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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 causeandeffect 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 experimentson 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 runis a test. More formally, we can define an experimentas 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 decisionmaking.
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 ȍ 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 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Includes bibliographical references and index ISBN 978-1-118-14692-7 Experimental design I Title QA279.M66 2013 519.5'7—dc23 2012000877 ISBN 978-1118-14692-7 10 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 – 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 (very quickly), most of Chapter 3, Chapter (excluding the material on incomplete blocks and only mentioning Latin squares briefly), and I discuss Chapters through 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 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 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: The supplemental text material described above Data sets from the book examples and homework problems, in electronic form Sample Student Projects Instructor Companion Site The instructor’s section of the textbook website contains the following: 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 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 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 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 Index Terms Links C Canonical analysis of a response surface 489 Canonical variables 494 Cause-and-effect diagram Cell plot 17 416 k Center points in a design 285 Center points in the central composite design 503 Central composite design 288 489 501 503 504 65 568 Central limit theorem 33 Characterization experiments see also screening experiments Characterization of a response surface 488 Chi-square distribution 33 Cochran’s theorem 74 Coded design factors 57 290 Coding data in ANOVA 76 Combined array design 558 Complete randomization 12 Completely randomized design (CRD) 561 567 66 69 188 233 234 Component axis 534 Component of interaction 397 398 70 574 Components of variance model Computer models 523 Conditional inference chart 264 Conference matrices 521 408 Confidence coefficient 43 Confidence intervals 36 43 57 59 78 109 251 251 467 468 597 600 Confidence interval on contrasts 93 Confidence interval on the mean response 468 Confidence intervals on effects 251 Confidence intervals on regression model coefficients 467 Confirmation experiments 15 This page has been reformatted by Knovel to provide easier navigation 252 20 333 Index Terms Links Confounding 306 402 k 306 313 k Confounding in the factorial design 402 Construction of optimal designs 514 Confounding in the factorial design Continuous probability distribution Contour plots 28 185 Contrasts 92 Contrasts and preplanned comparisons 95 Contrasts in a two-level design 236 Control-by-noise interaction in robust design 557 Controllable factors Coordinate exchange algorithm for design construction Correlation between residuals Correlation matrix 315 496 242 16 514 82 416 Covariance 30 Covariance matrix 116 Covariate 655 Critical region for a statistical test 37 Crossed array designs 56 454 Crossed factors, see factorial design Crossover designs 164 Cubodial versus spherical region of interest 504 D Data snooping 95 Defining contrast for a blocked design 308 Defining relation for a fold-over design 356 Defining relation for a fractional factorial design 322 Definitive screening designs 520 Degrees of freedom 314 403 334 408 32 Design generator 321 334 Design resolution 323 340 Designs balanced for residual effects 164 Designs for robust design 567 Desirability function optimization in RSM 498 Deterministic versus stochastic computer (simulation) models 523 Different error structures in the split-plot design 623 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Discovery experiments 15 Discrete probability distribution 28 Dispersion effects 114 253 271 338 D-optimal designs 283 Dot diagram 513 26 Dunnett’s test for comparing means with a control 101 Duplicate measurements on the response 274 E Effect heredity 326 Effect magnitude and direction 236 Effect of a factor 183 Effect of outliers in unreplicated designs 267 Effects