EIND 455 | 554 – Design of Experiments for Engineers
Course Project Requirements
Project Overview – Your project assignment is to design and implement a design of experiments
statistical analysis for a real world problem, selected by your team. The expectation is that
this experiment will include the task of collecting data, and then performing analysis to
determine the best available way to run the process. You may use data collected for projects
in other courses, but only with the written approval of the instructors for both courses.
Text
An important part of the learning process is designing a system to collect data and then
performing the actual collection. For this reason, use of publicly available historical data is
not permitted for this assignment.
As noted in the course syllabus, 10% (20% for students in 554) of your course grade is based
on the project. At the end of the semester, you are required to present your project in class
and submit a formal project report. The report is required to use a modified APA format
(example posted in D2L) with table and figures embedded within the manuscript.
Project Objectives
Apply statistical design of experiment and modeling techniques in order to gain an
understanding of how to apply concepts learned in the course to a non-textbook setting.
Gain experience with the problems associated with collecting data.
Gain experience making a formal presentation while working with a partner.
For EIND 554 – Apply materials from the text supplements or other sources beyond what
was covered in the course materials to your design and analysis.
Project Teams
You are responsible for building your own team of 2 people (3 allowed only with prior
approval from the instructor, no single person ‘teams’ allowed) and submitting the team
composition and presentation day preference to wschell@montana.edu by 27 February 12:00
noon. Any members not on a team by this time will be assigned one by the instructor.
Project Proposal – Due 8 March by 12:00 noon through the assignment DropBox on D2L.
One page summary
Must clearly identify the problem that is being solved or opportunity that will be better
understood through completion of the project.
Define the data collection plan and corresponding initial design for the experiment.
State a list of tasks that will be undertaken to complete the project. (project plan)
Identify the roles (with timelines) that each member of the group will hold.
Final Deliverables
Class presentation – 8 minute limit, 6 minute presentation + 2 minute Q&A – 8, 10, or
12 April. Presentation will go in order of team preference (set by order of arrival) unless
otherwise arranged by agreement with another team.
Final report – 15 April printed copy in class, e-copy (MS Word format) in DropBox on
D2L. 5 – 8 pages not including cover page or appendices. 1” margins with 1.5 line
spacing. Appendices may only be included if referenced in the body of the document.
Papers must include representative examples of analysis within figures or charts in the
body of the document.
EIND 455 | 554 – Design of Experiments for Engineers
Course Project Requirements
Project Overview – Your project assignment is to design and implement a design of experiments
statistical analysis for a real world problem, selected by your team. The expectation is that
this experiment will include the task of collecting data, and then performing analysis to
determine the best available way to run the process. You may use data collected for projects
in other courses, but only with the written approval of the instructors for both courses.
An important part of the learning process is designing a system to collect data and then
performing the actual collection. For this reason, use of publicly available historical data is
not permitted for this assignment.
As noted in the course syllabus, 10% (20% for students in 554) of your course grade is based
on the project. At the end of the semester, you are required to present your project in class
and submit a formal project report. The report is required to use a modified APA format
(example posted in D2L) with table and figures embedded within the manuscript.
Project Objectives
Apply statistical design of experiment and modeling techniques in order to gain an
understanding of how to apply concepts learned in the course to a non-textbook setting.
Gain experience with the problems associated with collecting data.
Gain experience making a formal presentation while working with a partner.
For EIND 554 – Apply materials from the text supplements or other sources beyond what
was covered in the course materials to your design and analysis.
Project Teams
You are responsible for building your own team of 2 people (3 allowed only with prior
approval from the instructor, no single person ‘teams’ allowed) and submitting the team
composition and presentation day preference to wschell@montana.edu by 27 February 12:00
noon. Any members not on a team by this time will be assigned one by the instructor.
Project Proposal – Due 8 March by 12:00 noon through the assignment DropBox on D2L.
One page summary
Must clearly identify the problem that is being solved or opportunity that will be better
understood through completion of the project.
Define the data collection plan and corresponding initial design for the experiment.
State a list of tasks that will be undertaken to complete the project. (project plan)
Identify the roles (with timelines) that each member of the group will hold.
Final Deliverables
Class presentation – 8 minute limit, 6 minute presentation + 2 minute Q&A – 8, 10, or
12 April. Presentation will go in order of team preference (set by order of arrival) unless
otherwise arranged by agreement with another team.
Final report – 15 April printed copy in class, e-copy (MS Word format) in DropBox on
D2L. 5 – 8 pages not including cover page or appendices. 1” margins with 1.5 line
spacing. Appendices may only be included if referenced in the body of the document.
Papers must include representative examples of analysis within figures or charts in the
body of the document.
Design and Analysis
of Experiments
Eighth Edition
DOUGLAS C. MONTGOMERY
Arizona State University
John Wiley & Sons, Inc.
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Library of Congress Cataloging-in-Publication Data:
Montgomery, Douglas C.
Design and analysis of experiments / Douglas C. Montgomery. — Eighth edition.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-14692-7
1. Experimental design. I. Title.
QA279.M66 2013
519.5'7—dc23
2012000877
ISBN 978-1118-14692-7
10 9 8 7 6 5 4 3 2 1
Preface
Audience
This is an introductory textbook dealing with the design and analysis of experiments. It is based
on college-level courses in design of experiments that I have taught over nearly 40 years at
Arizona State University, the University of Washington, and the Georgia Institute of Technology.
It also reflects the methods that I have found useful in my own professional practice as an engineering and statistical consultant in many areas of science and engineering, including the research
and development activities required for successful technology commercialization and product
realization.
The book is intended for students who have completed a first course in statistical methods. This background course should include at least some techniques of descriptive statistics,
the standard sampling distributions, and an introduction to basic concepts of confidence
intervals and hypothesis testing for means and variances. Chapters 10, 11, and 12 require
some familiarity with matrix algebra.
Because the prerequisites are relatively modest, this book can be used in a second course
on statistics focusing on statistical design of experiments for undergraduate students in engineering, the physical and chemical sciences, statistics, mathematics, and other fields of science.
For many years I have taught a course from the book at the first-year graduate level in engineering. Students in this course come from all of the fields of engineering, materials science,
physics, chemistry, mathematics, operations research life sciences, and statistics. I have also
used this book as the basis of an industrial short course on design of experiments for practicing technical professionals with a wide variety of backgrounds. There are numerous examples
illustrating all of the design and analysis techniques. These examples are based on real-world
applications of experimental design and are drawn from many different fields of engineering and
the sciences. This adds a strong applications flavor to an academic course for engineers
and scientists and makes the book useful as a reference tool for experimenters in a variety
of disciplines.
v
vi
Preface
About the Book
The eighth edition is a major revision of the book. I have tried to maintain the balance
between design and analysis topics of previous editions; however, there are many new topics
and examples, and I have reorganized much of the material. There is much more emphasis on
the computer in this edition.
Design-Expert, JMP, and Minitab Software
During the last few years a number of excellent software products to assist experimenters in
both the design and analysis phases of this subject have appeared. I have included output from
three of these products, Design-Expert, JMP, and Minitab at many points in the text. Minitab
and JMP are widely available general-purpose statistical software packages that have good
data analysis capabilities and that handles the analysis of experiments with both fixed and random factors (including the mixed model). Design-Expert is a package focused exclusively on
experimental design. All three of these packages have many capabilities for construction and
evaluation of designs and extensive analysis features. Student versions of Design-Expert and
JMP are available as a packaging option with this book, and their use is highly recommended. I urge all instructors who use this book to incorporate computer software into your
course. (In my course, I bring a laptop computer and use a computer projector in every
lecture, and every design or analysis topic discussed in class is illustrated with the computer.)
To request this book with the student version of JMP or Design-Expert included, contact
your local Wiley representative. You can find your local Wiley representative by going to
www.wiley.com/college and clicking on the tab for “Who’s My Rep?”
Empirical Model
I have continued to focus on the connection between the experiment and the model that
the experimenter can develop from the results of the experiment. Engineers (and physical,
chemical and life scientists to a large extent) learn about physical mechanisms and their
underlying mechanistic models early in their academic training, and throughout much of
their professional careers they are involved with manipulation of these models.
Statistically designed experiments offer the engineer a valid basis for developing an
empirical model of the system being investigated. This empirical model can then be
manipulated (perhaps through a response surface or contour plot, or perhaps mathematically) just as any other engineering model. I have discovered through many years of teaching
that this viewpoint is very effective in creating enthusiasm in the engineering community
for statistically designed experiments. Therefore, the notion of an underlying empirical
model for the experiment and response surfaces appears early in the book and receives
much more emphasis.
Factorial Designs
I have expanded the material on factorial and fractional factorial designs (Chapters 5 – 9) in
an effort to make the material flow more effectively from both the reader’s and the instructor’s viewpoint and to place more emphasis on the empirical model. There is new material
on a number of important topics, including follow-up experimentation following a fractional
factorial, nonregular and nonorthogonal designs, and small, efficient resolution IV and V
designs. Nonregular fractions as alternatives to traditional minimum aberration fractions in
16 runs and analysis methods for these design are discussed and illustrated.
Preface
vii
Additional Important Changes
I have added a lot of material on optimal designs and their application. The chapter on response
surfaces (Chapter 11) has several new topics and problems. I have expanded Chapter 12 on
robust parameter design and process robustness experiments. Chapters 13 and 14 discuss
experiments involving random effects and some applications of these concepts to nested and
split-plot designs. The residual maximum likelihood method is now widely available in software and I have emphasized this technique throughout the book. Because there is expanding
industrial interest in nested and split-plot designs, Chapters 13 and 14 have several new topics.
Chapter 15 is an overview of important design and analysis topics: nonnormality of the
response, the Box – Cox method for selecting the form of a transformation, and other alternatives; unbalanced factorial experiments; the analysis of covariance, including covariates in a
factorial design, and repeated measures. I have also added new examples and problems from
various fields, including biochemistry and biotechnology.
