Jmp Doe Guide

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D

ESIGN

OF

E

XPERIMENTS

SAS Institute Inc. SAS Campus DriveCary, NC 27513

JMP® Design of Experiments, Version 4

Copyright © 2000 by SAS Institute Inc., Cary, NC, USA

ISBN: 1-58025-631-7

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JMP Design Of ExperimentsContents

Credits and Acknowledgments ............................................................................................ v

Chapter 1 Design of Experiments (DOE) ....................................................................... 1

DOE Choices................................................................................................................... 3A Simple DOE Example ................................................................................................. 6The DOE Dialog.............................................................................................................. 7The JMP DOE Data Table............................................................................................. 11DOE Utility Commands................................................................................................ 12

Chapter 2 Introduction to Custom Designs................................................................. 17

Getting Started............................................................................................................... 19Modify a Design Interactively ....................................................................................... 23Introducing the Prediction Profiler................................................................................ 24Routine Screening Using Custom Designs ................................................................... 29How the Custom Designer Works................................................................................. 32

Chapter 3 Custom Design: Beyond the Textbook ....................................................... 33

Custom Situations ......................................................................................................... 35Flexible Block Sizes...................................................................................................... 36Response Surface Model with Categorical Factors....................................................... 38Fixed Covariate Factors................................................................................................. 43Mixtures with Nonmixture Factors ............................................................................... 45Factor Constraints ......................................................................................................... 48

Chapter 4 Screening Designs ....................................................................................... 53

Screening Design Types................................................................................................ 55A Screening Example.................................................................................................... 58Loading and Saving Responses and Factors (Optional)................................................ 66A Simple Effect Screening Analysis............................................................................. 67

Chapter 5 Response Surface Designs ........................................................................... 69

Response Surface Designs............................................................................................. 71A Box-Behnken Design: The Tennis Ball Example ..................................................... 76

Chapter 6 Full Factorial Designs.................................................................................. 85

The Factorial Dialog...................................................................................................... 87The Five-Factor Reactor Example................................................................................. 88

Chapter 7 Taguchi Designs ........................................................................................... 97

The Taguchi Design Approach ..................................................................................... 99Taguchi Design Example .............................................................................................. 99Analyze the Byrne-Taguchi Data................................................................................ 103

Chapter 8 Mixture Designs ......................................................................................... 105

The Mixture Design Dialog......................................................................................... 107Mixture Designs.......................................................................................................... 108Extreme Vertices Design for Constrained Factors...................................................... 113Adding Linear Constraints to Mixture Designs........................................................... 114Ternary and Tetrary Plots............................................................................................ 115Fitting Mixture Designs............................................................................................... 116Chemical Mixture Example......................................................................................... 118Plotting a Mixture Response Surface.......................................................................... 119

Chapter 9 Augmented Designs ................................................................................... 121

The Augment Design Interface ................................................................................... 123The Reactor Example Re-visited................................................................................. 126

Chapter 10 Prosective Power and Sample Size......................................................... 135

Prospective Power Analysis........................................................................................ 137Launch the Sample Size and Power facility ................................................................ 137

References...................................................................................................................... 145

Index ............................................................................................................................... 149

OriginJMP was developed by SAS Institute Inc., Cary, N.C. JMP is not a part of the SAS System and isnot as portable as SAS. A SAS add-on product called SAS/INSIGHT is related to JMP in someways but has different conventions and capabilities. Portions of JMP were adapted from routinesin the SAS System, particularly for linear algebra and probability calculations. Version 1 of JMPwent into production in October, 1989

CreditsJMP 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, XanGregg, Eric Wasserman, Charles Soper, and Kevin Hardman. Ann Lehman coordinated productdevelopment, production, quality assurance, and documentation. In the SAS Institute TechnicalSupport division, Ryan Gilmore, Maureen Hayes, Craig Devault, Toby Trott, and Peter Ruzzaprovide technical support and conducted test site administration. Statistical technical support isprovided by Duane Hayes, Kathleen Kiernan, and Annette Sanders. Nicole Jones and JianfengDing provide ongoing quality assurance. Additional testing and technical support is done byKyoko 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, LisaRohloff, Bob McCall, Chuck Boiler, Nick Zagone and Bonnie Rigo. Additional support is providedby Kathy Jablonski and Jean Davis.

The JMP manuals were written by Ann Lehman, John Sall, Bradley Jones, and Erin Vang withcontributions from Annie Dudley and Brian Corcoran. Editing was done by Lee Bumgarner, BradKellam, and Lee Creighton, design and production by Creative Solutions. Lee Creightonimplemented the online help system and online documentation with contribution from TimothyChristensen.

Special thanks to Jim Goodnight for supporting a product outside the usual traditions and to DaveDeLong for valuable ideas and advice on statistical and computational matters.

Thanks also to Robert N. Rodriguez, Ying So, Duane Hayes, Mark Bailey, Donna Woodward, andMike Stockstill for statistical editorial support and statistical QC advice. Thanks to GeorgesGuirguis, Warren Sarle, Randall Tobias, Gordon Johnston, Ying So, Wolfgang Hartmann, RussellWolfinger, and Warren Kuhfeld for statistical R&D support.

AcknowledgmentsWe owe special gratitude to the people that encouraged us to start JMP, to the alpha and betatesters 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, RamonLeon, Tom Lange, Homer Hegedus, Skip Weed, Michael Emptage, Pat Spagan, John Frei, PaulWenz, Mike Bowen, Lori Gates, Georgia Morgan, David Coleman, Linda Blazek, MichaelFriendly, Joe Hockman, Frank Shen, J.H. Goodman, David Ikle, Lou Valente, Robert Mee, BarryHembree, Dan Obermiller, Lynn Vanatta, and Kris Ghosh. Also, we thank Dick DeVeaux, GrayMcQuarrie, Robert Stein, George Fraction, Al Fulmer, Cary Tuckfield, Ron Thisted, DonnaFulenwider, Nancy McDermott, Veronica Czitrom, Tom Johnson, Avigdor Cahaner, and AndyMauromoustakos.

vi

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 multiplecomparisons methods (not all of which we were able to incorporate in JMP); Ralph O’Brien foradvice on homogeneity of variance tests; Ralph O’Brien and S. Paul Wright for advice onstatistical 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 ofmixed 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 thepollution data, and David Coleman for the pollen (eureka) data.

Past SupportMany people were important in the evolution of JMP. Special thanks Jeffrey Perkinson, Mary Cole, Kristin Nauta, AaronWalker, Ike Walker, Eric Gjertsen, Dave Tilley, Curt Yeo, Patricia Moell, Patrice Cherry, Mike Pezzoni, Mary AnnHansen, 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 3and 4 was done by Li Yang, Brenda Sun, Katrina Hauser, and Andrea Ritter. Thanks to Walt Martin for Postscript supportin documentation production.

Also thanks to Jenny Kendall, Elizabeth Shaw, and John Hansen, Eddie Routten, David Schlotzhauer, John Boling, andJames Mulherin, Thanks to Steve Shack, Greg Weier, and Maura Stokes for testing Version 1. Additional editorial supportwas 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, andWilliam Fehlner.

Technology License NoticesJMP software contains portions of the file translation library of MacLinkPlus, a product of DataViz Inc., 55 CorporateDrive, 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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IN NO EVENT WILL SAS INSTITUTE INC.’S LICENSORS AND THEIR DIRECTORS, OFFICERS, EMPLOYEESOR AGENTS ( COLLECTIVELY SAS INSTITUTE INC.’S LICENSOR) BE LIABLE TO YOU FOR ANYCONSEQUENTIAL, INCIDENTAL OR INDIRECT DAMAGES (INCLUDING DAMAGES FOR LOSS OF BUSINESSPROFITS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION, AND THE LIKE) ARISING OUT OFTHE USE OR INABILITY TO USE THE SOFTWARE EVEN IF SAS INSTITUTE INC.’S LICENSOR’S HAS BEENADVISED OF THE POSSIBILITY OF SUCH DAMAGES. BECAUSE SOME STATES DO NOT ALLOW THEEXCLUSION OR LIMITATION OF LIABILITY FOR CONSEQUENTIAL OR INCIDENTAL DAMAGES, THEABOVE LIMITATIONS MAY NOT APPLY TO YOU. SAS INSTITUTE INC.’S LICENSOR’S LIABILITY TO YOUFOR ACTUAL DAMAGES FOR ANY CAUSE WHATSOEVER, AND REGARDLESS OF THE FORM OF THEACTION (WHETHER IN CONTRACT, TORT (INCLUDING NEGLIGENCE), PRODUCT LIABILITY OROTHERWISE), WILL BE LIMITED TO $50.

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E 1

Chapter 1Design 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 of

experiments. A trial-and-error search for the vital few factors that most affect quality iscostly 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. The Response 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.

2

Chapter 1Contents

DOE Choices .............................................................................................................................. 3Custom Design .................................................................................................................... 4Screening Design ................................................................................................................. 4Response Surface Design .................................................................................................... 4Full Factorial Design ........................................................................................................... 5Taguchi Arrays .................................................................................................................... 5Mixture Design.................................................................................................................... 5Augment Design.................................................................................................................. 5Sample Size and Power ....................................................................................................... 6

A Simple DOE Example............................................................................................................. 6The DOE Dialog ......................................................................................................................... 7

Entering Responses ............................................................................................................. 8Entering Factors................................................................................................................... 9Select a Design Type ......................................................................................................... 10Modify a Design ................................................................................................................ 10

The JMP DOE Data Table........................................................................................................ 11DOE Utility Commands ........................................................................................................... 12

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EChapter 1 Design of Experiments 3

DOE ChoicesThe DOE 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.

4 Chapter 1 Design of Experiments

Custom DesignCustom 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 DesignAs 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 DesignResponse 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

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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 DesignA 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 ArraysThe 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 DesignThe 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 DesignThe 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 PowerThe 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 ExampleThe 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:

Operator, two technicians who operate the welding machine.

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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 DialogWhen 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. Aresponse 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 ResponsesBy 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) double click to highlight the response name and change it to Depth (In.).

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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 FactorsNext 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 TypeWhen you click Continue, the next section of the design dialog unfolds. This Choose aDesign 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 inthe 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 DesignSpecial 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 Order with 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.

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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 TableThe 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 1.3appears. 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,Speed, and Current, and all two factor interactions.


Figure 1.4 The Model Specification dialog Generated by the DOE Dialog

DOE Utility CommandsThe 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

The Save Responses command creates a JMP table from a

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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: LowerLimit, Upper Limit, ResponseGoal, 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 ConstraintState 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

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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.