coding 238 242 Effects model 69 141 166 19 20 188 Effects model for a Graeco-Latin square design 166 Effects model for a two-factor factorial design 188 Effects model for the Latin square design 160 Effects model for the RCBD 141 Empirical model 89 Engineering method Equiradial designs 505 Estimate 31 Estimating missing values in the RCBD 154 Estimating model parameters in a two-factor factorial 198 Estimating model parameters in the BIBD 172 Estimating model parameters in the RCBD 155 Estimating the overall mean in a random model 122 Estimating variance components 118 see also residual maximum likelihood method (REML) Estimation of parameters in ANOVA models 78 Estimator 31 Evolutionary operation (EVOP) Expected mean squares 540 73 This page has been reformatted by Knovel to provide easier navigation 152 Index Terms Links Expected value 29 Expected value operator 29 Experiment Experimental error 27 Experimental units 69 140 Experiments with computer models 10 523 Experimentwise error rates 98 Exponential distribution 646 Exponential family of distributions 646 Extra sum of squares method 465 F Face-centered cube design 504 Factor effect 234 Factorial design 183 187 206 233 Factorial experiment in a Latin square 223 Factorial experiment in a randomized complete block (RCBD) 219 Factorial experiments with covariates 667 Family of fractional factorial designs 323 F-distribution 35 First-order model 19 304 see also models for data from experiments First-order response surface designs 501 Fisher LSD procedure for comparing all pairs of means 99 Fixed factor effect 69 189 573 353 354 356 Fold over of a design 366 Fold over of a resolution IV design 368 Fold over of resolution III designs 353 Follow-up runs 20 see also confirmation experiments Formulation experiments Fraction of design space plot Fractional factorial design 11 285 506 320 Full cubic mixture model 533 Full fold over 354 This page has been reformatted by Knovel to provide easier navigation Index Terms Links G Gamma distribution 646 651 Gaussian process model 525 527 General factorial designs 206 Generalized interaction 314 Generalized linear models 645 G-optimal designs Graeco-Latin square designs 334 406 283 433 513 165 411 Graphical comparison of means 91 Graphical evaluation of designs 506 Guidelines for designing experiments 14 H Hadamard matrix designs 375 Half-normal plot of effects 262 Hall designs 418 Hat matrix in regression 470 Hidden replication 260 Hierarchical designs, see nested designs Histogram 27 Hybrid designs 506 Hypothesis testing 26 36 Hypothesis tests on variances 57 58 85 I I and J components of interaction 397 Identity element 245 Identity link 646 Immediacy 21 Incomplete block design 168 Independence assumption in ANOVA 82 Independent random variables 30 Influence on regression coefficients 473 Inner array in a crossed array 556 Integrated variance 28 This page has been reformatted by Knovel to provide easier navigation 306 84 Index Terms Links Interaction 234 244 Interaction and curvature 186 Interaction between treatments and blocks 150 Interblock analysis of the BIBD 174 Interclass correlation coefficient 121 Intrablock analysis of the BIBD 174 I-optimal design 283 Irregular design regions 511 Iterative experimentation 20 184 433 see also sequential experimentation J J component of interaction 397 K Kruskal-Wallis test 128 L Lack of fit 251 Latin hypercube designs 524 Latin square designs 158 Latin square designs and Sudoku puzzles 159 Least squares normal equations 125 k Lenth’s method for analyzing unreplicated designs Levels of a factor 473 223 409 126 452 36 66 262 25 see also treatments Levene’s test for equal variances 85 Leverage points 473 Linear mixture model 532 Linear predictor 646 Linear statistical model 69 see also models for data from experiments Link function 646 Log link 646 Logistic regression model 647 Logit link 647 This page has been reformatted by Knovel to provide easier navigation 651 Index Terms Links M Main effect of a factor 183 Maximum entropy designs 525 Maximum likelihood estimation of variance components 123 234 see also residual maximum likelihood method (REML) Mean of a distribution 29 Mean squares 72 Means model for a two-factor factorial design 189 Means model for the RCBD 141 Means model Measurement systems capability study Mechanistic model 69 141 575 582 189 Method of least squares 89 Method of unweighted means 56 