Experimental Design
Throughout the book I have stressed the importance of experimental design as a tool for engineers and scientists to use for product design and development as well as process development and improvement. The use of experimental design in developing products that are robust
to environmental factors and other sources of variability is illustrated. I believe that the use of
experimental design early in the product cycle can substantially reduce development lead time
and cost, leading to processes and products that perform better in the field and have higher
reliability than those developed using other approaches.
The book contains more material than can be covered comfortably in one course, and I
hope that instructors will be able to either vary the content of each course offering or discuss
some topics in greater depth, depending on class interest. There are problem sets at the end
of each chapter. These problems vary in scope from computational exercises, designed to
reinforce the fundamentals, to extensions or elaboration of basic principles.
Course Suggestions
My own course focuses extensively on factorial and fractional factorial designs. Consequently,
I usually cover Chapter 1, Chapter 2 (very quickly), most of Chapter 3, Chapter 4 (excluding
the material on incomplete blocks and only mentioning Latin squares briefly), and I discuss
Chapters 5 through 8 on factorials and two-level factorial and fractional factorial designs in
detail. To conclude the course, I introduce response surface methodology (Chapter 11) and give
an overview of random effects models (Chapter 13) and nested and split-plot designs (Chapter
14). I always require the students to complete a term project that involves designing, conducting, and presenting the results of a statistically designed experiment. I require them to do this
in teams because this is the way that much industrial experimentation is conducted. They must
present the results of this project, both orally and in written form.
The Supplemental Text Material
For the eighth edition I have prepared supplemental text material for each chapter of the book.
Often, this supplemental material elaborates on topics that could not be discussed in greater detail
in the book. I have also presented some subjects that do not appear directly in the book, but an
introduction to them could prove useful to some students and professional practitioners. Some of
this material is at a higher mathematical level than the text. I realize that instructors use this book
viii
Preface
with a wide array of audiences, and some more advanced design courses could possibly benefit
from including several of the supplemental text material topics. This material is in electronic form
on the World Wide Website for this book, located at www.wiley.com/college/montgomery.
Website
Current supporting material for instructors and students is available at the website
www.wiley.com/college/montgomery. This site will be used to communicate information
about innovations and recommendations for effectively using this text. The supplemental text
material described above is available at the site, along with electronic versions of data sets
used for examples and homework problems, a course syllabus, and some representative student term projects from the course at Arizona State University.
Student Companion Site
The student’s section of the textbook website contains the following:
1. The supplemental text material described above
2. Data sets from the book examples and homework problems, in electronic form
3. Sample Student Projects
Instructor Companion Site
The instructor’s section of the textbook website contains the following:
4.
5.
6.
7.
8.
9.
10.
Solutions to the text problems
The supplemental text material described above
PowerPoint lecture slides
Figures from the text in electronic format, for easy inclusion in lecture slides
Data sets from the book examples and homework problems, in electronic form
Sample Syllabus
Sample Student Projects
The instructor’s section is for instructor use only, and is password-protected. Visit the
Instructor Companion Site portion of the website, located at www.wiley.com/college/
montgomery, to register for a password.
Student Solutions Manual
The purpose of the Student Solutions Manual is to provide the student with an in-depth understanding of how to apply the concepts presented in the textbook. Along with detailed instructions on how to solve the selected chapter exercises, insights from practical applications are
also shared.
Solutions have been provided for problems selected by the author of the text.
Occasionally a group of “continued exercises” is presented and provides the student with a
full solution for a specific data set. Problems that are included in the Student Solutions
Manual are indicated by an icon appearing in the text margin next to the problem statement.
This is an excellent study aid that many text users will find extremely helpful. The
Student Solutions Manual may be ordered in a set with the text, or purchased separately.
Contact your local Wiley representative to request the set for your bookstore, or purchase the
Student Solutions Manual from the Wiley website.
Preface
ix
Acknowledgments
I express my appreciation to the many students, instructors, and colleagues who have used the six
earlier editions of this book and who have made helpful suggestions for its revision. The contributions of Dr. Raymond H. Myers, Dr. G. Geoffrey Vining, Dr. Brad Jones,
Dr. Christine Anderson-Cook, Dr. Connie M. Borror, Dr. Scott Kowalski, Dr. Dennis Lin,
Dr. John Ramberg, Dr. Joseph Pignatiello, Dr. Lloyd S. Nelson, Dr. Andre Khuri, Dr. Peter
Nelson, Dr. John A. Cornell, Dr. Saeed Maghsoodlo, Dr. Don Holcomb, Dr. George C. Runger,
Dr. Bert Keats, Dr. Dwayne Rollier, Dr. Norma Hubele, Dr. Murat Kulahci, Dr. Cynthia Lowry,
Dr. Russell G. Heikes, Dr. Harrison M. Wadsworth, Dr. William W. Hines, Dr. Arvind Shah,
Dr. Jane Ammons, Dr. Diane Schaub, Mr. Mark Anderson, Mr. Pat Whitcomb, Dr. Pat Spagon,
and Dr. William DuMouche were particularly valuable. My current and former Department
Chairs, Dr. Ron Askin and Dr. Gary Hogg, have provided an intellectually stimulating environment in which to work.
The contributions of the professional practitioners with whom I have worked have been
invaluable. It is impossible to mention everyone, but some of the major contributors include
Dr. Dan McCarville of Mindspeed Corporation, Dr. Lisa Custer of the George Group;
Dr. Richard Post of Intel; Mr. Tom Bingham, Mr. Dick Vaughn, Dr. Julian Anderson,
Mr. Richard Alkire, and Mr. Chase Neilson of the Boeing Company; Mr. Mike Goza, Mr. Don
Walton, Ms. Karen Madison, Mr. Jeff Stevens, and Mr. Bob Kohm of Alcoa; Dr. Jay Gardiner,
Mr. John Butora, Mr. Dana Lesher, Mr. Lolly Marwah, Mr. Leon Mason of IBM; Dr. Paul
Tobias of IBM and Sematech; Ms. Elizabeth A. Peck of The Coca-Cola Company; Dr. Sadri
Khalessi and Mr. Franz Wagner of Signetics; Mr. Robert V. Baxley of Monsanto Chemicals;
Mr. Harry Peterson-Nedry and Dr. Russell Boyles of Precision Castparts Corporation;
Mr. Bill New and Mr. Randy Schmid of Allied-Signal Aerospace; Mr. John M. Fluke, Jr. of
the John Fluke Manufacturing Company; Mr. Larry Newton and Mr. Kip Howlett of GeorgiaPacific; and Dr. Ernesto Ramos of BBN Software Products Corporation.
I am indebted to Professor E. S. Pearson and the Biometrika Trustees, John Wiley &
Sons, Prentice Hall, The American Statistical Association, The Institute of Mathematical
Statistics, and the editors of Biometrics for permission to use copyrighted material. Dr. Lisa
Custer and Dr. Dan McCorville did an excellent job of preparing the solutions that appear in
the Instructor’s Solutions Manual, and Dr. Cheryl Jennings and Dr. Sarah Streett provided
effective and very helpful proofreading assistance. I am grateful to NASA, the Office of
Naval Research, the National Science Foundation, the member companies of the
NSF/Industry/University Cooperative Research Center in Quality and Reliability Engineering
at Arizona State University, and the IBM Corporation for supporting much of my research
in engineering statistics and experimental design.
