PROJECT PROPOSAL | water quality test | two anova experiment (Doe engineering course)

User Generated

Arssqnjf

Mathematics

Description

SO I AM GOING TO DO A COMPARISON BETWEEN COUPLE BRANDS AND WHAT ARE THE VARIABLES THAT CAN BE IDENTIFIED THAT THIS WATER IS BETTER BY COMPARING ALL THE FACTORS THAT PLAYS A ROLE IN WATER QUALITY. EVEN THOUGH, I AM ABLE TO WRITE THE PROPOSAL BUT I WOULD LIKE TO HAVE SOME IDEAS TO START WITH FROM EXPERTS IN TWO ANOVA TEST. THE REQUIREMENT FOR THE PROPOSAL IS


    Project Overview – Your project assignment is to design and implement a design of experiments statistical analysis for a real world problem,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 assignmen

  • 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)


THANK YOU!



Unformatted Attachment Preview

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. VICE PRESIDENT AND PUBLISHER ACQUISITIONS EDITOR CONTENT MANAGER PRODUCTION EDITOR MARKETING MANAGER DESIGN DIRECTOR SENIOR DESIGNER EDITORIAL ASSISTANT PRODUCTION SERVICES COVER PHOTO COVER DESIGN Donald Fowley Linda Ratts Lucille Buonocore Anna Melhorn Christopher Ruel Harry Nolan Maureen Eide Christopher Teja Namit Grover/Thomson Digital Nik Wheeler/Corbis Images Wendy Lai This book was set in Times by Thomson Digital and printed and bound by Courier Westford. The cover was printed by Courier Westford. This book is printed on acid-free paper. ! Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. Copyright © 2013, 2009, 2005, 2001, 1997 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, website www.wiley.com/go/permissions. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. To order books or for customer service, please call 1-800-CALL WILEY (225-5945). 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...
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

Attached.

Running Head: PROJECT PROPOSAL

1

Project Proposal
Institutional Affiliation
Date

PROJECT PROPSAL

2

Problem Statement
In the current world, various companies are involved in the production and sale of drinking
water. One company is suspected to produce the same water and then market it separately as
two brands such that it is difficult to confirm or dismiss the suspicion. This project aims at
establishing if there is any relation between the two brands which can either link the two
brands to the same producer or dismiss the suspicion.
Data collection plan and initial design for the experiment
To address the project problem, there is a need to identify the various variables that can be
employed. The ...


Anonymous
Very useful material for studying!

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags