even though the resulting set of runs loses the mathematical optimality exhibited by the previous design. It is more important to solve problems than to run "optimal" designs. Figure 3.22 Add Two Center Points to Make a 14 Point Design.
When you compare the variance profile shown to the right to the one at the top in
Figure 3.21 you see that adding two center points has reduced the variance at the center of the design by more than a factor of two, an impressive improvement.