DOUGLAS C. MONTGOMERY
TEMPE, ARIZONA
Contents
Preface
v
1
Introduction
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1
8
11
14
21
22
23
Strategy of Experimentation
Some Typical Applications of Experimental Design
Basic Principles
Guidelines for Designing Experiments
A Brief History of Statistical Design
Summary: Using Statistical Techniques in Experimentation
Problems
2
Simple Comparative Experiments
2.1
2.2
2.3
2.4
2.5
2.6
2.7
25
Introduction
Basic Statistical Concepts
Sampling and Sampling Distributions
Inferences About the Differences in Means, Randomized Designs
25
27
30
36
2.4.1
2.4.2
2.4.3
2.4.4
2.4.5
2.4.6
2.4.7
36
43
44
48
50
50
51
Hypothesis Testing
Confidence Intervals
Choice of Sample Size
The Case Where !21 Z !22
The Case Where !21 and !22 Are Known
Comparing a Single Mean to a Specified Value
Summary
Inferences About the Differences in Means, Paired Comparison Designs
53
2.5.1
2.5.2
The Paired Comparison Problem
Advantages of the Paired Comparison Design
53
56
Inferences About the Variances of Normal Distributions
Problems
57
59
xi
xii
Contents
3
Experiments with a Single Factor:
The Analysis of Variance
3.1
3.2
3.3
3.4
3.5
An Example
The Analysis of Variance
Analysis of the Fixed Effects Model
66
68
70
3.3.1
3.3.2
3.3.3
3.3.4
71
73
78
79
3.8
3.9
Decomposition of the Total Sum of Squares
Statistical Analysis
Estimation of the Model Parameters
Unbalanced Data
Model Adequacy Checking
80
3.4.1
3.4.2
3.4.3
3.4.4
80
82
83
88
The Normality Assumption
Plot of Residuals in Time Sequence
Plot of Residuals Versus Fitted Values
Plots of Residuals Versus Other Variables
Practical Interpretation of Results
3.5.1
3.5.2
3.5.3
3.5.4
3.5.5
3.5.6
3.5.7
3.5.8
3.6
3.7
65
A Regression Model
Comparisons Among Treatment Means
Graphical Comparisons of Means
Contrasts
Orthogonal Contrasts
Scheffé’s Method for Comparing All Contrasts
Comparing Pairs of Treatment Means
Comparing Treatment Means with a Control
89
89
90
91
92
94
96
97
101
Sample Computer Output
Determining Sample Size
102
105
3.7.1
3.7.2
3.7.3
105
108
109
Operating Characteristic Curves
Specifying a Standard Deviation Increase
Confidence Interval Estimation Method
Other Examples of Single-Factor Experiments
110
3.8.1
3.8.2
3.8.3
110
110
114
Chocolate and Cardiovascular Health
A Real Economy Application of a Designed Experiment
Discovering Dispersion Effects
The Random Effects Model
116
3.9.1
3.9.2
3.9.3
116
117
118
A Single Random Factor
Analysis of Variance for the Random Model
Estimating the Model Parameters
3.10 The Regression Approach to the Analysis of Variance
125
3.10.1 Least Squares Estimation of the Model Parameters
3.10.2 The General Regression Significance Test
125
126
3.11 Nonparametric Methods in the Analysis of Variance
3.11.1 The Kruskal–Wallis Test
3.11.2 General Comments on the Rank Transformation
3.12 Problems
128
128
130
130
4
Randomized Blocks, Latin Squares,
and Related Designs
4 . 1 The Randomized Complete Block Design
4.1.1
4.1.2
Statistical Analysis of the RCBD
Model Adequacy Checking
139
139
141
149
Contents
4.1.3
4.1.4
4.2
4.3
4.4
4.5
Some Other Aspects of the Randomized Complete Block Design
Estimating Model Parameters and the General Regression
Significance Test
4.4.1
4.4.2
4.4.3
168
172
174
177
Statistical Analysis of the BIBD
Least Squares Estimation of the Parameters
Recovery of Interblock Information in the BIBD
Problems
183
Basic Definitions and Principles
The Advantage of Factorials
The Two-Factor Factorial Design
183
186
187
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
5.3.6
5.3.7
187
189
198
198
201
202
203
An Example
Statistical Analysis of the Fixed Effects Model
Model Adequacy Checking
Estimating the Model Parameters
Choice of Sample Size
The Assumption of No Interaction in a Two-Factor Model
One Observation per Cell
The General Factorial Design
Fitting Response Curves and Surfaces
Blocking in a Factorial Design
Problems
6
The 2k Factorial Design
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
155
158
165
168
Introduction to Factorial Designs
5.4
5.5
5.6
5.7
150
The Latin Square Design
The Graeco-Latin Square Design
Balanced Incomplete Block Designs
5
5.1
5.2
5.3
xiii
Introduction
The 22 Design
The 23 Design
The General 2k Design
A Single Replicate of the 2k Design
Additional Examples of Unreplicated 2k Design
2k Designs are Optimal Designs
The Addition of Center Points to the 2k Design
Why We Work with Coded Design Variables
Problems
206
211
219
225
233
233
234
241
253
255
268
280
285
290
292
7
Blocking and Confounding in the 2k
Factorial Design
7.1
7.2
7.3
Introduction
Blocking a Replicated 2k Factorial Design
Confounding in the 2k Factorial Design
304
304
305
306
xiv
Contents
7.4
7.5
7.6
7.7
7.8
7.9
Confounding the 2k Factorial Design in Two Blocks
Another Illustration of Why Blocking Is Important
Confounding the 2k Factorial Design in Four Blocks
Confounding the 2k Factorial Design in 2p Blocks
Partial Confounding
Problems
8
Two-Level Fractional Factorial Designs
8.1
8.2
8.3
8.4
8.5
8.6
320
Introduction
The One-Half Fraction of the 2k Design
320
321
8.2.1
8.2.2
8.2.3
321
323
324
Definitions and Basic Principles
Design Resolution
Construction and Analysis of the One-Half Fraction
The One-Quarter Fraction of the 2k Design
The General 2k!p Fractional Factorial Design
333
340
8.4.1
8.4.2
8.4.3
340
343
344
Choosing a Design
Analysis of 2k!p Fractional Factorials
Blocking Fractional Factorials
Alias Structures in Fractional Factorials
and other Designs
Resolution III Designs
8.6.1
8.6.2
Constructing Resolution III Designs
Fold Over of Resolution III Fractions to
Separate Aliased Effects
Plackett-Burman Designs
8.6.3
8.7
306
312
313
315
316
319
349
351
351
353
357
Resolution IV and V Designs
366
8.7.1
8.7.2
8.7.3
366
367
373
Resolution IV Designs
Sequential Experimentation with Resolution IV Designs
Resolution V Designs
8.8 Supersaturated Designs
8.9 Summary
8.10 Problems
374
375
376
9
Additional Design and Analysis Topics for Factorial
and Fractional Factorial Designs
9.1
k
The 3 Factorial Design
9.1.1
9.1.2
9.1.3
9.1.4
9.2
Confounding in the 3k Factorial Design
9.2.1
9.2.2
9.2.3
9.3
Notation and Motivation for the 3k Design
The 32 Design
The 33 Design
The General 3k Design
The 3k Factorial Design in Three Blocks
The 3k Factorial Design in Nine Blocks
The 3k Factorial Design in 3p Blocks
Fractional Replication of the 3k Factorial Design
9.3.1
9.3.2
The One-Third Fraction of the 3k Factorial Design
Other 3k!p Fractional Factorial Designs
394
395
395
396
397
402
402
403
406
407
408
408
410
Contents
9.4
9.5
9.6
9.7
Factorials with Mixed Levels
412
9.4.1
9.4.2
412
414
Factors at Two and Three Levels
Factors at Two and Four Levels
Nonregular Fractional Factorial Designs
415
9.5.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs
9.5.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs
9.5.3 Analysis of Nonregular Fractional Factorial Designs
418
425
427
Constructing Factorial and Fractional Factorial Designs Using
an Optimal Design Tool
431
9.6.1
9.6.2
9.6.3
433
433
443
Design Optimality Criteria
Examples of Optimal Designs
Extensions of the Optimal Design Approach
Problems
10
Fitting Regression Models
10.1
10.2
10.3
10.4
444
449
Introduction
Linear Regression Models
Estimation of the Parameters in Linear Regression Models
Hypothesis Testing in Multiple Regression
449
450
451
462
10.4.1 Test for Significance of Regression
10.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients
462
464
10.5 Confidence Intervals in Multiple Regression
10.5.1 Confidence Intervals on the Individual Regression Coefficients
10.5.2 Confidence Interval on the Mean Response
10.6 Prediction of New Response Observations
10.7 Regression Model Diagnostics
10.7.1 Scaled Residuals and PRESS
10.7.2 Influence Diagnostics
10.8 Testing for Lack of Fit
10.9 Problems
11
Response Surface Methods and Designs
11.1 Introduction to Response Surface Methodology
11.2 The Method of Steepest Ascent
11.3 Analysis of a Second-Order Response Surface
11.3.1
11.3.2
11.3.3
11.3.4
Location of the Stationary Point
Characterizing the Response Surface
Ridge Systems
Multiple Responses
11.4 Experimental Designs for Fitting Response Surfaces
11.4.1
11.4.2
11.4.3
11.4.4
11.5
11.6
11.7
11.8
xv
Designs for Fitting the First-Order Model
Designs for Fitting the Second-Order Model
Blocking in Response Surface Designs
Optimal Designs for Response Surfaces
Experiments with Computer Models
Mixture Experiments
Evolutionary Operation
Problems
467
467
468
468
470
470
472
473
475
478
478
480
486
486
488
495
496
500
501
501
507
511
523
530
540
544
xvi
Contents
12
Robust Parameter Design and Process
Robustness Studies
12.1
12.2
12.3
12.4
Introduction
Crossed Array Designs
Analysis of the Crossed Array Design
Combined Array Designs and the Response
Model Approach
12.5 Choice of Designs
12.6 Problems
13
Experiments with Random Factors
13.1
13.2
13.3
13.4
13.5
13.6
13.7
554
556
558
561
567
570
573
Random Effects Models
The Two-Factor Factorial with Random Factors
The Two-Factor Mixed Model
Sample Size Determination with Random Effects
Rules for Expected Mean Squares
Approximate F Tests
Some Additional Topics on Estimation of Variance Components
573
574
581
587
588
592
596
13.7.1 Approximate Confidence Intervals on Variance Components
13.7.2 The Modified Large-Sample Method
597
600
13.8 Problems
14
Nested and Split-Plot Designs
14.1 The Two-Stage Nested Design
14.1.1
14.1.2
14.1.3
14.1.4
14.2
14.3
14.4
14.5
554
Statistical Analysis
Diagnostic Checking
Variance Components
Staggered Nested Designs
601
604
604
605
609
611
612
The General m-Stage Nested Design
Designs with Both Nested and Factorial Factors
The Split-Plot Design
Other Variations of the Split-Plot Design
614
616
621
627
14.5.1 Split-Plot Designs with More Than Two Factors
14.5.2 The Split-Split-Plot Design
14.5.3 The Strip-Split-Plot Design
627
632
636
14.6 Problems
15
Other Design and Analysis Topics
15.1 Nonnormal Responses and Transformations
15.1.1 Selecting a Transformation: The Box–Cox Method
15.1.2 The Generalized Linear Model
637
642
643
643
645
Contents
15.2 Unbalanced Data in a Factorial Design
15.2.1 Proportional Data: An Easy Case
15.2.2 Approximate Methods
15.2.3 The Exact Method
15.3 The Analysis of Covariance
15.3.1
15.3.2
15.3.3
15.3.4
Description of the Procedure
Computer Solution
Development by the General Regression Significance Test
Factorial Experiments with Covariates
15.4 Repeated Measures
15.5 Problems
Appendix
Table I.
Table II.
Table III.
Table IV.
Table V.
Table VI.
Table VII.
Table VIII.
Table IX.
Table X.
xvii
652
652
654
655
655
656
664
665
667
677
679
683
Cumulative Standard Normal Distribution
Percentage Points of the t Distribution
Percentage Points of the "2 Distribution
Percentage Points of the F Distribution
Operating Characteristic Curves for the Fixed Effects Model
Analysis of Variance
Operating Characteristic Curves for the Random Effects Model
Analysis of Variance
Percentage Points of the Studentized Range Statistic
Critical Values for Dunnett’s Test for Comparing Treatments
with a Control
Coefficients of Orthogonal Polynomials
Alias Relationships for 2k!p Fractional Factorial Designs with k # 15
and n # 64
684
686
687
688
693
697
701
703
705
706
Bibliography
719
Index
725
C H A P T E R
1
Introduction
CHAPTER OUTLINE
1.1 STRATEGY OF EXPERIMENTATION
1.2 SOME TYPICAL APPLICATIONS
OF EXPERIMENTAL DESIGN
1.3 BASIC PRINCIPLES
1.4 GUIDELINES FOR DESIGNING EXPERIMENTS
1.5 A BRIEF HISTORY OF STATISTICAL DESIGN
1.6 SUMMARY: USING STATISTICAL TECHNIQUES
IN EXPERIMENTATION
SUPPLEMENTAL MATERIAL FOR CHAPTER 1
S1.1 More about Planning Experiments
S1.2 Blank Guide Sheets to Assist in Pre-Experimental
Planning
S1.3 Montgomery’s Theorems on Designed Experiments
The supplemental material is on the textbook website www.wiley.com/college/montgomery.
1.1
Strategy of Experimentation
Observing a system or process while it is in operation is an important part of the learning
process, and is an integral part of understanding and learning about how systems and
processes work. The great New York Yankees catcher Yogi Berra said that “. . . you can
observe a lot just by watching.” However, to understand what happens to a process when
you change certain input factors, you have to do more than just watch—you actually have
to change the factors. This means that to really understand cause-and-effect relationships in
a system you must deliberately change the input variables to the system and observe the
changes in the system output that these changes to the inputs produce. In other words, you
need to conduct experiments on the system. Observations on a system or process can lead
to theories or hypotheses about what makes the system work, but experiments of the type
described above are required to demonstrate that these theories are correct.
Investigators perform experiments in virtually all fields of inquiry, usually to discover
something about a particular process or system. Each experimental run is a test. More formally,
we can define an experiment as a test or series of runs in which purposeful changes are made
to the input variables of a process or system so that we may observe and identify the reasons
for changes that may be observed in the output response. We may want to determine which
input variables are responsible for the observed changes in the response, develop a model
relating the response to the important input variables and to use this model for process or system
improvement or other decision-making.
This book is about planning and conducting experiments and about analyzing the
resulting data so that valid and objective conclusions are obtained. Our focus is on experiments in engineering and science. Experimentation plays an important role in technology
1
2
Chapter 1 ■ Introduction
commercialization and product realization activities, which consist of new product design
and formulation, manufacturing process development, and process improvement. The objective in many cases may be to develop a robust process, that is, a process affected minimally
by external sources of variability. There are also many applications of designed experiments
in a nonmanufacturing or non-product-development setting, such as marketing, service operations, and general business operations.
As an example of an experiment, suppose that a metallurgical engineer is interested in
studying the effect of two different hardening processes, oil quenching and saltwater
quenching, on an aluminum alloy. Here the objective of the experimenter (the engineer) is
to determine which quenching solution produces the maximum hardness for this particular
alloy. The engineer decides to subject a number of alloy specimens or test coupons to each
quenching medium and measure the hardness of the specimens after quenching. The average hardness of the specimens treated in each quenching solution will be used to determine
which solution is best.
As we consider this simple experiment, a number of important questions come to mind:
1. Are these two solutions the only quenching media of potential interest?
2. Are there any other factors that might affect hardness that should be investigated or
controlled in this experiment (such as, the temperature of the quenching media)?
3. How many coupons of alloy should be tested in each quenching solution?
4. How should the test coupons be assigned to the quenching solutions, and in what
order should the data be collected?
5. What method of data analysis should be used?
6. What difference in average observed hardness between the two quenching media
will be considered important?
All of these questions, and perhaps many others, will have to be answered satisfactorily
before the experiment is performed.
Experimentation is a vital part of the scientific (or engineering) method. Now there are
certainly situations where the scientific phenomena are so well understood that useful results
including mathematical models can be developed directly by applying these well-understood
principles. The models of such phenomena that follow directly from the physical mechanism
are usually called mechanistic models. A simple example is the familiar equation for current
flow in an electrical circuit, Ohm’s law, E $ IR. However, most problems in science and engineering require observation of the system at work and experimentation to elucidate information about why and how it works. Well-designed experiments can often lead to a model of
system performance; such experimentally determined models are called empirical models.
Throughout this book, we will present techniques for turning the results of a designed experiment into an empirical model of the system under study. These empirical models can be
manipulated by a scientist or an engineer just as a mechanistic model can.
A well-designed experiment is important because the results and conclusions that can
be drawn from the experiment depend to a large extent on the manner in which the data were
collected. To illustrate this point, suppose that the metallurgical engineer in the above experiment used specimens from one heat in the oil quench and specimens from a second heat in
the saltwater quench. Now, when the mean hardness is compared, the engineer is unable to
say how much of the observed difference is the result of the quenching media and how much
is the result of inherent differences between the heats.1 Thus, the method of data collection
has adversely affected the conclusions that can be drawn from the experiment.
1
A specialist in experimental design would say that the effect of quenching media and heat were confounded; that is, the effects of
these two factors cannot be separated.
1.1 Strategy of Experimentation
FIGURE 1.1
process or system
Controllable factors
x1
Inputs
x2
z2
General model of a
Output
y
Process
z1
■
xp
3
zq
Uncontrollable factors
In general, experiments are used to study the performance of processes and systems.
The process or system can be represented by the model shown in Figure 1.1. We can usually
visualize the process as a combination of operations, machines, methods, people, and other
resources that transforms some input (often a material) into an output that has one or more
observable response variables. Some of the process variables and material properties x1,
x2, . . . , xp are controllable, whereas other variables z1, z2, . . . , zq are uncontrollable
(although they may be controllable for purposes of a test). The objectives of the experiment
may include the following:
1. Determining which variables are most influential on the response y
2. Determining where to set the influential x’s so that y is almost always near the
desired nominal value
3. Determining where to set the influential x’s so that variability in y is small
4. Determining where to set the influential x’s so that the effects of the uncontrollable
variables z1, z2, . . . , zq are minimized.
As you can see from the foregoing discussion, experiments often involve several factors.
Usually, an objective of the experimenter is to determine the influence that these factors have
on the output response of the system. The general approach to planning and conducting the
experiment is called the strategy of experimentation. An experimenter can use several strategies. We will illustrate some of these with a very simple example.
I really like to play golf. Unfortunately, I do not enjoy practicing, so I am always looking for a simpler solution to lowering my score. Some of the factors that I think may be important, or that may influence my golf score, are as follows:
1.
2.
3.
4.
5.
6.
7.
8.
The type of driver used (oversized or regular sized)
The type of ball used (balata or three piece)
Walking and carrying the golf clubs or riding in a golf cart
Drinking water or drinking “something else” while playing
Playing in the morning or playing in the afternoon
Playing when it is cool or playing when it is hot
The type of golf shoe spike worn (metal or soft)
Playing on a windy day or playing on a calm day.
Obviously, many other factors could be considered, but let’s assume that these are the ones of primary interest. Furthermore, based on long experience with the game, I decide that factors 5
through 8 can be ignored; that is, these factors are not important because their effects are so small
Chapter 1 ■ Introduction
R
O
Driver
■
FIGURE 1.2
T
B
Ball
Score
Score
Score
that they have no practical value. Engineers, scientists, and business analysts, often must make
these types of decisions about some of the factors they are considering in real experiments.
Now, let’s consider how factors 1 through 4 could be experimentally tested to determine
their effect on my golf score. Suppose that a maximum of eight rounds of golf can be played
over the course of the experiment. One approach would be to select an arbitrary combination
of these factors, test them, and see what happens. For example, suppose the oversized driver,
balata ball, golf cart, and water combination is selected, and the resulting score is 87. During
the round, however, I noticed several wayward shots with the big driver (long is not always
good in golf), and, as a result, I decide to play another round with the regular-sized driver,
holding the other factors at the same levels used previously. This approach could be continued almost indefinitely, switching the levels of one or two (or perhaps several) factors for the
next test, based on the outcome of the current test. This strategy of experimentation, which
we call the best-guess approach, is frequently used in practice by engineers and scientists. It
often works reasonably well, too, because the experimenters often have a great deal of technical or theoretical knowledge of the system they are studying, as well as considerable practical experience. The best-guess approach has at least two disadvantages. First, suppose the
initial best-guess does not produce the desired results. Now the experimenter has to take
another guess at the correct combination of factor levels. This could continue for a long time,
without any guarantee of success. Second, suppose the initial best-guess produces an acceptable result. Now the experimenter is tempted to stop testing, although there is no guarantee
that the best solution has been found.
Another strategy of experimentation that is used extensively in practice is the onefactor-at-a-time (OFAT) approach. The OFAT method consists of selecting a starting point,
or baseline set of levels, for each factor, and then successively varying each factor over its
range with the other factors held constant at the baseline level. After all tests are performed,
a series of graphs are usually constructed showing how the response variable is affected by
varying each factor with all other factors held constant. Figure 1.2 shows a set of these graphs
for the golf experiment, using the oversized driver, balata ball, walking, and drinking water
levels of the four factors as the baseline. The interpretation of this graph is straightforward;
for example, because the slope of the mode of travel curve is negative, we would conclude
that riding improves the score. Using these one-factor-at-a-time graphs, we would select the
optimal combination to be the regular-sized driver, riding, and drinking water. The type of
golf ball seems unimportant.
The major disadvantage of the OFAT strategy is that it fails to consider any possible
interaction between the factors. An interaction is the failure of one factor to produce the same
effect on the response at different levels of another factor. Figure 1.3 shows an interaction
between the type of driver and the beverage factors for the golf experiment. Notice that if I use
the regular-sized driver, the type of beverage consumed has virtually no effect on the score, but
if I use the oversized driver, much better results are obtained by drinking water instead of beer.
Interactions between factors are very common, and if they occur, the one-factor-at-a-time strategy will usually produce poor results. Many people do not recognize this, and, consequently,
Score
4
R
W
Mode of travel
SE
W
Beverage
Results of the one-factor-at-a-time strategy for the golf experiment
1.1 Strategy of Experimentation
T
Type of ball
Score
Oversized
driver
5
Regular-sized
driver
B
B
W
R
O
Beverage type
Type of driver
F I G U R E 1 . 3 Interaction between
type of driver and type of beverage for
the golf experiment
■
F I G U R E 1 . 4 A two-factor
factorial experiment involving type
of driver and type of ball
■
OFAT experiments are run frequently in practice. (Some individuals actually think that this
strategy is related to the scientific method or that it is a “sound” engineering principle.) Onefactor-at-a-time experiments are always less efficient than other methods based on a statistical
approach to design. We will discuss this in more detail in Chapter 5.
The correct approach to dealing with several factors is to conduct a factorial experiment. This is an experimental strategy in which factors are varied together, instead of one
at a time. The factorial experimental design concept is extremely important, and several
chapters in this book are devoted to presenting basic factorial experiments and a number of
useful variations and special cases.
To illustrate how a factorial experiment is conducted, consider the golf experiment and
suppose that only two factors, type of driver and type of ball, are of interest. Figure 1.4 shows
a two-factor factorial experiment for studying the joint effects of these two factors on my golf
score. Notice that this factorial experiment has both factors at two levels and that all possible
combinations of the two factors across their levels are used in the design. Geometrically, the
four runs form the corners of a square. This particular type of factorial experiment is called a
22 factorial design (two factors, each at two levels). Because I can reasonably expect to play
eight rounds of golf to investigate these factors, a reasonable plan would be to play two
rounds of golf at each combination of factor levels shown in Figure 1.4. An experimental
designer would say that we have replicated the design twice. This experimental design would
enable the experimenter to investigate the individual effects of each factor (or the main
effects) and to determine whether the factors interact.
Figure 1.5a shows the results of performing the factorial experiment in Figure 1.4. The
scores from each round of golf played at the four test combinations are shown at the corners
of the square. Notice that there are four rounds of golf that provide information about using
the regular-sized driver and four rounds that provide information about using the oversized
driver. By finding the average difference in the scores on the right- and left-hand sides of the
square (as in Figure 1.5b), we have a measure of the effect of switching from the oversized
driver to the regular-sized driver, or
92 % 94 % 93 % 91 88 % 91 % 88 % 90
!
4
4
$ 3.25
Driver effect $
That is, on average, switching from the oversized to the regular-sized driver increases the
score by 3.25 strokes per round. Similarly, the average difference in the four scores at the top
Chapter 1 ■ Introduction
88, 91
92, 94
88, 90
93, 91
Type of ball
T
B
O
R
Type of driver
Type of ball
+
–
T
B
+
+
+
–
–
T
B
–
+
B
–
+
T
Type of ball
(a) Scores from the golf experiment
Type of ball
6
–
O
R
Type of driver
O
R
Type of driver
O
R
Type of driver
(b) Comparison of scores leading
to the driver effect
(c) Comparison of scores
leading to the ball effect
(d) Comparison of scores
leading to the ball–driver
interaction effect
FIGURE 1.5
factor effects
■
Scores from the golf experiment in Figure 1.4 and calculation of the
of the square and the four scores at the bottom measures the effect of the type of ball used
(see Figure 1.5c):
88 % 91 % 92 % 94 88 % 90 % 93 % 91
!
4
4
$ 0.75
Ball effect $
Finally, a measure of the interaction effect between the type of ball and the type of driver can
be obtained by subtracting the average scores on the left-to-right diagonal in the square from
the average scores on the right-to-left diagonal (see Figure 1.5d), resulting in
92 % 94 % 88 % 90 88 % 91 % 93 % 91
!
4
4
$ 0.25
Ball–driver interaction effect $
The results of this factorial experiment indicate that driver effect is larger than either the
ball effect or the interaction. Statistical testing could be used to determine whether any of
these effects differ from zero. In fact, it turns out that there is reasonably strong statistical evidence that the driver effect differs from zero and the other two effects do not. Therefore, this
experiment indicates that I should always play with the oversized driver.
One very important feature of the factorial experiment is evident from this simple
example; namely, factorials make the most efficient use of the experimental data. Notice that
this experiment included eight observations, and all eight observations are used to calculate
the driver, ball, and interaction effects. No other strategy of experimentation makes such an
efficient use of the data. This is an important and useful feature of factorials.
We can extend the factorial experiment concept to three factors. Suppose that I wish
to study the effects of type of driver, type of ball, and the type of beverage consumed on my
golf score. Assuming that all three factors have two levels, a factorial design can be set up
1.1 Strategy of Experimentation
7
FIGURE 1.6
A three-factor
factorial experiment involving type of
driver, type of ball, and type of beverage
Beverage
■
Ball
Driver
as shown in Figure 1.6. Notice that there are eight test combinations of these three factors
across the two levels of each and that these eight trials can be represented geometrically as
the corners of a cube. This is an example of a 23 factorial design. Because I only want to
play eight rounds of golf, this experiment would require that one round be played at each
combination of factors represented by the eight corners of the cube in Figure 1.6. However,
if we compare this to the two-factor factorial in Figure 1.4, the 23 factorial design would provide the same information about the factor effects. For example, there are four tests in both
designs that provide information about the regular-sized driver and four tests that provide
information about the oversized driver, assuming that each run in the two-factor design in
Figure 1.4 is replicated twice.
Figure 1.7 illustrates how all four factors—driver, ball, beverage, and mode of travel
(walking or riding)—could be investigated in a 24 factorial design. As in any factorial design,
all possible combinations of the levels of the factors are used. Because all four factors are at
two levels, this experimental design can still be represented geometrically as a cube (actually
a hypercube).
Generally, if there are k factors, each at two levels, the factorial design would require 2k
runs. For example, the experiment in Figure 1.7 requires 16 runs. Clearly, as the number of
factors of interest increases, the number of runs required increases rapidly; for instance, a
10-factor experiment with all factors at two levels would require 1024 runs. This quickly
becomes infeasible from a time and resource viewpoint. In the golf experiment, I can only
play eight rounds of golf, so even the experiment in Figure 1.7 is too large.
Fortunately, if there are four to five or more factors, it is usually unnecessary to run all
possible combinations of factor levels. A fractional factorial experiment is a variation of the
basic factorial design in which only a subset of the runs is used. Figure 1.8 shows a fractional
factorial design for the four-factor version of the golf experiment. This design requires only
8 runs instead of the original 16 and would be called a one-half fraction. If I can play only
eight rounds of golf, this is an excellent design in which to study all four factors. It will provide
good information about the main effects of the four factors as well as some information about
how these factors interact.
Mode of travel
Ride
Beverage
Walk
Ball
Driver
F I G U R E 1 . 7 A four-factor factorial experiment involving type
of driver, type of ball, type of beverage, and mode of travel
■
8
Chapter 1 ■ Introduction
Mode of travel
Ride
Beverage
Walk
Ball
Driver
F I G U R E 1 . 8 A four-factor fractional factorial experiment involving
type of driver, type of ball, type of beverage, and mode of travel
■
Fractional factorial designs are used extensively in industrial research and development,
and for process improvement. These designs will be discussed in Chapters 8 and 9.
1.2
Some Typical Applications of Experimental Design
Experimental design methods have found broad application in many disciplines. As noted
previously, we may view experimentation as part of the scientific process and as one of the
ways by which we learn about how systems or processes work. Generally, we learn through
a series of activities in which we make conjectures about a process, perform experiments to
generate data from the process, and then use the information from the experiment to establish
new conjectures, which lead to new experiments, and so on.
Experimental design is a critically important tool in the scientific and engineering
world for improving the product realization process. Critical components of these activities
are in new manufacturing process design and development, and process management. The
application of experimental design techniques early in process development can result in
1.
2.
3.
4.
Improved process yields
Reduced variability and closer conformance to nominal or target requirements
Reduced development time
Reduced overall costs.
Experimental design methods are also of fundamental importance in engineering
design activities, where new products are developed and existing ones improved. Some applications of experimental design in engineering design include
1. Evaluation and comparison of basic design configurations
2. Evaluation of material alternatives
3. Selection of design parameters so that the product will work well under a wide variety of field conditions, that is, so that the product is robust
4. Determination of key product design parameters that impact product performance
5. Formulation of new products.
The use of experimental design in product realization can result in products that are easier
to manufacture and that have enhanced field performance and reliability, lower product
cost, and shorter product design and development time. Designed experiments also have
extensive applications in marketing, market research, transactional and service operations,
and general business operations. We now present several examples that illustrate some of
these ideas.
1.2 Some Typical Applications of Experimental Design
EXAMPLE 1.1
Characterizing a Process
A flow solder machine is used in the manufacturing process
for printed circuit boards. The machine cleans the boards in
a flux, preheats the boards, and then moves them along a
conveyor through a wave of molten solder. This solder
process makes the electrical and mechanical connections
for the leaded components on the board.
The process currently operates around the 1 percent defective level. That is, about 1 percent of the solder joints on a
board are defective and require manual retouching. However,
because the average printed circuit board contains over 2000
solder joints, even a 1 percent defective level results in far too
many solder joints requiring rework. The process engineer
responsible for this area would like to use a designed experiment to determine which machine parameters are influential
in the occurrence of solder defects and which adjustments
should be made to those variables to reduce solder defects.
The flow solder machine has several variables that can
be controlled. They include
1.
2.
3.
4.
5.
6.
7.
Solder temperature
Preheat temperature
Conveyor speed
Flux type
Flux specific gravity
Solder wave depth
Conveyor angle.
In addition to these controllable factors, several other factors
cannot be easily controlled during routine manufacturing,
although they could be controlled for the purposes of a test.
They are
1. Thickness of the printed circuit board
2. Types of components used on the board
EXAMPLE 1.2
3. Layout of the components on the board
4. Operator
5. Production rate.
In this situation, engineers are interested in characterizing the flow solder machine; that is, they want to determine which factors (both controllable and uncontrollable)
affect the occurrence of defects on the printed circuit
boards. To accomplish this, they can design an experiment
that will enable them to estimate the magnitude and direction of the factor effects; that is, how much does the
response variable (defects per unit) change when each factor is changed, and does changing the factors together
produce different results than are obtained from individual
factor adjustments—that is, do the factors interact?
Sometimes we call an experiment such as this a screening
experiment. Typically, screening or characterization experiments involve using fractional factorial designs, such as in
the golf example in Figure 1.8.
The information from this screening or characterization
experiment will be used to identify the critical process factors and to determine the direction of adjustment for these
factors to reduce further the number of defects per unit. The
experiment may also provide information about which factors should be more carefully controlled during routine manufacturing to prevent high defect levels and erratic process
performance. Thus, one result of the experiment could be the
application of techniques such as control charts to one or
more process variables (such as solder temperature), in
addition to control charts on process output. Over time, if the
process is improved enough, it may be possible to base most
of the process control plan on controlling process input variables instead of control charting the output.
Optimizing a Process
In a characterization experiment, we are usually interested
in determining which process variables affect the response.
A logical next step is to optimize, that is, to determine the
region in the important factors that leads to the best possible response. For example, if the response is yield, we
would look for a region of maximum yield, whereas if the
response is variability in a critical product dimension, we
would seek a region of minimum variability.
Suppose that we are interested in improving the yield
of a chemical process. We know from the results of a characterization experiment that the two most important
process variables that influence the yield are operating
temperature and reaction time. The process currently runs
at 145°F and 2.1 hours of reaction time, producing yields
of around 80 percent. Figure 1.9 shows a view of the
time–temperature region from above. In this graph, the
lines of constant yield are connected to form response
contours, and we have shown the contour lines for yields
of 60, 70, 80, 90, and 95 percent. These contours are projections on the time–temperature region of cross sections
of the yield surface corresponding to the aforementioned
percent yields. This surface is sometimes called a
response surface. The true response surface in Figure 1.9
is unknown to the process personnel, so experimental
methods will be required to optimize the yield with
respect to time and temperature.
9
10
Chapter 1 ■ Introduction
F I G U R E 1 . 9 Contour plot of yield as a
function of reaction time and reaction temperature,
illustrating experimentation to optimize a process
■
Second optimization experiment
200
Path leading
to region of
higher yield
Temperature (°F)
190
180
95%
170
90% 80%
160
150
140
82
Initial
optimization
experiment
78
80
Current
operating
conditions
0.5
70%
75
70
60%
1.0
1.5
2.0
2.5
To locate the optimum, it is necessary to perform an
experiment that varies both time and temperature together,
that is, a factorial experiment. The results of an initial factorial experiment with both time and temperature run at two
levels is shown in Figure 1.9. The responses observed at the
four corners of the square indicate that we should move in
the general direction of increased temperature and decreased
reaction time to increase yield. A few additional runs would
be performed in this direction, and this additional experimentation would lead us to the region of maximum yield.
Once we have found the region of the optimum, a second
experiment would typically be performed. The objective of
this second experiment is to develop an empirical model of
the process and to obtain a more precise estimate of the optimum operating conditions for time and temperature. This
approach to process optimization is called response surface
methodology, and it is explored in detail in Chapter 11. The
second design illustrated in Figure 1.9 is a central composite design, one of the most important experimental designs
used in process optimization studies.
Time (hours)
EXAMPLE 1.3
Designing a Product—I
A biomedical engineer is designing a new pump for the
intravenous delivery of a drug. The pump should deliver a
constant quantity or dose of the drug over a specified period of time. She must specify a number of variables or
design parameters. Among these are the diameter and
length of the cylinder, the fit between the cylinder and the
plunger, the plunger length, the diameter and wall thickness
of the tube connecting the pump and the needle inserted
into the patient’s vein, the material to use for fabricating
EXAMPLE 1.4
both the cylinder and the tube, and the nominal pressure at
which the system must operate. The impact of some of
these parameters on the design can be evaluated by building prototypes in which these factors can be varied over
appropriate ranges. Experiments can then be designed and
the prototypes tested to investigate which design parameters are most influential on pump performance. Analysis of
this information will assist the engineer in arriving at a
design that provides reliable and consistent drug delivery.
Designing a Product—II
An engineer is designing an aircraft engine. The engine is a
commercial turbofan, intended to operate in the cruise configuration at 40,000 ft and 0.8 Mach. The design parameters
include inlet flow, fan pressure ratio, overall pressure, stator outlet temperature, and many other factors. The output
response variables in this system are specific fuel consumption and engine thrust. In designing this system, it would be
prohibitive to build prototypes or actual test articles early in
the design process, so the engineers use a computer model
of the system that allows them to focus on the key design
parameters of the engine and to vary them in an effort to
optimize the performance of the engine. Designed experiments can be employed with the computer model of the
engine to determine the most important design parameters
and their optimal settings.
1.3 Basic Principles
11
Designers frequently use computer models to assist them in carrying out their activities.
Examples include finite element models for many aspects of structural and mechanical
design, electrical circuit simulators for integrated circuit design, factory or enterprise-level
models for scheduling and capacity planning or supply chain management, and computer
models of complex chemical processes. Statistically designed experiments can be applied to
these models just as easily and successfully as they can to actual physical systems and will
result in reduced development lead time and better designs.
EXAMPLE 1.5
Formulating a Product
A biochemist is formulating a diagnostic product to detect
the presence of a certain disease. The product is a mixture
of biological materials, chemical reagents, and other materials that when combined with human blood react to provide a diagnostic indication. The type of experiment used
here is a mixture experiment, because various ingredients
that are combined to form the diagnostic make up 100 percent of the mixture composition (on a volume, weight, or
EXAMPLE 1.6
Designing a Web Page
A lot of business today is conducted via the World Wide
Web. Consequently, the design of a business’ web page has
potentially important economic impact. Suppose that the
Web site has the following components: (1) a photoflash
image, (2) a main headline, (3) a subheadline, (4) a main
text copy, (5) a main image on the right side, (6) a background design, and (7) a footer. We are interested in finding
the factors that influence the click-through rate; that is, the
number of visitors who click through into the site divided by
the total number of visitors to the site. Proper selection of
the important factors can lead to an optimal web page
design. Suppose that there are four choices for the photoflash image, eight choices for the main headline, six choices for the subheadline, five choices for the main text copy,
1.3
mole ratio basis), and the response is a function of the mixture proportions that are present in the product. Mixture
experiments are a special type of response surface experiment that we will study in Chapter 11. They are very useful
in designing biotechnology products, pharmaceuticals,
foods and beverages, paints and coatings, consumer products such as detergents, soaps, and other personal care
products, and a wide variety of other products.
four choices for the main image, three choices for the background design, and seven choices for the footer. If we use a
factorial design, web pages for all possible combinations of
these factor levels must be constructed and tested. This is a
total of 4 & 8 & 6 & 5 & 4 & 3 & 7 $ 80,640 web
pages. Obviously, it is not feasible to design and test this
many combinations of web pages, so a complete factorial
experiment cannot be considered. However, a fractional factorial experiment that uses a small number of the possible
web page designs would likely be successful. This experiment would require a fractional factorial where the factors
have different numbers of levels. We will discuss how to
construct these designs in Chapter 9.
Basic Principles
If an experiment such as the ones described in Examples 1.1 through 1.6 is to be performed
most efficiently, a scientific approach to planning the experiment must be employed.
Statistical design of experiments refers to the process of planning the experiment so that
appropriate data will be collected and analyzed by statistical methods, resulting in valid
and objective conclusions. The statistical approach to experimental design is necessary if we
wish to draw meaningful conclusions from the data. When the problem involves data that are
subject to experimental errors, statistical methods are the only objective approach to analysis.
Thus, there are two aspects to any experimental problem: the design of the experiment and
the statistical analysis of the data. These two subjects are closely related because the method
12
Chapter 1 ■ Introduction
of analysis depends directly on the design employed. Both topics will be addressed in this
book.
The three basic principles of experimental design are randomization, replication, and
blocking. Sometimes we add the factorial principle to these three. Randomization is the cornerstone underlying the use of statistical methods in experimental design. By randomization
we mean that both the allocation of the experimental material and the order in which the individual runs of the experiment are to be performed are randomly determined. Statistical methods require that the observations (or errors) be independently distributed random variables.
Randomization usually makes this assumption valid. By properly randomizing the experiment, we also assist in “averaging out” the effects of extraneous factors that may be present.
For example, suppose that the specimens in the hardness experiment are of slightly different
thicknesses and that the effectiveness of the quenching medium may be affected by specimen
thickness. If all the specimens subjected to the oil quench are thicker than those subjected to
the saltwater quench, we may be introducing systematic bias into the experimental results.
This bias handicaps one of the quenching media and consequently invalidates our results.
Randomly assigning the specimens to the quenching media alleviates this problem.