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17

Chapter 2Introduction to Custom Designs

The DOE platform in JMP has the

following 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.

PredictFitCollectDescribe Design

Chapter 3, “Custom Design: Beyond the Textbook," uses a case study approach to introduce

the advanced capabilities of the Custom Design personality.

18

Chapter 2Contents

Getting Started .......................................................................................................................... 19Define Factors in the Factors Panel ................................................................................... 19Describe the Model in the Model Panel ............................................................................ 20The Design Generation Panel ............................................................................................ 20The Design Panel and Output Options .............................................................................. 21Make Table........................................................................................................................ 22

Modify a Design Interactively .................................................................................................. 23Introducing the Prediction Variance Profiler ........................................................................... 24

A Quadratic Model ............................................................................................................ 24A Cubic Model .................................................................................................................. 26

Routine Screening Using Custom Designs ............................................................................... 28Main Effects Only ............................................................................................................. 28All Two-Factor Interactions Involving Only One Factor.................................................. 30All Two-Factor Interactions .............................................................................................. 31

How the Custom Designer Works ............................................................................................ 32

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Chapter 2 Custom Designs 19

Getting StartedThe 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 PanelWhen 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.

20 Chapter 2 Custom Designs

Figure 2.1 Select Custom Design and Enter Factors

Describe the Model in the Model PanelWhen 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 PanelAs 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

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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. Itsvalue 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.

User Specified 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 OptionsBefore 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 Designby default.

Make TableWhen 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 ModelYou 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 2.4Comparison of

Prediction Variance

Profiles.

These profiles are formiddle variance andlowest variance, for

sample sizes 6 (topcharts) 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 ModelThe 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

prediction variance profile than the equally spaced design with the same number of runs.

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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 Script from the Edit menu. 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-Optimaldesign.


Figure 2.6Comparison of Prediction

Variance Profiles For

Cubic Design with

Unequally Spaced Points

and Augmented to Have

Equally Spaced Points

Routine Screening Using Custom DesignsYou can use the 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 OnlyFirst, 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 FactorSometimes 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.


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.

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Figure 2.9 Two-factor Interactions that Involve Only One of the Factors

All Two-Factor InteractionsIn 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.


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 WorksThe 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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33

Chapter 3Custom Design: Beyond the Textbook

No list of pre-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 Factor popup 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.

34

Chapter 3Contents

Custom Situations ..................................................................................................................... 35Flexible Block Sizes ................................................................................................................. 36Response Surface Model with Categorical Factors .................................................................. 38Fixed Covariate Factors............................................................................................................ 43Mixtures with Nonmixture Factors........................................................................................... 45Factor Constraints ..................................................................................................................... 48

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Custom SituationsWhen 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 Surface designer 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.

This chapter consists of five examples addressing these custom situations.


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..

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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 FactorsIt 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 Minimumoption 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.

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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 3.7Prediction 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.


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.


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 X1and X2 levels are well balanced over the weight values.

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Figure 3.12

Three-dimensional Spinning Plot of

Two Design Factors, Observed

Covariate Values and Predicted

Covariate Grid

Mixtures with Nonmixture FactorsThis 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 DonevMixture 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.

Figure 3.16 Prediction Variance Plot for Ten- Run Design.


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 ConstraintsSometimes 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 Cu

stom

ized 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 3rd from 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

3 Cu

stom

ized II


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.

4 Screen

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Chapter 4 Screening Designs 53

Chapter 4 Screening Designs

Screening designs are the most popular designs for industrial experimentation. They are

attractive because they are a cheap and efficient way to begin improving a process.

The purpose of screening experiments is to identify the key factors affecting a response.

Compared to other design methods, screening designs require fewer experimental runs,

which is why they are cheap.

The efficiency of screening designs depends on the critical assumption of effect sparcity.

Effect sparcity results because ‘real-world’ processes usually have only a few driving

factors; other factors are relatively unimportant. To understand the importance of effect

sparcity, you can contrast screening designs to full factorial designs.

A full factorial consists 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•3•4=24 runs.

By contrast screening designs reduce the number of runs in two ways:

❿ restricting the factors to two (or three) levels.

❿ performing only a fraction of the full factorial design

Applying these to the case described above, you can restrict the factors to two levels, which

yields 2•2•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 reduces the 24-run

experiment to 4 runs.

Of course, there is a price for this reduction. This chapter discusses the screening approach

in detail, showing both pros and cons.

54 Chapter 4 Screening Designs

Chapter 4Contents

Screening Design Types ........................................................................................................... 55Two-Level Full Factorial ................................................................................................... 55Two-Level Fractional Factorial ......................................................................................... 55Plackett-Burman Designs .................................................................................................. 56Mixed-Level Designs ........................................................................................................ 57Cotter Designs ................................................................................................................... 57

A Screening Example ............................................................................................................... 58Two-Level Design Selection and Description................................................................... 59Design Output Options ...................................................................................................... 60The Coded Design and Factor Generators......................................................................... 61Aliasing of Effects ............................................................................................................. 63Output Options for the JMP Design Table ........................................................................ 63The Design Data Table...................................................................................................... 64

Loading and Saving Responses and Factors (Optional) ........................................................... 66A Simple Effect Screening Analysis ........................................................................................ 67

Main Effects Report Options ............................................................................................. 67The Actual-by-Predicted Plot ............................................................................................ 68The Scaled Estimates Report ............................................................................................. 68

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Screening Design TypesThe design list for the Screening designer features four types of designs. The discussion

below compares and contrasts these design types.

Two-Level Full FactorialA full factorial design contains 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

1, 1, 1

–1. –1, –1

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.

Two-Level Fractional FactorialA 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.

Like the full factorial, fractional factorial designs are

orthogonal.

The big trade-off in screening designs is between the

number of runs and what is often referred to as the

resolution of the design. If price is no object, you–1. –1, 1

1, 1, 1

–1. 1, –1 1,– 1, –1

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 planning which interaction effects are confounded with the other

interaction effects.

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

Main effects are not confounded with other main effects. They 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

Main effects are not confounded with either other main effects or two-factor

interactions. However, two-factor interactions can be confounded with other two-factor

interactions.

resolution ≥ 5

There is no confounding between main effects, between two-factor interactions, or

between main effects and two-factor interactions.

All the fractional factorial designs are minimum aberration designs. A minimum aberration

design is one in which there are a minimum number of confoundings for a given resolution.

Plackett-Burman DesignsPlackett-Burman designs are an alternative to fractional factorials for screening. One useful

characteristic is that the sample size is a multiple of 4 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 3 fractional factorial where two-factor

interactions are indistinguishable from main effects.

In cases of effect sparcity, a stepwise regression approach can allow for removing some

insignificant main effects while adding highly significant and only somewhat correlated

two-factor interactions.

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Mixed-Level DesignsIf 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.

Table 4.1 Types of Mixed-Level Designs

Design Two–Level Factors Three–Level Factors

L18 John 1 7

L18 Chakravarty 3 6

L18 Hunter 8 6

L36 11 12

If you have less than or equal to the number of factors for a design listed in Table 4.1, 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.

Cotter DesignsCotter 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, JMP 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:

AOdd = A + ABC AEven = AB + AC

BOdd = B + ABC BEven = AB + BC

COdd = C + ABC CEven = AC + BC

It turns out that 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 recommend against

them.

A Screening ExampleExperiments 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.

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 following example, adapted from Meyer, et. al. (1996), demonstrates how to use the

JMP Screening designer. In this study, a chemical engineer investigates the effects of five

factors on the percent reaction of a chemical process. The factors are:

❿ feed rate, the amount of raw material added to the reaction chamber in liters per minute

❿ percentage of catalyst

❿ stir rate, the RPMs of a propeller in the chamber

❿ reaction temperature in degrees Celsius

❿ concentration of reactant.

To begin, choose Screening Design from the DOE tab on the JMP Starter or from the DOE

main menu.

4 Screen

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Two-Level Design Selection and DescriptionWhen you choose Screening Design the dialog shown in Figure 4.1 appears. Fill in the

number of factors (up to 31). For the reactor example add 5 factors. Then, modify the factor

names and give them high and low values. To edit the names of factors, double click on the

text and type new names.

❿ Change the default names (X1-X5) to Feed Rate, Catalyst, Stir Rate, Temperature,

and Concentration.

❿ Enter the high and low values as shown in Figure 4.1.

Figure 4.1 Factor Names and Values

Note that the Responses outline level is closed. Click the disclosure diamond to open it.

You see one default response called Y. Double click on the name and change it to

Percent Reacted.

In this experiment the goal is to maximize the response, which is

the default goal. To see the popup list of other goal choices shown

to the right, click on the word Maximize.

Change the minimum acceptable reaction percentage to 90 as shown in Figure 4.2. When

you complete these changes, click Continue. (See Figure 4.1 ).


Figure 4.2

Response Name andGoal

Now, JMP lists the designs for the number of factors you specified, as shown to the left in

Figure 4.3. Select the first item in the list, which is an 8-run fractional factorial design.

Click Continue again to see the Design Output Options panel on the right in

Figure 4.3.

Figure 4.3 Two-level Screening Design (left) and Design Output Options (right)

Design Output OptionsThe Design Output Options Panel supplies ways to describe and modify a design.

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.

4 Screen

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Coded Design

Shows the pattern of high and low values for the factors in each run.

Run Order Choice

Controls sorting or randomization through the Run Order Choice popup menu.

Center Points

Add center points by entering the number you want in the edit box. The default is zero.

Replicates

Add the desired number of replicates in the edit box. One replicate doubles the number

of runs.

Make Table

Creates a JMP table of the design with columns for the factors and responses.

Backup

Removes the Design Output Options Panel and re-displays the list of designs.

The Coded Design and Factor GeneratorsOpen Coded Design to see the pattern of high and low levels for each run as shown to the

left in Figure 4.4. Each row is a run. Plus signs designate high levels and minus signs

represent low levels.

Note that 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.

The Change Generating Rules table to the right in Figure 4.4 also shows how the last

two columns are constructed in terms of the first three columns. The check marks for

Temperature show it is a function of Feed Rate, Catalyst, and Stir Rate. The check

marks for Concentration show it is a function of Catalyst and Stir Rate.


Figure 4.4 Default Coded Designs and Generating Rules

You can change the check marks in the Change Generating Rules panel to change the

coded design. For example, if you enter check marks as in Figure 4.5 and click Apply, the

Coded Design changes as shown. The first three columns of the coded design remain a

full factorial for the first three factors (Feed Rate, Catalyst, and Stir Rate).