Minimum aberration design 341 Minimum run resolution IV designs 435 Minimum run resolution V designs 366 Minimum variance estimator 125 451 415 438 31 Missing value problems in the RCBD 154 158 Mixed level fractional factorials 412 414 Mixed model 581 Mixture designs for constrained regions 535 Mixture experiments 530 Model adequacy checking, see residual plots Model independent estimate of error Models for data from experiments 474 36 53 69 89 141 160 166 188 189 238 247 285 479 533 534 574 581 583 646 657 667 Modified large-sample method for finding confidence intervals on variance components 600 Moment estimators of variance components 119 m-stage nested designs 614 Multiple comparisons 90 Multiple comparisons in a factorial experiment 194 This page has been reformatted by Knovel to provide easier navigation 575 98 Index Terms Links Multiple comparisons in the RCBD 146 Multiple linear regression model 450 see also regression models Multiple responses in RSM 496 498 N Nested and factorial factors 616 Nested designs 574 604 605 612 614 616 No interaction in a factorial model 202 No-confounding designs 420 425 Noise factors 16 556 Noise reduction from blocking 56 146 Nongeometric designs 357 Nonisomorphic designs 418 Nonlinear programming 498 Nonnormal response distributions 84 87 269 374 415 643 Nonparametric ANOVA 128 Nonregular fractional factorial designs 359 425 Nonstandard models 512 Normal distribution 32 Normal probability plot 41 Normal probability plot of effects 646 257 Normal probability plot of residuals 81 Normality assumption in ANOVA 80 Nuisance factors 13 Null hypothesis 37 139 O Ockham’s razor 326 Odds ratio 647 One replicate of a factorial experiment 203 One-factor-at-a-time (OFAT) experiments One-half fraction One-sided alternative hypothesis One-step RSM designs 38 520 This page has been reformatted by Knovel to provide easier navigation 255 321 Index Terms Links Operating characteristic curve Optimal designs with covariates Optimal designs 45 105 201 587 153 672 21 280 374 431 511 535 14 17 672 Optimal response surface designs Optimization experiment 511 Optimization with contour plots 496 Orthogonal blocking 507 Orthogonal coding 238 Orthogonal contrasts 242 94 Orthogonal design 238 242 Orthogonal Latin squares 165 396 Outer array in a crossed array 556 Outliers 82 459 267 P Paired comparison tests 53 Partial aliasing 358 411 Partial confounding 309 316 Partial F test 466 Partial fold over 371 Path of steepest ascent 481 Placket-Burman designs 357 Point exchange algorithms for design construction 513 Poisson distribution 646 Pooled estimate of variance 72 Power curve 45 Power family transformations Power of a statistical test 485 486 643 37 Prediction interval on a future observation 468 Prediction profile plot 264 Prediction variance profiler 283 Pre-experimental planning 18 107 PRESS statistic 251 470 Principal block 308 403 Principal fraction 323 This page has been reformatted by Knovel to provide easier navigation 471 Index Terms Links Probability distributions 28 Process robustness study 544 Product design 10 k Projection of a 260 Projection of fractional factorial designs 325 Projection of Plackett-Burman designs 359 Propagation of error 563 Proportional data in ANOVA 652 Pseudocomponents 536 Pure quadratic curvature 286 P-values 343 359 40 Q Quadratic mixture model Quadratic model 532 90 see also second-order model Qualitative factors 233 289 399 Quantitative factors 185 233 285 395 399 R R2 251 R for prediction 251 Random effects model 116 573 Random factor effect 69 116 Random sample 30 Random treatments and blocks 151 Random variable 27 Randomization 12 139 143 159 Randomization tests 43 77 Randomized block design 56 Randomized complete block design (RCBD) 140 Rank transformation in ANOVA 130 Ranks 128 Recovery of interblock information in the BIBD 174 Reference distribution Regression approach to ANOVA 574 38 125 This page has been reformatted by Knovel to provide easier navigation 141 Index Terms Links Regression model for a factorial Regression models 238 89 185 238 449 450 451 152 222 13 645 Regular fractional factorial designs 359 Relationship between coded and natural variables 128 REML 123 579 Repeated measures designs 677 Replicated design Replication 12 66 106 Replication of Latin squares 163 Replication versus repeated measurements 13 Residual plots 80 81 82 83 88 146 149 198 239 261 609 662 80 146 