Computer software programs are widely used to assist experimenters in selecting and
constructing experimental designs. These programs often present the runs in the experimental
design in random order. This random order is created by using a random number generator.
Even with such a computer program, it is still often necessary to assign units of experimental
material (such as the specimens in the hardness example mentioned above), operators, gauges
or measurement devices, and so forth for use in the experiment.
Sometimes experimenters encounter situations where randomization of some aspect of
the experiment is difficult. For example, in a chemical process, temperature may be a very
hard-to-change variable as we may want to change it less often than we change the levels of
other factors. In an experiment of this type, complete randomization would be difficult
because it would add time and cost. There are statistical design methods for dealing with
restrictions on randomization. Some of these approaches will be discussed in subsequent
chapters (see in particular Chapter 14).
By replication we mean an independent repeat run of each factor combination. In the
metallurgical experiment discussed in Section 1.1, replication would consist of treating a
specimen by oil quenching and treating a specimen by saltwater quenching. Thus, if five
specimens are treated in each quenching medium, we say that five replicates have been
obtained. Each of the 10 observations should be run in random order. Replication has two
important properties. First, it allows the experimenter to obtain an estimate of the experimental error. This estimate of error becomes a basic unit of measurement for determining
whether observed differences in the data are really statistically different. Second, if the sample mean (y) is used to estimate the true mean response for one of the factor levels in the
experiment, replication permits the experimenter to obtain a more precise estimate of this
parameter. For example; if ! 2 is the variance of an individual observation and there are
n replicates, the variance of the sample mean is
!2
!y!2 $ n
The practical implication of this is that if we had n $ 1 replicates and observed
y1 $ 145 (oil quench) and y2 $ 147 (saltwater quench), we would probably be unable to
make satisfactory inferences about the effect of the quenching medium—that is, the
observed difference could be the result of experimental error. The point is that without
replication we have no way of knowing why the two observations are different. On the
other hand, if n was reasonably large and the experimental error was sufficiently small and
if we observed sample averages y1 < y2, we would be reasonably safe in concluding that
1.3 Basic Principles
13
saltwater quenching produces a higher hardness in this particular aluminum alloy than
does oil quenching.
Often when the runs in an experiment are randomized, two (or more) consecutive runs
will have exactly the same levels for some of the factors. For example, suppose we have three
factors in an experiment: pressure, temperature, and time. When the experimental runs are
randomized, we find the following:
Run number
Pressure (psi)
Temperature ('C)
Time (min)
i
i%1
i%2
30
30
40
100
125
125
30
45
45
Notice that between runs i and i % 1, the levels of pressure are identical and between runs
i % 1 and i % 2, the levels of both temperature and time are identical. To obtain a true replicate, the experimenter needs to “twist the pressure knob” to an intermediate setting between
runs i and i % 1, and reset pressure to 30 psi for run i % 1. Similarly, temperature and time
should be reset to intermediate levels between runs i % 1 and i % 2 before being set to their
design levels for run i % 2. Part of the experimental error is the variability associated with hitting and holding factor levels.
There is an important distinction between replication and repeated measurements.
For example, suppose that a silicon wafer is etched in a single-wafer plasma etching process,
and a critical dimension (CD) on this wafer is measured three times. These measurements are
not replicates; they are a form of repeated measurements, and in this case the observed variability in the three repeated measurements is a direct reflection of the inherent variability in
the measurement system or gauge and possibly the variability in this CD at different locations
on the wafer where the measurement were taken. As another illustration, suppose that as part
of an experiment in semiconductor manufacturing four wafers are processed simultaneously
in an oxidation furnace at a particular gas flow rate and time and then a measurement is taken
on the oxide thickness of each wafer. Once again, the measurements on the four wafers are
not replicates but repeated measurements. In this case, they reflect differences among the
wafers and other sources of variability within that particular furnace run. Replication reflects
sources of variability both between runs and (potentially) within runs.
Blocking is a design technique used to improve the precision with which comparisons
among the factors of interest are made. Often blocking is used to reduce or eliminate the variability transmitted from nuisance factors—that is, factors that may influence the experimental response but in which we are not directly interested. For example, an experiment in a
chemical process may require two batches of raw material to make all the required runs.
However, there could be differences between the batches due to supplier-to-supplier variability, and if we are not specifically interested in this effect, we would think of the batches of
raw material as a nuisance factor. Generally, a block is a set of relatively homogeneous experimental conditions. In the chemical process example, each batch of raw material would form
a block, because the variability within a batch would be expected to be smaller than the variability between batches. Typically, as in this example, each level of the nuisance factor
becomes a block. Then the experimenter divides the observations from the statistical design
into groups that are run in each block. We study blocking in detail in several places in the text,
including Chapters 4, 5, 7, 8, 9, 11, and 13. A simple example illustrating the blocking principal is given in Section 2.5.1.
The three basic principles of experimental design, randomization, replication, and
blocking are part of every experiment. We will illustrate and emphasize them repeatedly
throughout this book.
14
1.4
Chapter 1 ■ Introduction
Guidelines for Designing Experiments
To use the statistical approach in designing and analyzing an experiment, it is necessary for
everyone involved in the experiment to have a clear idea in advance of exactly what is to be studied, how the data are to be collected, and at least a qualitative understanding of how these data
are to be analyzed. An outline of the recommended procedure is shown in Table 1.1. We now
give a brief discussion of this outline and elaborate on some of the key points. For more details,
see Coleman and Montgomery (1993), and the references therein. The supplemental text
material for this chapter is also useful.
1. Recognition of and statement of the problem. This may seem to be a rather obvious point, but in practice often neither it is simple to realize that a problem requiring
experimentation exists, nor is it simple to develop a clear and generally accepted statement of this problem. It is necessary to develop all ideas about the objectives of the
experiment. Usually, it is important to solicit input from all concerned parties: engineering, quality assurance, manufacturing, marketing, management, customer, and
operating personnel (who usually have much insight and who are too often ignored).
For this reason, a team approach to designing experiments is recommended.
It is usually helpful to prepare a list of specific problems or questions that are
to be addressed by the experiment. A clear statement of the problem often contributes
substantially to better understanding of the phenomenon being studied and the final
solution of the problem.
It is also important to keep the overall objectives of the experiment in mind.
There are several broad reasons for running experiments and each type of experiment
will generate its own list of specific questions that need to be addressed. Some (but
by no means all) of the reasons for running experiments include:
a. Factor screening or characterization. When a system or process is new,
it is usually important to learn which factors have the most influence on
the response(s) of interest. Often there are a lot of factors. This usually
indicates that the experimenters do not know much about the system so
screening is essential if we are to efficiently get the desired performance
from the system. Screening experiments are extremely important when
working with new systems or technologies so that valuable resources will
not be wasted using best guess and OFAT approaches.
b. Optimization. After the system has been characterized and we are reasonably certain that the important factors have been identified, the next
objective is usually optimization, that is, find the settings or levels of
TA B L E 1 . 1
Guidelines for Designing an Experiment
■
1. Recognition of and statement of the problem
2. Selection of the response variablea
3. Choice of factors, levels, and rangesa
4. Choice of experimental design
5. Performing the experiment
6. Statistical analysis of the data
7. Conclusions and recommendations
a
Pre-experimental
planning
In practice, steps 2 and 3 are often done simultaneously or in reverse order.
1.4 Guidelines for Designing Experiments
15
the important factors that result in desirable values of the response. For
example, if a screening experiment on a chemical process results in the
identification of time and temperature as the two most important factors, the optimization experiment may have as its objective finding the
levels of time and temperature that maximize yield, or perhaps maximize yield while keeping some product property that is critical to the
customer within specifications. An optimization experiment is usually
a follow-up to a screening experiment. It would be very unusual for a
screening experiment to produce the optimal settings of the important
factors.
c. Confirmation. In a confirmation experiment, the experimenter is usually
trying to verify that the system operates or behaves in a manner that is
consistent with some theory or past experience. For example, if theory
or experience indicates that a particular new material is equivalent to the
one currently in use and the new material is desirable (perhaps less
expensive, or easier to work with in some way), then a confirmation
experiment would be conducted to verify that substituting the new material results in no change in product characteristics that impact its use.
Moving a new manufacturing process to full-scale production based on
results found during experimentation at a pilot plant or development site
is another situation that often results in confirmation experiments—that
is, are the same factors and settings that were determined during development work appropriate for the full-scale process?
d. Discovery. In discovery experiments, the experimenters are usually trying
to determine what happens when we explore new materials, or new factors, or new ranges for factors. In the pharmaceutical industry, scientists
are constantly conducting discovery experiments to find new materials or
combinations of materials that will be effective in treating disease.
e. Robustness. These experiments often address questions such as under
what conditions do the response variables of interest seriously degrade?
Or what conditions would lead to unacceptable variability in the response
variables? A variation of this is determining how we can set the factors in
the system that we can control to minimize the variability transmitted into
the response from factors that we cannot control very well. We will discuss some experiments of this type in Chapter 12.
Obviously, the specific questions to be addressed in the experiment relate
directly to the overall objectives. An important aspect of problem formulation is the
recognition that one large comprehensive experiment is unlikely to answer the key
questions satisfactorily. A single comprehensive experiment requires the experimenters to know the answers to a lot of questions, and if they are wrong, the results
will be disappointing. This leads to wasting time, materials, and other resources and
may result in never answering the original research questions satisfactorily. A
sequential approach employing a series of smaller experiments, each with a specific
objective, such as factor screening, is a better strategy.