Note: Be sure to click Apply to switch to the new generating rules.

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 4.5 Modified Coded Designs and Generating Rules

4 Screen

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Aliasing of EffectsA full factorial with 5 factors requires 25=32 runs. Eight runs can only accommodate a full

factorial with three, 2-level factors. As described above, 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 8 is effect aliasing (confounding).

Confounding is the direct result of the assignment of new factor values to products of the

coded design columns. For example, the values for Temperature are the product of the

values for Feed Rate and Catalyst. This means you can’t tell the difference of the effect

of Temperature and the synergistic (interactive) effect of Feed Rate and Catalyst.

The Aliasing of Effects panel shows which effects are confounded with which other effects.

It shows effects and confounding up to two-factor interactions. In the example shown in

Figure 4.6 all the main effects are confounded with two-factor interactions. This is

characteristic of resolution 3 designs.

Figure 4.6

Aliasing of Effects

Panel

Output Options for the JMP Design TableThe design dialog has options to modify the final design table as follows:

Run Order

gives the popup menu (shown next), which determines the order of runs as they will

appear in the JMP data table.


Number of Center Points

lets you add as many

additional center points as

you want.

Number of Replicates

lets you repeat the complete set experimental runs a specified number of times.

The Design Data TableWhen you click Make Table JMP creates and displays the data table shown in

Figure 4.7 that lists the runs for the design you selected. In addition, it has a column called

Y for recording experimental results, as shown to the right of the data table.

The high and low values you specified show for each run. If you don’t enter values in the

Design Specification dialog, the default is –1 and 1 for the low and high values of each

factor. The column called Pattern shows the pattern of low values denoted “–” and high

values denoted “+”. Pattern is especially suitable to use as a label variable in plots.

Figure 4.7 Modified Coded Designs and Generating Rules

4 Screen

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The Design of Experiments facility in JMP automatically

generates a JMP data table with a JSL script that creates a

Model Specification dialog with the appropriate model for the

analysis of the specified design. If you double click on the

Table Property name, Model, the dialog shown here appears

with the JSL script generated

by the DOE facility.

The model generated by this

example contains all the

main effects and two estimable interaction terms, as shown in Figure 4.8. The two-factor

interactions in the model actually represent a group of aliased interactions. Any predictions

made using this model implicitly assume that these interactions are active rather than the

others in the group.

Figure 4.8

Model

Specification

Dialog

Generated by

the Design

Table with

Interaction

Term Added


Loading and Saving Responses and Factors (Optional)

If you plan to do further experiments with factors you have

given meaningful names and values, it is convenient to save the

factor information and load the stored information directly into

the Factors panel. The popup menu on the Design Experiment

title bar has commands to save the information you entered, and

retrieve it later to reconstruct a design table. The reactor data is

a good example. The names and values of the 5 factors shown

in the dialog can be saved to a JMP data table with the Save

Factors command in the platform popup menu.

Save Factors creates the JMP

Data table shown here. The

data table contains a column

for each factor, and a row for

each factor level. You use the

Save Factors command to

name the table and save it.

To load the factor names and

level values into the DOE

dialog:

❿ open the data table that contains the factor names and levels

❿ select the design type you want from the DOE menu

❿ choose Load Factors from the Design dialog menu.

Use the same steps to save and reload information about Responses.

See Chapter 1, “Design of Experiments (DOE)” for a description of all the platform

commands.

4 Screen

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A Simple Effect Screening AnalysisOf the five factors in the reaction time experiment, you expect a few to stand out in

comparison to the others. The next sections show an approach to an analysis that looks for

active effects, using the table generated previously by the DOE facility and the model in

Figure 4.7.

Open the sample data table Reactor 8 Runs.jmp to run the model generated by the data, as

shown previously in Figure 4.6-7. You can choose the Model script stored as a Table

Property (automatically generated by the DOE facility) to see the Model Specification

dialog, or choose Fit Model from the Analyze menu and the model saved as a Table

Property by the DOE facility automatically fills the Model Specification dialog.

Main Effects Report OptionsThe Fit Model report consists of the outline shown to the left in Figure 4.9. The Factor

Profiling command in the platform menu shown to the right in Figure 4.9 accesses these

effect profiling tools:

❿ Profiler shows how a predicted response changes as you change any factor.

❿ Interaction Plots gives multiple profile plots across one factor under different settings

of another factor.

❿ Contour Profiler shows how predicted values change with respect to changing factors

two at a time.

❿ Cube Plots show predicted values in the corners of the factor space.

❿ Box Cox Transformation finds a power transformation of the response that would fit

best.

Figure 4.9 Platform Commands for Fit Model Report


The Actual-by-Predicted PlotThe Actual-by-Predicted plot is at the top

of the report. The pattern variable in the

data table shows as the label for each

point.

The mean line falls inside the bounds of

the 95% confidence curves, which tells

you that the model is not significant. The

model p-value, R-square 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. In this case the RMSE is

14.199, which is much larger than expected. This suggests that effects other than the main

effects of each factor are important. Because of the confounding between two-factor

interactions and main effects in this design, it is impossible to determine which two-factor

interactions are important without performing more experimental runs.

The Scaled Estimates ReportThis report shows 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. None of the factor effects are significant, but the Catalyst effect is large enough to

be interesting if it is real. At this stage the results are not clear, but this does not mean that

the experiment has failed. It means that some follow-up runs are necessary.

If you want to find out how this story ends, look ahead in the Augmented Designs chapter.

For comparison, Chapter 6, “Full Factorial Designs” has the complete 32-run factorial

experimental data and analysis.

5 Su

rface 69

Chapter 5Response Surface Designs

Response surface designs are useful for modeling a curved surface (quadratic) to continuous

factors. If a minimum or maximum response exists inside the factor region, a response

surface model can pinpoint it. 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 by the

axialpoints

centerpoints

fractional factorial points

�

�

�

�

�

��

��

��

� ��

diagram. It combines a two-level fractional

factorial and two other kinds of points:

❿ Center points, for which all the factor

values are at the zero (or midrange)

value.

❿ Axial (or star) points, for which all but

one factor set at zero (midrange) and

one factor set at outer (axial) values.

��

��

��

��

The Box-Behnken design, shown to the

left, 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.

70

Chapter 5Contents

Response Surface Designs........................................................................................................ 71The Response Surface Design Dialog ............................................................................... 71The Design Table .............................................................................................................. 72Axial Scaling Options ....................................................................................................... 73A Central Composite Design ............................................................................................. 74Fitting the Model ............................................................................................................... 75

A Box-Behnken Design: The Tennis Ball Example ................................................................. 76Geometry of a Box-Behnken Design ................................................................................ 78Analysis of Response Surface Models .............................................................................. 78

5 Su

rfaceChapter 5 Response Surface Designs 71

Response Surface Designs

The Response Surface Design DialogThe Response Surface Design command on the DOE main

menu (or DOE JMP Starter tab page) displays the dialog, shown

to the left in Figure 5.1, for you to enter factors and responses.

When you click Continue the list of design selections shown on

the right appears. The response surface design list has a Box-

Behnken design and two types of central composite design,

called 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:

❿ Uniform precision means that the number of center points is chosen so that the

prediction variance at the center is approximately the same as at the design vertices.

❿ 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.

Figure 5.1 Design Dialogs to Specify Factors and Choose Design Type

To complete the dialog, enter the number of factors (up to eight) and click Continue. In the

table shown to the right in Figure 5.1, the 15- run Box-Behnken design is selected. Click

Continue to use this design.

72 Chapter 5 Response Surface Designs

The left panel in Figure 5.2 shows the next step of the dialog. To reproduce the right panel

of Figure 5.2 specify 1 replicate with 2 center points per replicate, and change the run order

popup choice to Randomize. When you finish specifying the output options you want, click

Make Table.

Figure 5.2 Design Dialog to Modify Order of Runs and Simulate Responses

The Design TableThe JMP data table (Figure 5.3) lists the design runs specified in Figure 5.2. Note that the

design table also has a column called Y for recording experimental results.

Figure 5.3

The JMP DesignFacility

AutomaticallyGenerates a JMP

Data Table

5 Su


Axial Scaling OptionsWhen you select a central composite design and then click Continue, you see the dialog on

the right in Figure 5.4. The dialog supplies default axial scaling information but you can

use the options described next and enter the values you want.

Figure 5.4 CCD Design With a Specified Type of Axial Scaling

The axial scaling options control how far out the axial points are:

Rotatable

makes the variance of prediction depend only on the scaled distance from the center of

the design.

Orthogonal

makes the effects orthogonal in the analysis.

In both previous cases the axial points are more extreme than the –1 or 1 representing the

range of the factor. If this factor range cannot be practically achieved, then you can choose

either of the following options:

On Face

is the default. These designs leave the axial points at the end of the -1 and 1 ranges.

User Defined

uses the value entered by the user, which can be any value greater than zero.

Inscribe

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 in from that).


A Central Composite DesignThe generated design, shown in the JMP data table in Figure 5.3, lists the runs for the

design specified in Figure 5.2. Note that the design table also has a column called Y for

recording experimental results.

Figure 5.5 shows the specification and design table for a 20-run 6-block Central Composite

design with simulated responses.

Figure 5.5 Central Composite Response Surface Design

The column called Pattern identifies the coding of the factors. The Pattern column shows

all the factor codings with “+” for high, “–” for low, “a” and “A” for low and high axial

5 Su


values, and “0” for midrange. If the Pattern variable is a label column, then when you click

on a point in a plot of the factors, the pattern value shows the factor coding of the point.

Note: The resulting data table has a Table Variable called Designthat contains the design type. This variable appears as a note at the

top of the Tables panel to the left of the data grid. In this example,

Design says CCD-Orthogonal Blocks. The table also contains a

model script stored as a Table Property, and displayed as a menu

icon labeled Model.

.

Fitting the ModelWhen you click the Table Property icon for the model (in the Tables panel to the left of the

data grid), a popup menu appears with the Run Script command. The Run Script command

opens the Model Specification dialog window and lists the appropriate effects for the model

you selected. This example has the main effects and interactions as seen in Figure 5.6.

When you collect data, you can key or paste them into the design table and run this model.

The model is permanently stored with the data table.

Figure 5.6 Model Specification dialog for Response Surface Design

&RS

&RS

&RS


A Box-Behnken Design: The Tennis Ball Example

The Bounce Data.jmp sample data file has the response surface data inspired by the tire

tread data described in Derringer and Suich (1980). The objective is to match a standardized

target value, given as 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 begin, select Response Surface Design from the DOE menu. The responses and factors

information is in existing JMP files found in the Design Experiment Sample Data folder.