198 239 260 261 453 609 662 323 351 353 408 415 324 366 415 373 415 500 Residuals Resolution III designs Resolution IV designs 435 Resolution V designs 324 438 Response curves 211 Response model approach to robust design 562 Response surface designs 395 479 501 520 478 481 488 496 185 211 240 261 15 Response surface methodology (RSM) Response surface plots Response variable Restricted form of the mixed model 581 Ridge systems in response surfaces 495 This page has been reformatted by Knovel to provide easier navigation 486 214 Index Terms Links Rising ridge 495 Robust parameter design 554 557 567 106 108 109 153 201 Robustness 15 Rotatability 502 Rotatable central composite design 503 R-student 472 Rules for determining expected mean squares 588 Run S Sample mean 30 Sample size determination 587 Sample standard deviation 31 Sample variance 30 Sampling distribution 30 Saturated fractional factorial design 351 Scaled prediction variance (SPV) 506 Scatter diagram 67 Scheffe’s method for comparing all contrasts 96 Scientific method 32 Screening experiments 14 17 233 Second-order model 19 90 Second-order response surface model 285 479 Sequences of fractional factorials 331 332 15 20 21 23 288 331 367 480 501 see also models for data from experiments Sequential experimentation 524 Signal-to-noise ratios 558 Significance level of a statistical test 37 Simplex centroid design 532 Simplex design in RSM 501 Simplex lattice design 531 Simplex mixture designs 531 Simultaneous confidence intervals 79 This page has been reformatted by Knovel to provide easier navigation 38 96 Index Terms Links Single factor experiment 68 k Single replicate of a 255 Single-factor fold over 354 Small composite designs 505 Space-filling designs 524 Sparsity of effects principle 255 Special cubic mixture model 533 Sphere packing designs 525 Spherical central composite design 503 Split-plot designs 574 621 627 632 Split-split-plot designs 632 Staggered nested designs 612 Standard error 38 Standard error of a regression coefficient 454 Standard Latin square 162 Standard normal distribution k Standard order in a design 33 237 Standardized contrasts 94 Standardized residual 470 Stationary point on a response surface 486 Stationary ridge 495 Statistic 30 Statistical approach to designing experiments 11 Steepest ascent 96 253 480 Strategy of experimentation Strip-split-plot designs 636 Strong heredity 326 Studentized range statistic 98 Studentized residual 470 Subplot error 622 Subplot treatments 621 Subplots 621 Subsampling 626 Supersaturated designs 374 Symmetric BIBD 169 This page has been reformatted by Knovel to provide easier navigation 471 625 Index Terms Links T t-distribution 34 Test for significance of regression 462 Test statistic 37 Tests of hypotheses on regression model coefficients 46 Total effect of a factor 35 236 Transformations to correct violations of assumptions 84 87 269 643 Transmission of error Treatments 561 25 Trilinear coordinates 531 Tukey’s additivity test 204 Tukey’s test for comparing all pairs of means 98 Tukey-Kramer test 98 Two-factor factorial design 187 Two-sample t-test 38 Two-sample t-test with unequal variances 48 Two-sided alternative hypothesis 37 Two-stage nested designs Types of factors in experiments 68 41 604 16 U Unbalanced data in ANOVA 79 Unbiased estimator 31 Uncontrollable factors Uniform designs 652 16 556 574 611 525 k Unreplicated designs 255 Unrestricted form of the mixed model 583 Unscaled prediction variance 283 Unusual sample size requirements 513 V Variability 27 Variance components 116 Variance dispersion graph 506 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Variance modeling 559 Variance of a distribution 29 Variance operator 29 V-optimality 513 W W component of interaction 398 Weak heredity 327 Weighted squares of means method 655 Whole plot error 622 Whole plot treatments 621 Whole plots 621 X X component of interaction 398 Y Y component of interaction 398 Yates’s order 237 Z Z component of interaction Z-tests on means 398 50 This page has been reformatted by Knovel to provide easier navigation 57 ... 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. .. Recognition of and statement of the problem Selection of the response variablea Choice of factors, levels, and rangesa Choice of experimental design Performing the experiment Statistical analysis of the... 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