2. Selection of the response variable. In selecting the response variable, the experimenter should be certain that this variable really provides useful information about
the process under study. Most often, the average or standard deviation (or both) of
the measured characteristic will be the response variable. Multiple responses are
not unusual. The experimenters must decide how each response will be measured,
and address issues such as how will any measurement system be calibrated and
16
Chapter 1 ■ Introduction
how this calibration will be maintained during the experiment. The gauge or measurement system capability (or measurement error) is also an important factor. If
gauge capability is inadequate, only relatively large factor effects will be detected
by the experiment or perhaps additional replication will be required. In some situations where gauge capability is poor, the experimenter may decide to measure
each experimental unit several times and use the average of the repeated measurements as the observed response. It is usually critically important to identify issues
related to defining the responses of interest and how they are to be measured before
conducting the experiment. Sometimes designed experiments are employed to
study and improve the performance of measurement systems. For an example, see
Chapter 13.
3. Choice of factors, levels, and range. (As noted in Table 1.1, steps 2 and 3 are often
done simultaneously or in the reverse order.) When considering the factors that may
influence the performance of a process or system, the experimenter usually discovers that these factors can be classified as either potential design factors or nuisance
factors. The potential design factors are those factors that the experimenter may wish
to vary in the experiment. Often we find that there are a lot of potential design factors, and some further classification of them is helpful. Some useful classifications
are design factors, held-constant factors, and allowed-to-vary factors. The design
factors are the factors actually selected for study in the experiment. Held-constant
factors are variables that may exert some effect on the response, but for purposes of
the present experiment these factors are not of interest, so they will be held at a specific level. For example, in an etching experiment in the semiconductor industry,
there may be an effect that is unique to the specific plasma etch tool used in the
experiment. However, this factor would be very difficult to vary in an experiment, so
the experimenter may decide to perform all experimental runs on one particular (ideally “typical”) etcher. Thus, this factor has been held constant. As an example of
allowed-to-vary factors, the experimental units or the “materials” to which the design
factors are applied are usually nonhomogeneous, yet we often ignore this unit-to-unit
variability and rely on randomization to balance out any material or experimental
unit effect. We often assume that the effects of held-constant factors and allowed-tovary factors are relatively small.
Nuisance factors, on the other hand, may have large effects that must be
accounted for, yet we may not be interested in them in the context of the present experiment. Nuisance factors are often classified as controllable, uncontrollable, or noise
factors. A controllable nuisance factor is one whose levels may be set by the experimenter. For example, the experimenter can select different batches of raw material
or different days of the week when conducting the experiment. The blocking principle, discussed in the previous section, is often useful in dealing with controllable nuisance factors. If a nuisance factor is uncontrollable in the experiment, but it can be
measured, an analysis procedure called the analysis of covariance can often be used
to compensate for its effect. For example, the relative humidity in the process environment may affect process performance, and if the humidity cannot be controlled,
it probably can be measured and treated as a covariate. When a factor that varies naturally and uncontrollably in the process can be controlled for purposes of an experiment, we often call it a noise factor. In such situations, our objective is usually to find
the settings of the controllable design factors that minimize the variability transmitted from the noise factors. This is sometimes called a process robustness study or a
robust design problem. Blocking, analysis of covariance, and process robustness
studies are discussed later in the text.
1.4 Guidelines for Designing Experiments
17
Once the experimenter has selected the design factors, he or she must choose
the ranges over which these factors will be varied and the specific levels at which runs
will be made. Thought must also be given to how these factors are to be controlled at
the desired values and how they are to be measured. For instance, in the flow solder
experiment, the engineer has defined 12 variables that may affect the occurrence of
solder defects. The experimenter will also have to decide on a region of interest for
each variable (that is, the range over which each factor will be varied) and on how
many levels of each variable to use. Process knowledge is required to do this. This
process knowledge is usually a combination of practical experience and theoretical
understanding. It is important to investigate all factors that may be of importance and
to be not overly influenced by past experience, particularly when we are in the early
stages of experimentation or when the process is not very mature.
When the objective of the experiment is factor screening or process characterization, it is usually best to keep the number of factor levels low. Generally, two
levels work very well in factor screening studies. Choosing the region of interest is
also important. In factor screening, the region of interest should be relatively large—
that is, the range over which the factors are varied should be broad. As we learn more
about which variables are important and which levels produce the best results, the
region of interest in subsequent experiments will usually become narrower.
The cause-and-effect diagram can be a useful technique for organizing
some of the information generated in pre-experimental planning. Figure 1.10 is the
cause-and-effect diagram constructed while planning an experiment to resolve
problems with wafer charging (a charge accumulation on the wafers) encountered
in an etching tool used in semiconductor manufacturing. The cause-and-effect diagram is also known as a fishbone diagram because the “effect” of interest or the
response variable is drawn along the spine of the diagram and the potential causes
or design factors are organized in a series of ribs. The cause-and-effect diagram
uses the traditional causes of measurement, materials, people, environment, methods, and machines to organize the information and potential design factors. Notice
that some of the individual causes will probably lead directly to a design factor that
Measurement
Materials
Charge monitor
calibration
Charge monitor
wafer probe failure
Faulty hardware
readings
People
Incorrect part
materials
Parts condition
Environment
Flood gun rebuild
procedure
Methods
■ FIGURE 1.10
experiment
Improper procedures
Wafer charging
Flood gun
installation
Time parts exposed
to atmosphere
Parts cleaning
procedure
Humid/Temp
Unfamiliarity with normal
wear conditions
Water flow to flood gun
Wheel speed
Gas flow
Vacuum
Machines
A cause-and-effect diagram for the etching process
18
Chapter 1 ■ Introduction
Uncontrollable
factors
Controllable design
factors
x-axis shift
Spindle differences
Ambient temp
y-axis shift
z-axis shift
Spindle speed
Titanium properties
Fixture height
Feed rate
Viscosity of
cutting fluid
Operators
Tool vendor
Nuisance (blocking)
factors
■ FIGURE 1.11
machine experiment
Blade profile,
surface finish,
defects
Temp of cutting
fluid
Held-constant
factors
A cause-and-effect diagram for the CNC
will be included in the experiment (such as wheel speed, gas flow, and vacuum),
while others represent potential areas that will need further study to turn them into
design factors (such as operators following improper procedures), and still others
will probably lead to either factors that will be held constant during the experiment
or blocked (such as temperature and relative humidity). Figure 1.11 is a cause-andeffect diagram for an experiment to study the effect of several factors on the turbine blades produced on a computer-numerical-controlled (CNC) machine. This
experiment has three response variables: blade profile, blade surface finish, and
surface finish defects in the finished blade. The causes are organized into groups
of controllable factors from which the design factors for the experiment may be
selected, uncontrollable factors whose effects will probably be balanced out by
randomization, nuisance factors that may be blocked, and factors that may be held
constant when the experiment is conducted. It is not unusual for experimenters to
construct several different cause-and-effect diagrams to assist and guide them during preexperimental planning. For more information on the CNC machine experiment and further discussion of graphical methods that are useful in preexperimental
planning, see the supplemental text material for this chapter.
We reiterate how crucial it is to bring out all points of view and process information in steps 1 through 3. We refer to this as pre-experimental planning. Coleman
and Montgomery (1993) provide worksheets that can be useful in pre-experimental
planning. Also see the supplemental text material for more details and an example
of using these worksheets. It is unlikely that one person has all the knowledge required
to do this adequately in many situations. Therefore, we strongly argue for a team effort
in planning the experiment. Most of your success will hinge on how well the preexperimental planning is done.
4. Choice of experimental design. If the above pre-experimental planning activities are
done correctly, this step is relatively easy. Choice of design involves consideration of
sample size (number of replicates), selection of a suitable run order for the experimental trials, and determination of whether or not blocking or other randomization
restrictions are involved. This book discusses some of the more important types of
1.4 Guidelines for Designing Experiments
19
experimental designs, and it can ultimately be used as a guide for selecting an appropriate experimental design for a wide variety of problems.
There are also several interactive statistical software packages that support this
phase of experimental design. The experimenter can enter information about the number of factors, levels, and ranges, and these programs will either present a selection of
designs for consideration or recommend a particular design. (We usually prefer to see
several alternatives instead of relying entirely on a computer recommendation in most
cases.) Most software packages also provide some diagnostic information about how
each design will perform. This is useful in evaluation of different design alternatives for
the experiment. These programs will usually also provide a worksheet (with the order
of the runs randomized) for use in conducting the experiment.
Design selection also involves thinking about and selecting a tentative empirical
model to describe the results. The model is just a quantitative relationship (equation)
between the response and the important design factors. In many cases, a low-order
polynomial model will be appropriate. A first-order model in two variables is
y $ "0 % "1x1 % "2x2 % #
where y is the response, the x’s are the design factors, the (’s are unknown parameters that will be estimated from the data in the experiment, and # is a random error
term that accounts for the experimental error in the system that is being studied. The
first-order model is also sometimes called a main effects model. First-order models
are used extensively in screening or characterization experiments. A common extension of the first-order model is to add an interaction term, say
y $ "0 % "1x1 % "2x2 % "12x1x2 % #
where the cross-product term x1x2 represents the two-factor interaction between the
design factors. Because interactions between factors is relatively common, the firstorder model with interaction is widely used. Higher-order interactions can also be
included in experiments with more than two factors if necessary. Another widely used
model is the second-order model
y $ "0 % "1x1 % "2x2 % "12x1x2 % "11x211 % "22x22 % #
Second-order models are often used in optimization experiments.
In selecting the design, it is important to keep the experimental objectives in
mind. In many engineering experiments, we already know at the outset that some of
the factor levels will result in different values for the response. Consequently, we are
interested in identifying which factors cause this difference and in estimating the magnitude of the response change. In other situations, we may be more interested in verifying uniformity. For example, two production conditions A and B may be compared,
A being the standard and B being a more cost-effective alternative. The experimenter
will then be interested in demonstrating that, say, there is no difference in yield
between the two conditions.
5. Performing the experiment. When running the experiment, it is vital to monitor
the process carefully to ensure that everything is being done according to plan.
Errors in experimental procedure at this stage will usually destroy experimental
validity. On...
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