Use the Load Responses and Load Factors commands in the popup menu on the RSM

Design title bar to load the response file called Bounce Response.jmp and the factor file

called Bounce Factor.jmp. Figure 5.7 shows the completed Response panel and Factors

panel.

Figure 5.7

Response and Factors

For Bounce Data

After the response data and factors data loads, the Response Surface Design Choice dialog

lists the designs in Figure 5.8.

5 Su


Figure 5.8 Response Surface Design Selection

The Box-Behnken design selected for three effects generates the design table of 15 runs

shown in Figure 5.9. The data are in the Bounce Data.jmp sample data table. The Table

Variable (Model) runs a script to launch the Model Specification dialog.

After the experiment is conducted, the responses are entered into the JMP table.

Figure 5.9

JMP Table for a

Three-Factor Box-

Behnken Design


Geometry of a Box-Behnken DesignThe geometric structure of a design with three effects is seen by using the Spinning Plot

platform. The spinning plot shown in Figure 5.10 illustrates the three Box-Behnken design

columns. Options available on the spin platform draw rays from the center to each point,

inscribe the points in a box, and suppress the x, y, and z axes. You can clearly see the 12

points midway between the vertices, leaving three points in the center.

Figure 5.10

Spinning Plot

of a Box-

Behnken

Design for

Three Effects

Analysis of Response Surface ModelsTo analyze response surface designs, select the Fit Model command from the Analyze

menu and designate the surface effects in the Model Specification dialog. To do this, select

the surface effects in the dialog variable selection list and add them to the Effects in Model

list. Then select Response Surface from the Effect Attributes popup menu (see

Figure 5.6).

However, if the table to be analyzed was generated by the DOE Response Surface

designer, then the Run Model table variable script automatically assigns the response

surface attribute to the factors, as previously illustrated in Figure 5.6.

Analysis ReportsThe standard analysis results appear in tables shown in Figure 5.11, with parameter

estimates for all surface and crossed effects in the model.

5 Su


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.

Figure 5.11 JMP Statistical Reports for a Response Surface Analysis of Bounce Data

See Chapter 9, “Standard Least Squares: Introduction“ in the JMP Statistics and Graphics

Guide for more information about interpretation of the tables in Figure 5.11.

The Response Surface report also has the tables shown in Figure 5.12:

❿ The Response Surface table is a summary of the parameter estimates.

❿ The Solution table lists the critical values of the surface factors and tells the kind of

solution (maximum, minimum, or saddlepoint).

❿ The Canonical Curvature table shows eigenvalues and eigenvectors of the effects.

Note that the solution for the Bounce example is a saddlepoint. The Solution table also

warns that the critical values given by the solution are outside the range of data values.

See Chapter 11, “Standard Least Squares: Exploring the Prediction Equation“ in the JMP

Statistics and Graphics Guide for details about the response surface analysis tables in

Figure 5.12.


Figure 5.12 Statistical Reports for a Response Surface Analysis

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 5.13.

The Prediction ProfilerThe response Prediction Profiler gives you a closer look at the response surface to find the

best settings that produce the response target. It is a way of changing one variable at a time

and looking at the effects on the predicted response.

Open the Prediction Profiler with the

Profiler command from the Factor

Profiling popup menu on the Response

title bar. The Profiler displays 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 first profile in Figure 5.13 show initial settings for the factors Silica, Silane, and

Sulfur, which result in a value for Stretch of 396, which is close to the specified target of

450. However, you can adjust the prediction traces of the factors and find a Stretch value

that is closer to the target.

The next step is to choose Desirability Functions from the popup menu on the Profiler title

bar. This command appends a new row of plots to the bottom of the plot matrix, which graph

5 Su


desirability on a scale from 0 to 1. The row has a plot for each factor, showing its desirability

trace, as illustrated by the second profiler in Figure 5.13. The Desirability Functions

command also adds a column that has an adjustable desirability function for each Y variable.

The overall desirability measure appears to the left of the row of desirability traces.

The response goal for Stretch is a target value of 450, as illustrated by the desirability

function in Figure 5.13. If needed, you can drag the middle handle on the desirability

function vertically to change the target value. The range of acceptable values is determined

by the positions of the upper and lower handles. See Chapter 11, “Standard Least Squares:

Exploring the Prediction Equation“ in the JMP Statistics and Graphics Guide for further

discussion of the Prediction Profiler.

The overall desirability shows to the left of the row of desirability traces. However, note in

this example that the desirability function is set to 450, the target value. 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 5.13 Prediction Profiler for a Response Surface Analysis


You can adjust the factor traces by hand to change the predicted value of Stretch. Another

convenient way to find good factor settings is to select Maximize Desirability from the

Prediction Profiler popup menu. This command adjusts the profile traces to produce the

response value closest to the specified target (the target given by the desirability function).

Figure 5.14 shows the result of the most desirable settings. Changing the settings of Silicafrom 1.2 to 0.94512, Silane from 50 to 50.0038, and Sulfur from 2.3 to 2.11515 raised the

predicted response from 396 to the target value of 450.


A Response Surface PlotAnother way to look at the response surface is to use the Contour Profiler. The Contour

Profiler command in the Factor Profiling menu brings up the interactive contour profiling

facility as shown in Figure 5.15. It is useful for optimizing response surfaces graphically,

especially when there are multiple responses. This example shows the profile to Silica and

Silane for a fixed value of Sulphur.

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, as shown in the contour plot in Figure 5.15.

The sliders for each factor set values for Current X and Current Y. The surface plots (mesh

plots) at the bottom of the report illustrate the effect on the response surface when you set

Sulphur to its minimum (40) and then to its maximum (60). This change in the surface

shape clearly shows that there is interaction between Sulfur and the other factors .

5 Su



Silane=40 Silane=60

Figure 5.16 shows the Contour profile when the Current X values have the most desirable

settings as shown at the bottom in Figure 5.14 .


Figure 5.16 Prediction Profiler with High and Low Limits

The Prediction Profiler and the Contour Profiler are discussed in more detail in Chapter 11

of the Statistics and Graphics Guide, “Standard Least Squares: Exploring the Prediction

Equation.”

6 Facto

rial85

Chapter 6Full Factorial Designs

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•3•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 on the left. For more

factors, the design point lie on 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.

86

Chapter 6Contents

The Factorial Dialog ................................................................................................................. 87The Five-Factor Reactor Example............................................................................................ 88

6 Facto

rialChapter 6 Factorial Designs 87

The Factorial DialogTo start, select Full Factorial Design in the DOE main menu, or click the Full FactorialDesign button on the JMP Starter DOE tab page. The popup menu on the right in

Figure 6.1 illustrates the way to specify categorical factors with 2 to 9 levels. Add a

continuous factor and two categorical factors with three and four levels respectively.

Change the levels to those shown at the left in Figure 6.1.

Figure 6.1 Full Factorial Factor Panel

When you finish adding factors, click Continue. to

see a panel of output options (as shown to the right).

When you click Make Table, the table shown

in Figure 6.2 appears. Note that the values in the

Pattern column describe the run each row

represents. For continuous variables, plus or minus

signs represent high and low levels. Level numbers

represent values of of categorical variables.

88 Chapter 6 Factorial Designs

Figure 6.2 2x3x4 Full Factorial Design Table

inus sign for ow level of ontinuous factor

lus sign for igh level of ontinuous actor

evel number for ategorical ariable

The Five-Factor Reactor ExampleResults from the reactor experiment described in Chapter 4, “Screening Designs” can be

found in the Reactor 32 Runs.jmp sample data folder, (Box, Hunter, and Hunter 1978, pp

374-390). The variables have the same names: Feed Rate, Catalyst, Stir Rate,

Temperature, and Concentration. These are all two-level continuous factors.

To create the design yourself, select Full Factorial Design from the DOE main menu (or

toolbar), or click Full Factorial Design on the DOE tab page of the JMP Starter window. Do

the following to complete the Response panel and the Factors panel:

❿ Use the Load Responses command from the popup menu on the Full Factorial Design

title bar and open the Reactor Response.jmp file to get the response specifications.

❿ Likewise, use the Load Factors command and open the Reactor Factors.jmp file to

get the Factors panel.

You should see the completed dialog shown in Figure 6.3.

6 Facto


Figure 6.3

Full-Factorial Example

Response and Factors

Panels

A full factorial design includes runs for all

combinations of high and low factors for the five

variables, giving 32 runs. Click Continue to see

Output Options panel shown to the right.

When you click Make Table, the JMP Table in

Figure 6.4 is constructed with a run for every

combination of high and low values for the five

variables, and an empty Y column for

entering response values when the experiment is complete. The table has 32 rows, which

cover all combinations of a five factors with two levels each. The Reactor 32 Runs.jmpsample data file has these experimental runs and the results from the Box, Hunter, and

Hunter study. Figure 6.4 shows the runs and the response data.


Figure 6.4 25 Factorial Reactor Data (Reactor 32.jmp sample data)

6 Facto


Begin the analysis with a quick

look at the data before fitting the

factorial model. The plot on the

right shows a distribution of the

response, Percent Reacted,

using the Normal Quantile plot

option on the Distribution

command on the Analyze menu.

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 that includes main effects and all two factor interactions, and brings up

the Stepwise control panel. To do this, choose Run Script from the Fit Model popup menu

on the title bar of the Reactor 32 Run.jmp table.

The Stepwise Regression Control Panel appears with a

preliminary Current Estimates report. The probability to enter a

factor into the model is 0.05 (the default is 0.25), and the

probability to remove a factor is 0.1.

A useful way to use Stepwise is to check

all the main effects in the Current

Estimates table, and then use Mixed as

the Direction for the stepwise process,

which can both include or exclude

factors in the model.

Change from default settings:Prob to Enter Factor is .05 Prob to Leave factor is .10Mixed direction instead of Forward or Backward

To do this, click the check boxes for the main effects of the factors as shown in

Figure 6.5, and click Go on the Stepwise control panel.


Figure 6.5 Starting Model For Stepwise Process

The Mixed stepwise procedure removes insignificant main effects and adds important

interactions. The end result is shown in Figure 6.6. Note that the Feed Rate and Stir Ratefactors are no longer in the model.

Figure 6.6 Model After Mixed Stepwise Regression

Click the Make Model button to generate a new model dialog. The Model Specification

dialog automatically has the effects identified by the stepwise model (Figure 6.7).

6 Facto


Figure 6.7

Model Dialog

for Fitting a

Prediction

Model

Click Run Model to see the analysis for a candidate

prediction model. The figure to the right shows the

whole model leverage plot. 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.

Figure 6.8 shows a table of model coefficients and their standard errors. All effects selected

by the stepwise process are highly significant.

Figure 6.8Prediction Model

Estimates


The factor Prediction Profiler also gives

you a way to compare the factors and find

optimal settings.

Open the Prediction Profiler with the

Profiler command on the Factor Profiling

submenu on the Response title bar. The Prediction Profiler is discussed in more detail in

Chapter 5, “Response Surface Models” in this book, and Chapter 11, “Standard Least

Squares: Exploring the Prediction Equation” of the JMP Statistics and Graphics Guide.”

The top profile in Figure 6.9 shows the initial settings. An easy way to find optimal settings

is to choose Desirability Functions from the popup menu on the profiler title bar.

Then select Maximize Desirability,

as shown here. These selections

give the bottom profile in

Figure 6.9. The plot of Desirabilityversus Percent Reacted shows

that the goal is to

maximize Percent Reacted. The reaction is unfeasible economically unless the PercentReacted is above 90%, therefore the Desirability for values less than 90% is 0.

Desirability increases linearly as the Percent Reacted increases.

The maximum Desirability is 0.9445 when Catalyst and Temperature are at their highest

settings, and Concentration is at its lowest setting. Percent Reacted increases from 65.5

at the center of the factor ranges to 95.2875 at the most desirable setting.

6 Facto


Figure 6.9 Initial Profiler Settings and Optimal Settings

7 Tag

uch

i 97

Chapter 7Taguchi 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.

Dividing system variables into signal and noise factor roles is a key ingredient in robust

engineering. Signal factors are system control inputs. Noise factors are variables that are

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 the DOE platform supports signal and noise factors, inner and

outer arrays, and signal-to-noise ratios as Taguchi specifies.

98

Chapter 7Contents

The Taguchi Design Approach ................................................................................................. 99Taguchi Design Example ......................................................................................................... 99Analyze the Byrne-Taguchi Data ........................................................................................... 103

7 Tag

uch

iChapter 7 Taguchi Arrays 99

The Taguchi Design ApproachThe 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 7.1 lists the

recommended performance statistics.

Table 7.1 Taguchi's Signal to Noise Ratios

Goal S/N Ratio Formula

nominal is best Y2

sSN =–10log

2

larger-is-better (maximize) =–10logS

NLTB1n Σ

i Yi2

1

smaller-is-better (minimize) =–10log 1n

Yi2Σ

i

SNSTB

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.

The data are in the Byrne Taguchi Data.jmp data table in the Sample Data folder, but you

can generate the original design table with the Taguchi designer of the JMP DOE facility.

Table 7.2 shows the signal and noise factors for this example.

100 Chapter 7 Taguchi Arrays

Table 7.2 Definition of Adhesiveness Experiment Effects

Factor Name Type Levels Comment

Interfer control 3 tubing and connector interference

Wall control 3 the wall thickness of the connector

IDepth control 3 insertion depth of the tubing into the connector

Adhesive control 3 percent adhesive

Time noise 2 the conditioning time

Temp noise 2 temperature

Humidity noise 2 the relative humidity

The factors for the example are in the JMP file called Byrne Taguchi Factors.jmp, found

in the DOE Sample Data folder. To start this example,

1) open the factors table.

2) choose Taguchi from the DOE main menu or toolbar, or click the Taguchi button on the

DOE tab page of the JMP Starter.

3) Select Load Factors in the platform popup menu as

shown here. The factors panel then shows the four three-

level control (signal) factors and three noise factors listed

in Figure 7.1.

Figure 7.1

Response, and Signal and

Noise Factors for the

Byrne-Taguchi Example

7 Tag

uch


When you click Continue, the list of

available inner and outer array designs

appears. This example uses the designs

highlighted in the design choice panel

shown to the right. L9-Taguchi gives the

L9 orthogonal array for the inner design.

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.

Click Make Table to create the design table shown in

Figure 7.2.

The pull-off adhesive force measures are collected and entered into the new columns,

shown in the bottom table of Figure 7.3. As a notational convenience, the Y column names

are ‘Y’ appended with 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.

Figure 7.2 Taguchi Design Before Data Entry


Figure 7.3 Complete Taguchi Design Table

The column called 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.

–10Log10 Mean1 1 1 1 1 1 1 1

y- - + 2,

y - - - 2,

y+- - 2,

y+- +2,

y++-2,y+++2

,,y - -+ 2

,2y - + +

This expression is large when all of the individual Y values are small.

7 Tag

uch


Analyze the Byrne-Taguchi Data

The data are now ready to analyze. The

Table Property called Model in the Tables

panel runs a JSL script that launches the

Fit Model platform shown to the right.

The default model includes the main

effects of the four Signal factors. The two

responses are the mean and S/N Ratio over

the outer array. The goal of the analysis is

to find factor settings that maximize both

the mean and the S/N Ratio.

The prediction profiler is a quick way to

find settings that give the highest signal-

to-noise ratio for this experiment. The

default prediction profile has all the

factors set to low levels as shown in the

top of Figure 7.4. The profile traces indicate that different settings of the first three factors

would increase SN Ratio Y.

The Prediction Profiler has a popup menu with options to help find the best settings for a

given Desirability Function. The Desirability Functions option adds the row of traces and

column of function settings to the profiler, as shown at the bottom in Figure 7.4. The

default desirability functions are set to larger-is-better, which is what you want in this

experiment. See Chapter 11, ”Standard Least Squares: Perspectives on the Estimates,” in

The JMP Statistics and Graphics Guide for more details about the Prediction Profiler.

After the Desirability Functions option is in effect, you can

choose Maximum Desirable, which automatically sets the

prediction traces to give the best results according to the

desirability functions. In this example you can see that the

settings for Interfer and Wall changed from L1 to L2. The

Depth setting changed from L1 to L3. There was no change in

Adhesive. These new settings increased the signal-to-noise

ratio from 24.0253 to 29.9075.


Figure 7.4 Best Factor Settings for Byrne Taguchi Data

8 Mixtu

re 105

Chapter 8Mixture Designs

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 0 and 1. The sum of the

proportions in any mixture recipe is 1 (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. The interpretation of screening

experiments is 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.

x3

x1

x2

1

0

1

1

triangular feasible region

Because the proportions sum to one, mixture designs

have an interesting geometry. The feasible region for a

mixture 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, as illustrated in the diagram to the left.

You can rotate the plane to see the triangle face-on and

see the points in the form of a ternary plot.

Design Experiment offers the following types of designs for mixtures:

❿ simplex centroid

❿ simplex lattice

❿ extreme vertices

❿ ABCD designs.

The extreme vertices design is the most flexible, since it handles constraints on the values

of the factors.

106

Chapter 8Contents

The Mixture Design Dialog .................................................................................................... 107Mixture Designs ..................................................................................................................... 108

Simplex Centroid Design ................................................................................................ 108Simplex Lattice Design ................................................................................................... 110Extreme Vertices ............................................................................................................. 112

Extreme Vertices Design for Constrained Factors ................................................................. 113Adding Linear Constraints to Mixture Designs...................................................................... 114

Details on Extreme Vertices Method for Linear Constraints .......................................... 115Ternary and Tetrary Plots ....................................................................................................... 115Fitting Mixture Designs.......................................................................................................... 116

Whole Model Test and Anova Report ............................................................................. 117Response Surface Reports ............................................................................................... 117

Chemical Mixture Example.................................................................................................... 118Plotting a Mixture Response Surface ..................................................................................... 119

8 Mixtu

reChapter 8 Mixture Designs 107

The Mixture Design DialogThe Mixture Design command on the DOE main menu or

JMP Starter DOE tab page displays the standard Add Factors

panel. When you click Continue, the Mixture dialog shown

Figure 8.1, lets you select one of the following types of

design:

Simplex Centroid

You specify the degree up to which the factor

combinations are to be made.

Simplex Lattice

You specify how many levels you want on each edge of the grid.

Extreme Vertices

You specify linear constraints or restrict the upper and lower bounds to be within the 0

to 1 range.

ABCD Design

This approach by Snee (1975) generates a screening design for mixtures.

Figure 8.1 Mixture Design Selection Dialog

• For Simplex Centroid - enter K• Simplex Lattice - enter Levels• Extreme Vertices - enter Degree

The design table appears when you click a design type button. The following sections show

examples of each mixture design type.

108 Chapter 8 Mixture Designs

Mixture Designs

If the process of interest is determined by a mixture of components, the relative proportions

of the ingredients, rather than the absolute amounts, needs to be studied. In mixture designs

all the factors sum to 1.

Simplex Centroid DesignA simplex centroid design of degree k with nf factors is composed of mixture runs with

❿ all one factor

❿ all combinations of two factors at equal levels

❿ all combinations of three factors at equal levels

❿ 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.

The table of runs for a design of degree 1 with three factors (left in Figure 8.2) 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 8.2 Three-Factor Simplex Centroid Designs of Degrees 1 and 2

Run X1 X2 X3 Run X1 X2 X3

1 1 0 0 1 1 0 0

2 0 1 0 2 0 1 0

3 0 0 1 3 0 0 1

4 0.333 0.333 0.333 4 0.5 0.5 0

5 0.5 0 0.5

6 0 0.5 0.5

7 0.333 0.333 0.333

8 Mixtu


To generate the set of runs in Figure 8.2, choose the Mixture Design command from the

DOE menu and enter three continuous factors. You should see the designs in Figure 8.3.

Figure 8.3 Create Simplex Centroid Designs of Degrees 1 and 2

Simplex Centroid, K=1 Simplex Centroid, K=2

As another example, enter 5 for the number of factors and click Continue. When the

Mixture Design dialog appears, the default value of K is 4, which is fine for this example.

Click Simplex Centroid. When the design appears, click Make Table to see the 31-run JMP

data table shown in Figure 8.4. Note that the first five runs have only one factor. The next

ten runs have all the combinations of two factors. Then, there are ten runs for three-factor

combinations, five runs for four-factor combinations, and (as always) the last run with all

factors.


Figure 8.4 Data Table of Runs for Five-Factor Simplex Centroid Design

Simplex Lattice DesignThe 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, specify the number of levels you want in the design

dialog (Figure 7.1) and click Simplex Lattice. Figure 8.5 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 center point.

Figure 8.6 lists the runs for a simplex lattice of degree 3 for five effects. In the five-level

example, you can see the runs creep across the hyper-triangular region and fill the space in

a grid-like manner.

8 Mixtu


Figure 8.5 Three-Factor Simplex Lattice Designs for Factor Levels 3, 4, and 5

Figure 8.6 JMP Design Table for Simplex Lattice, Order (Degree) 3


Extreme VerticesThe extreme vertices design incorporates limits on factors into the design and picks the

vertices and their averages formed by these limits as the design points. The additional limits

are usually in the form of range constraints, upper bounds, and lower bounds on the factor

values.

The following example design

table is for five factors with the

constraints shown here, where the

ranges are smaller than the default

0 to 1 range. Click Continue and

enter 4 as the Degree.

Figure 8.7 shows a partial listing

of the JMP design table.

Figure 8.7 JMP Design Table for Extreme Vertices with Range Constraints

Details on Extreme Vertices Method for Range ConstraintsIf the only constraints are 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.

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

8 Mixtu


based on the restriction that the factors’ values must sum to 1. 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.

Extreme Vertices Design for Constrained FactorsThe 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 1 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 neighboring vertices joined by a line is a 2nd degree

centroid because a line is two dimensional. The centroid point for vertices sharing a plane is

a 3rd 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.


Adding Linear Constraints to Mixture Designs

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:

X1 ≥ 0.1X1 ≤ 0.5X2 ≥ 0.1X2 ≤ 0.7X3 ≤ 0.7

90 ≤ 85*X1 + 90*X2 + 100*X385*X1 + 90*X2 + 100*X3 ≤ 95

.4 ≤ 0.7*X1 + X3

You first enter the upper and

lower limits in the factors panel

as shown here. Click Continue to

see the Mixture Design dialog.

The Extreme Vertices selection

on the Mixture Design dialog has

an additional button to add linear constraints. Click the Linear Constraints button for each

constraint you have. In this example you need three constraint dialogs.

Figure 8.8 shows constraints panels completed for each of the constraints given previously.

After the constraints are entered, click Extreme Vertices, then Make Table to see the JMP

table in Figure 8.8.

Figure 8.8

Constraints and Table of Runs for Snee(1979)

Mixture Model Example

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Details on Extreme Vertices Method for Linear ConstraintsThe extreme vertices implementation for linear constraints is based on the CONSIM algo-

rithm 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 for doing centroid points is by Piepel (1988), named

CONAEV.

If there are only range constraints, check Add Linear Constraints to see the results of the

CONSIM method, rather than the results from the XVERT method normally used by JMP.

Ternary PlotsThe Piepel (1979) example is best understood by the ternary plot shown in Figure 8.9. Each

constraint is a line. The area that satisfies all constraints is the shaded feasible area. There

are six active constraints, six vertices, and six centroid points shown on the plot, as well as

two inactive (redundant) constraints.

Figure 8.9

Ternary Plot

Showing Piepel

Example

Constraints2,32,4

3,5

4,6Center

1,51,6 2(1)

2(2)

2(3)

2(4)

2(5)2(6)

1, 01, 0

.4, .6

.5, .5

.6, .4

.7, .3

.8, .2

.9, .1

0, 1

.1, .9

.2, .8

.3, .7

0, 1 .1, .9 .2, .8 .3, .7 .4, 6 .5, .5 .6, .4 .7, .3 .8, .2 .9, .1 1, 0

.4, .6

.5, .5

.6, .4

.7,.3

.8, .2

.9, .1

0, 1

.1, .9

.2, .8

.3, .7

X1

X2 X3

X1 ≥ .1

X1 ≤ .5

X2 ≥ .1

X2 ≤ .7 X3 ≤ .7

90 ≤ 85*X1 + 90*X2 + 100*X3

85*X1 + 90*X2 + 100*X3 ≤ 95

.4 ≤ .7*X1 + X3


A mixture problem in three components can be represented in two dimensions because the

third component is a linear function of the others. This ternary plot shows how close to 1 a

given component is by how close it is to the vertex of that variable in the triangle. The plot

in Figure 8.10 illustrates a ternary plot.

Figure 8.10

Ternary Plot for Mixture Design

Fitting Mixture DesignsWhen 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 model to fit a mixture response

surface is

❿ to suppress the intercept

❿ to include all the linear main-effect terms

❿ to exclude all the square terms (like X1*X1)

❿ to include all the cross terms (like X1*X2)

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This model is called the Scheffe polynomial (Scheffe 1958).

This is the model JMP DOE creates and stores with the data

table as a Table Property. This Table Property, called Model,

runs the script to launch the Model Specification dialog, which is

automatically filled with the saved model.

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 is all one factor.

The coefficients on the cross terms indicate the curvature across each edge of the factor space.

Whole Model Test and Anova ReportIn the whole-model Anova 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 R-square to be reported with respect to the

intercept model, rather than reported as missing.

Response Surface ReportsWhen JMP encounters effects that are marked as response surface effects “&RS”, it creates

additional reports that analyze the resulting 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. It can do this for any combination

yielding a constant and involving the last response surface term.

Unfortunately, the contour-plot feature of these reports does not fold to handle mixtures. If

you want a contour plot of the surface, you can either refit using a full response surface that

omits the last factor, or use the Contour Plot platform in the Graph menu, and add points to

make the plot less granular.


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.

To create Cornell’s mixture design used in JMP:

❿ Select Mixture Design from the DOE menu or JMP Starter DOE tab page.

❿ In the Factors panel, request 3 factors.

Name them p1, p2, and p3, and enter the

high and low constraints as shown here.

❿ Click Continue, then specify a degree of

three in Mixture Design Type dialog

for an Extreme Vertices design.

❿ When you click Make Design, then Generate Table,

JMP generates a table with the first 9 runs as shown

here to the right.

For this problem, the experimenter added an extra 5

design runs by duplicating the vertex points and center

point shown highlighted in the table, giving a total of 14

rows in the design table. After the experiment is

complete, the results of the experiment (thickness) are

entered in the Y column. Use the Plasticizer.jmp sample

data to see the experimental results (Y values).

To run the mixture model either use the Table Property

called Model, which runs a script that creates the completed Model Specification dialog, or

choose Fit Model from the Analyze menu, select p1, p2 and p3 as mixture response surface

effects, and Y as the Y variable. Then click Run Model, and when the model has run,

choose Save Prediction Formula from the Save commands in the platform popup menu.

The predicted values show as a new column in the data table. To see the prediction formula,

open the formula for that column:

0–50.1465*p1–282.1982*p2–911.6484*p3+p2*317.363

+p3*p1*1464.3298+p3*p2*1846.2177

Note: These results correct the coefficients reported in Cornell[1990].

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When you fit the response surface model, the

Response Surface Solution report shows that a

maximum predicted value of 19.570299 occurs

at point (0.63505, .015568, 0.20927).

You can visualize the results of a mixture

design with the Profiler in the Fit Model

platform, and a Ternary plot, as described in the

next section.

Plotting a Mixture Response SurfaceThe Fit Model platform automatically displays a Prediction Profiler when the analysis

emphasis is effect screening. If the Profiler is not visible, you can select the Profiler

command from the Factor Profiling popup menu to display it.

The Profiler to the right, for

the chemical mixture

example, shows optimal

settings of 0.6615 for p1,

0.126 for p2, and 0.21225 for

p3, which give an estimated

response of 19.26923.

The crossed effects show as curvature in the prediction traces. When you drag one of the

vertical reference lines, the other two move in the opposite direction maintaining their ratio.

To plot a mixture response surface choose Ternary from the Graph menu (or toolbar), or

click Ternary on the Graph tab page of the JMP Starter. Specify plot variables in the

Launch dialog shown in Figure 8.11

Optionally, you can identify a contour variable if there is one. The contour variable must

have a prediction formula to form the contour lines as shown by the Ternary Plots at the

bottom in Figure 8.11. The Ternary platform only shows points if there is no prediction

formula. The prediction equation is often the result of using the Save Prediction Formula

command after fitting the response surface mixture.


Figure 8.11 Termary Plot of a Mixture Response Surface

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121

Chapter 9Augmented Designs

It is best to treat experimentation as an iterative process. That way 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 supports the following four ways to extend previous experimental

work:

Add Centerpoints

Adding centerpoints is useful to check for curvature and to reduce the prediction error

in the center of the factor region.

Replication

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.

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.

D-optimal Augmentation

D-optimal augmentation is a power 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.

This chapter provides an overview of the interface of the Augment designer. It also

presents a case study of design augmentation using the reactor example from

Chapter 4, “Screening Designs.”

122

Chapter 9Contents

The Augment Design Interface............................................................................................... 123Replicate Design ............................................................................................................. 124Add Centerpoints ............................................................................................................. 125Fold Over ......................................................................................................................... 125

The Reactor Example Re-visited ............................................................................................ 126Interface for D-Optimal Augmentation ........................................................................... 126Analyze the Augmented Design...................................................................................... 130

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Chapter 9 Augmented Designs 123

The Augment Design InterfaceThe augment design feature of JMP DOE gives the ability to modify an existing design data

table. If you do not have an open JMP table when you select Augment Design from the

DOE menu, or from the DOE tab on the JMP Starter, the File Open dialog for your

computer appears as in Figure 9.1. Select a data set that you want to augment. For this

example, use the Reactor 8 Runs.jmp data in the Design Experiment sample data folder.

This table was generated previously in Chapter 4, “Screening Designs.”

Figure 9.1

File Open Dialog to Open

a Design Data Table

After the file opens, the dialogs in Figure 9.2 prompt you to identify the factors and

responses you want to use for the augmented design.

Figure 9.2 Choose Columns for Factors and Responses

124 Chapter 9 Augmented Designs

Select the columns that are model factors and click OK. Then select the column or columns

that are responses. When you click OK again, the dialog below appears with the list of

factors and factor values that were saved with the design data table. Buttons on the dialog

give four choices for

augmenting a design:

❿ Replicate

❿ Add Centerpoints

❿ Fold Over

❿ Augment

The next sections describe

how to use these augment-

ation choices.

Replicate DesignThe Replicate button displays the dialog shown

here. Enter the number of times to perform each

run. Enter two (2) in the dialog text entry to

specify that you want each run to appear twice

in the resulting design. This is the same as one replicate. Figure 9.3 shows the Reactordata with one replicate.

Figure 9.3

Design With

One Replicate

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Add CenterpointsWhen you click Add

Centerpoints, a dialog

appears for you to enter

the number of center-

points you want. The

table shown to the right

is the design table for

the reactor data with

two center points

appended to the end of the table.

Fold OverWhen you select Foldover and click Make Data Table, the JMP Table that results has an

extra column called Block as shown in Figure 9.4. The first set of runs is block 1 and the

new (foldover) runs are block 2.

Note: Adding centerpoints or replicating the design also generates an additional Blockcolumn in the JMP Table.

Figure 9.4 Listing of a Foldover Design for the Reactor Data


The Reactor Example Re-visited

The factors in the previous section were from the reactor example in the Chapter 4,

“Screening Designs.” This section returns to that example, which had ambiguous results. To

begin, open the Reactor 8 Runs.jmp table from the Design Experiment sample data folder

(if it is not already open).

Interface for D-Optimal AugmentationAfter you identify the factors

and response and click OK, the

Augment Design dialog shown

to the right appears.

Now click Augment on this

dialog to see the display shown

in Figure 9.5.

This display is the same as the one for Custom Design, except that the only factor control

is the Add Block Factor button. Click Add Block Factor to add a two-level block factor to

the factors panel. The original runs are the first block level and the new runs that result from

augmenting the design are the second level. Choosing this option means that the augment

design algorithm will optimally block the new runs versus the original runs.

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Figure 9.5 Augment User Interface


To continue with the reactor analysis, choose 2nd from the Interactions popup menu as

shown on the left in Figure 9.6, which adds all the two-factor interactions to the model. The

minimum number of runs given the specified model is 16, as shown in the Design

Generation text edit box. You can increase this number by clicking in the box and typing a

new number.

Figure 9.6 Augmented Model

When you click Make Design, the DOE facility computes D-optimally augmented factor

settings, as shown in Figure 9.7.

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Figure 9.7 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 9.7, (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 to the right shows the

random number (1859832026)

used to generate the runs in

Figure 9.7.


Figure 9.8 is the data table data from the corresponding runs in the Reactor Example from

Chapter 6, "Full Factorial Designs." The Reactor Augment Data.jmp sample data file

contains these runs. The example analysis in the next section uses this data table.

Figure 9.8 Completed Augmented Experiment

Analyze the Augmented DesignTo start the analysis, run the Fit Model script stored as a table property

with the data table. This table property contains the JSL commands that

display the stepwise regression control panel shown in Figure 9.9. Click

the check boxes for all the main effect terms.

Note: If you generate a data table using the design dialog, the table property automatically

generated by the DOE facility is called Model and contains a standard least squares fit

model script. This data table has a script written specifically to do a stepwise regression.

The stepwise regression can then do a standard least squares model fit after selecting

effects.

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Figure 9.9 Initial Stepwise Model

Click Go to see the stepwise regression process begin and continues 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 9.10 shows the result of this example analysis.

Note that Feed Rate and Stir Rate are out of the model while the Temperature*Catalystand the Temperature*Concentration interactions have entered the model.


Figure 9.10 Completed Stepwise Model

After Stepwise is finished, click Make Model on the Stepwise control panel to generate this

reduced model, as shown in Figure 9.11. You can now fit the reduced model to do

additional diagnostic work, make predictions, and find the optimal factor settings.

Figure 9.11 New Prediction Model Dialog

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The ANOVA and Lack of Fit Tests in Figure 9.12 indicate a highly significant regression

model with no evidence of Lack of Fit.

Figure 9.12

Prediction Model

Analysis of Variance and

Lack of Fit Tests

The Scaled Estimates table in Figure 9.13 show that Catalyst has the largest main effect.

However, the significant 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 Chapter 4,

“Screening Designs,” had ambiguous results.

Figure 9.13 Prediction Model Estimates Plot

It is desirable to maximize the percent reaction. The prediction profile plot in

Figure 9.14 shows that maximum occurs at the high levels of Catalyst and Temperatureand the low level of Concentration. When you drag the prediction traces for each factor to

their maximum settings, the estimate of Percent Reacted increases from 65.375 to

95.6635.


Figure 9.14

Maximum Percent

Reacted

To summarize, compare the analysis of 16 runs with the analyses of reactor data from

previous chapters:

❿ In Chapter 4, “Screening Designs,” 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.

❿ Chapter 6, “Full Factorial Designs,” 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.

❿ 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.

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135

Chapter 10Prospective Power and Sample Size

Prospective analysis helps 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 a prospective power analysis, an estimates of the group means and sample sizes

in a data table and an estimate of the within-group standard deviation (σ) are required in the

Power Details dialog.

The Sample Size, Power command in the DOE menu determine how large a sample is

needed to be reasonably likely that an experiment or sample will yield a significant result,

given that he true effect size is at least a certain size.

The Sample Size, Power platform handles the following cases:

¨ Testing one sample's mean is different from a hypothesized value.

¨ Testing two samples have the same mean

¨ Testing that there are differences in the means among k samples.

The Power and Sample Size facility assumes that there are equal numbers of units in each

group.

You can also apply this facility to more general experimental designs, if they are balanced,

and a number-of-parameters adjustment is specified.

136

Chapter 10Contents

Prospective Power Analysis ................................................................................................... 137Launch the Sample Size and Power facility ........................................................................... 137

Single-Sample Mean ....................................................................................................... 139Two-Sample Means......................................................................................................... 141k-Sample Means .............................................................................................................. 142

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Chapter 10 Power and Sample Size 137

Prospective Power AnalysisThe following five values have an important relationship in a statistical test on means:

¨ Alpha is the significance level that prevents declaring a zero effect significant more

than alpha portion of the time.

¨ Error Standard Deviation is the unexplained random variation around the means.

¨ Sample Size is how many experimental units (runs, or samples) are involved in the

experiment.

¨ Power is the probability of declaring a significant result.

¨ Effect Size is how different the means are from each other or from the hypothesized

value.

The Sample Size and Power facility in JMP helps estimate in advance either the sample size

needed, power expected, or the effect size expected in the experimental situation where

there is a single mean comparison, a two sample comparison, or when comparing k sample

means.

The Sample Size, Power

command is on the DOE main

menu (or toolbar), or on the

DOE tab page of the JMP

Starter. When you launch this

facility, the dialog shown here

appears with a button

selection for three experimental situations. Each of these selections then displays its own

dialog that prompts for estimated parameter values and the desired computation.

Launch the Sample Size and Power FacilityAfter you click either One Sample Mean , Two Sample Means , or k Sample Means in the

initial dialog (shown above), the next dialog asks for the anticipated experimental values.

The values you enter depend on your initial choice. As an example, consider the two-

sample situation.

138 Chapter 10 Power and Sample Size

The Two Sample Means choice in the

initial power dialog always requires values

for Alpha and the error standard deviation

(Error Std Dev) , as shown here, and one or

two of the other three values: Difference todetect, Sample Size, and Power. The

power facility then calculates the missing

item. If there are two unspecified fields, the

power facility constructs a plot that shows

the relationship between those 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 dialog asks for the values depending the first choice of design:

Alpha

is the significance level, usually .05. This implies willingness to accept (if the true

difference between groups is zero) that 5% (alpha) of the time a significant difference

will be incorrectly declared.

Error Std Deviation

is the true residual error. Even though the true error is not known, the power

calculations are an exercise in probability that calculates what might happen if the true

values were as specified.

Extra Params

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 the 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—two parameters for the extra main effects, and three parameters for

the interactions. In practice, it isn’t very important what values you enter here unless

the experiment is in a range where there is very few degrees of freedom for error.

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Difference to Detect

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.

Sample Size

is the total number of observations (runs, experimental units, or samples). Sample size

is not the number per group, but the total over all groups. Computed sample size

numbers can have fractional values, which you need to adjust to real units. This is

usually done by increasing the estimated sample size to the smallest number evenly

divisible by the number of groups.

Power

is the probability of getting a statistic that will be declared statistically significant.

Bigger power is better, but the cost is higher in sample size. Power is equal to alpha

when the specified effect size is zero. You should go for powers of at least .90 or .95 if

you can afford it. If an experiment requires considerable effort, plan so that the

experimental design has the power to detect a sizable effect, when there is one.

Continue

evaluates at the entered values.

Backup

means go back to the previous dialog.

Single-Sample MeanSuppose there is a single sample and the goal is to detect a difference of 2 where the error

variance is .9, as shown in the left-hand dialog in Figure 10.1 To calculate the power when

the sample size is 10, leave Power missing in the dialog and click Continue . The dialog on

the right in Figure 10.1 shows the power is calculated to be .99998, rounding to 1.


Figure 10.1 A One-Sample Example

To see a plot of the relationship of power and sample size, leave both Sample Size and

Power missing and click Continue .

Double click on the horizontal axis to get any desired scale. The right-hand graph in

Figure 10.2 shows a range of sample sizes for which the power varies from about 0.2 to

.95. Change the range of the curve by changing the range of the horizontal axis. For

example, the plot on the right in Figure 10.2 has the horizontal axis scaled from 1 to 8,

which gives a more typical looking power curve.

Figure 10.2 A One-Sample Example

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When only Sample Size, is specified (Figure 10.3) and Difference to Detect and Powerare left blank, a plot of power by difference appears.

Figure 10.3 Plot of Power by Difference to Detect for a Given Sample Size

Two-Sample MeansThe dialogs work similarly for two samples; the Difference to Detect is the difference

between two means. Suppose the error variance is .9 (as before), the desired detectable

difference is 1, and the sample size is 16.

Leave Power blank and click Continue to see the power calculation, 0.5433, as shown in

the dialog on the left in Figure 10.4. This is considerably lower than in the single sample

because each mean has only half the sample size. The comparison is between two random

samples instead of one.

To increase the power requires a larger sample. To find out how large, click Backup on the

Power Calculation dialog. Leave Sample Size and Power both blank and examine the plot

shown on the right in Figure 10.4. The crosshair tool estimates that a sample size of about

35 is needed to obtain a power of 0.9.


Figure 10.4 Plot of Power by Difference to Detect for a Given Sample Size

k-Sample MeansThe k-sample situation can examine up to 10 kinds of means. The next example considers a

situation where 4 levels of means are expected to be about 10 to 13, and the Error Std Devis 0.9. When a sample size of 16 is entered the power calculation is 0.95.

As before, if you leave both Sample Size and Power are left blank, the power facility

produces the power curve shown on the right in Figure 10.5. 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+.(–5)2+.052+1.52, which is the square root of 5.

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Figure 10.5 Prospective Power for k-Means and Plot of Power by Sample Size

Referen

cesReferences 145

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Bose, R.C., (1947) "Mathematical Theory of the Symmetrical Factorial Design"Sankhya: The Indian Journal of Statistics, Vol 8, Part 2, pp. 107-166.

Box, G.E.P. and Meyer, R.D. (1986), “An analysis of Unreplicated Fractional Factorials,”Technometrics 28, 11–18.

Box, G.E.P. and Draper, N.R. (1987), Empirical Model–Building and Response Surfaces,New York: John Wiley and Sons.

Box, G.E.P. (1988), “Signal–to–Noise Ratio, Performance Criteria, andTransformations,” Technometrics 30, 1–40.

Box, G.E.P., Hunter,W.G., and Hunter, J.S. (1978), Statistics for Experimenters, NewYork: John Wiley and Sons, Inc.

Byrne, D.M. and Taguchi, G. (1986), ASQC 40th Anniversary Quality Control CongressTransactions, Milwaukee, WI: American Society of Quality Control, 168–177.

Chen, J., Sun, D.X., and Wu, C.F.J. (1993), “A Catalogue of Two-level and Three-LevelFractional Factorial Designs with Small Runs,” International Statistical Review, 61,1, p131-145, International Statistical Institute.

Cochran, W.G. and Cox, G.M. (1957), Experimental Designs, Second Edition, NewYork: John Wiley and Sons.

Cornell, J.A. (1990), Experiments with Mixtures, Second Edition New York: John Wiley& Sons.

Daniel, C. (1959), "Use of Half–normal Plots in Interpreting Factorial Two–levelExperiments," Technometrics, 1, 311–314.

Daniel C. and Wood, F. (1980), Fitting Equations to Data, Revised Edition, New York:John Wiley and Sons, Inc.

Derringer, D. and Suich, R. (1980), “Simultaneous Optimization of Several ResponseVariables”, Journal of Quality Technology, Oct 1980, 12:4, 214–219.

Haaland, P.D. (1989), Experimental Design in Biotechnology, New York: Marcel Dekker,Inc.

146 References

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John, P.W.M. (1972), Statistical Design and Analysis of Experiments, New York:Macmillan Publishing Company, Inc.

Johnson, M.E. and Nachtsheim, C.J. (1983), “Some Guidelines for Constructing ExactD–Optimal Designs on Convex Design Spaces,” Technometrics 25, 271–277.

Jones, Bradley (1991), “An Interactive Graph For Exploring Multidimensional RespnseSurfaces,” 1991 Joint Statistical Meetings, Atlanta, Georgia

Khuri, A.I. and Cornell J.A. (1987) Response Surfaces: Design and Analysis, New York:Marcel Dekker.

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Mahalanobis, P.C. (1947), "Sankhya," The Indian Journal of Statistics, Vol 8, Part 2,April.

Myers, R.H. (1976) Response Surface Methodology, Boston: Allyn and Bacon.

Meyers, R.H. (1988), Response Surface Methodology, Virginia Polytechnic and StateUniversity.

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Meyer, R.D., Steinberg, D.M., and Box, G.(1996), Follow-up Designs to ResolveConfounding in Multifactor Experiments, Technometrics , Vol. 38, #4, p307.

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Plackett, R.L. and Burman, J.P. (1947), “The Design of Optimum MultifactorialExperiments,” Biometrika, 33, 305–325.

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Snee, R.D., and Marquardt D. (1975), "Extreme vertices designs for linear mixturemodels", Technometrics 16 399-408.

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Snee, R.D. (1979), “Experimental Designs for Mixture Systems with MulticomponentConstraints,” Commun. Statistics, A8(4), 303–326.

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Ind

esIndex 149

Index

AABCD, mixture design 105actual-by-predicted plot 68add center points, augment design 125aliasing of effects 60, 63analysis example

augmented design 130-134mixture design 116response surface design 78-84screening design 67-68

augment design 121-134add center points 121, 125analysis example 130-134block factor 126D-optimal 121, 126data table 125, 129foldover design 121, 125interface 123, 126-128Model Specification dialog 130random number seed 129replicate design 121, 124stepwise regression 130

axial points, RSM 69, 73axial scaling options 73backup button 61

BBig Class.jmp sample data 43BounceData.jmp sample data 76BounceFactor.jmp sample data 76BounceResponse.jmp sample data 76Box-Behnken, RSM 69, 71, 76-78Box-Cox transformation 67Byrne Taguchi Data.jmp sample data 99Byrne Taguchi Factors.jmp sample data 100

Ccanonical curvature, RSM 80center points 63, 69, 71central composite design, RSM 69, 74coded design 60, 61column property (data table) 14constraints, loading and saving 14contour profiler

response su rface design 83screening design 67

covariate factors 43cube plot 67cubic model, custom design 26Cubic Model.jsl sample script 27custom design 17, 33-51

all two-factor interactions 31all two-factor interactions involving

only one factor 30cubic model 26data table 22design generation panel 20-21dialog 19-23factor constraints 48factors, defining 19fixed covariate factors 43-45flexible block sizes 36-38internal details 32JSL scripting example 27main effects only 28mixture with nonmixture factors 47model panel 20modify design interactively 23number of runs 21output options 21

150 Index

prediction variance profiler 24quadratic model 24random number seed 15RSM with categorical factors 38-42screening design examples 28-31

DD-Optimal augmentation 126-129data table 11, 13

augmented design 125, 129custom design 22design role 14extreme vertices mixture design 112full factorial design 90pattern variable 63replicates 63response surface design 72run script command 75screening design 61, 63simplex centroid mixture design 110simplex lattice mixture design 111simulated response 63table property 75Taguchi arrays 101variable constraint state (DOE) 14

design choicesmixture design 107response surface design 77screening designs 10, 60Taguchi arrays 101

design output options 60-61desirability trace, prediction variance

profiler 81Diamond Constraints.jmp sample data 48DOE Example 1.jmp sample data 11DOE main menu 3-6

Augment Design 5, 121-134Custom Design 4, 17-32, 33-51Full Factorial Design 5, 85-95

Mixture Design 5, 105-120Response Surface Design 4, 69-84Sample Size, Power 6, 135-143Screening Design 4, 53-68Taguchi Arrays 5, 97-104

Donev Mixture Factors.jmp sample data 45

Eeffect sparcity 53, 56extreme vertices mixture design 105, 112

Ffactors 13

constraints 48-51entering into dialog 9generators 61profiling 67saving and loading 13

factors panelcustom design 19screening design 58Taguchi design 100

foldover design, augment design 125full factorial design 85, 93-95

5-factor example 88analysis example 91data table 90dialog 87load responses and factors 88prediction variance profiler 94sample size 85stepwise regression 91

Iinner array, Taguchi arrays 97interaction plot 67

JJMP Starter DOE tab 3

Ind

exIndex 151

LL18, L36 screening designs 57loading constraints 15loading factors and responses 66, 76, 88

MMain menu, DOE 3mixture design 105-120

analysis example 116constrained factors 113-115data table 110, 111design choices 107dialog 107extreme vertices 105, 112-113factor constraints 51prediction variance profiler 119response surface reports 117simplex centroid 105, 107, 109-110simplex lattice 105, 110ternary plot 115, 120

Model Specification dialog 11, 65augmented design 130full factorial design 93response surface model 75stepwise regression 132Taguchi arrays 103

Nnon-estimable effect 56

Oorthogonal axial scaling 73orthogonal design 55, 57, 71outer array, Taguchi arrays 97

Ppattern variable 63, 68, 74Plasticizer.jmp sample data 118power analysis 135-143

alpha 137, 138difference to detect 139

effect size 137error standard deviation 137extra parms 138k-sample means 142plotting 140, 142, 143power 137, 138, 139single sample 139standard error deviation 137two-sample means 141

prediction variance profileraugmented design 134custom design 24desirability function 94desirability trace 81full factorial design 94mixture model analysis 47prediction trace 81response surface design 81-82Taguchi arrays 103

prospective power analysis 6, 135-143

Qquadratic model, custom design 24

Rrandom number seed 15, 129Reactor 32 Runs.jmp, sample data 88Reactor 8 Runs.jmp sample data 67, 123,

126Reactor Augment Data.jmp sample data 130Reactor Factors.jmp sample data 88Reactor Response.jmp sample data 88replicate design, augment design 124replicates 60, 63resolution 56response surface design 69-84

3-d geormentric view 78analysis example 78-84analysis reports 79axial points 69

152 Index

axial scaling 72Box-Behnken 69, 71, 76-78canonical curvature 80categorical factors 38-42central composite 69, 74contour profiler 83data table 72design choices 77dialog 71factor constraints 48load responses and factors 76Model Specification dialogorthogonal 71pattern variable 74plotting 82-84prediction variance profiler 81run script command 75simulate response 74solution 80star points 69uniform precision 71

response surface reports, mixture design 117responses 8-12, 13

entering into dialog 8saving and loading 13simulate 15

rotatable axial scaling 73run order 60

Ssample data

Big Class 43BounceData 76BounceFactor 76BounceResponse 76Byrne Taguchi Data 99Cubic Model.jsl 27Diamond Constraints 48DOE Example 1 11

Donev Mixture factors 45Plasticizer 118Reactor 32 Runs 88Reactor 8 Runs 67, 123, 126Reactor Augment Data 130Reactor Factors 88Reactor Response 88

Sample Size, Power command 135-143sample size, prospective 6saving constraints 15saving factors and responses 66scaled estimates report 68screening design 53-68

aliasing of effects 62analysis example 67center points 63coded design 61Cotter Design 57data table 64design choices 60dialog 7, 58example 58-65factor generators 61factors panel 58L18, L36 mixed-level designs 57loading factors and responses 66mixed-level designs 57Model Specification dialog 65non-estimable effect 56orthogonal 55output options 60, 63Plackett-Burman design 56replicates 63resolution 56response panel 58saving factors and responses 66simulate response 61, 63two-level fractional factorial 55

Ind

exIndex 153

two-level full factorial 55types 55-57

signal-to-noise ratio, Taguchi arrays 97signal-to-noise ratio, Taguchiarrays 99simplex centroid, mixture design 105, 109simplex lattice, mixture design 105, 111simulate responses 15, 61, 63, 74single sample power analysis 139star points, RSM 69stepwise regression

augmented design 130full factorial design 91

TTaguchi arrays 97-104

contour profiler 103data table 101design choices 101desirability function 103example 99-102inner array 97outer array 97signal-to-noise ratio 97-99

ternary plot, mixture design 115, 120

Uuniform precision, RSM 71utility functions 12

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