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Planning, Construction, and Statistical Analysis of Comparative Experiments

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: David J. Balding, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Geert Molenberghs, Louise M. Ryan, David W. Scott, Adrian F. M. Smith, JozefL. Teugels Editors Emeriti: Vic Barnett, J. Stuart Hunter, David G. Kendall

A complete list of the titles in this series appears at the end of this volume.

Planning, Construction, and Statistical Analysis of Comparative Experiments

FRANCIS G. GIESBRECHT MARCIA L. GUMPERTZ

,WILEY~ INTERSCIENCE

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2004 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Giesbrecht, Francis G., 1935-Planning, construction, and statistical analysis of comparative experiments / Francis G.

Giesbrecht, Marcia L. Gumpertz. p. cm.—(Wiley series in probability and statistics)

Includes bibliographical references and index. ISBN 0-471-21395-0 (acid-free paper) 1. Experimental design. I. Gumpertz, Marcia L., 1952- II. Title. III. Series.

QA279.G52 2004 519.5—dc21

2003053838

1 0 9 8 7 6 5 4 3 2 1

Contents

Preface xüi

1. Introduction 1

1.1 Role of Statistics in Experiment Design 1 1.2 Organization of This Book 3 1.3 Representativeness and Experimental Units 4 1.4 Replication and Handling Unexplained Variability 7 1.5 Randomization: Why and How 8 1.6 Ethical Considerations 10 Review Exercises 12

2. Completely Randomized Design 13

2.1 Introduction 13 2.2 Completely Randomized Design 13 2.3 Assumption of Additivity 15 2.4 Factorial Treatment Combinations 18 2.5 Nested Factors 23 Review Exercises 26

3. Linear Models for Designed Experiments 28

3.1 Introduction 28 3.2 Linear Model 29 3.3 Principle of Least Squares 30 3.4 Parameterizations for Row-Column Models 37 Review Exercises 41 Appendix 3A: Linear Combinations of Random Variables 43 Appendix 3B: Simulating Random Samples 44

CONTENTS

Testing Hypotheses and Determining Sample Size 45

4.1 Introduction 45 4.2 Testing Hypotheses in Linear Models with Normally

Distributed Errors 46 4.3 Kruskal-Wallis Test 53 4.4 Randomization Tests 54 4.5 Power and Sample Size 58 4.6 Sample Size for Binomial Proportions 68 4.7 Confidence Interval Width and Sample Size 72 4.8 Alternative Analysis: Selecting and Screening 74 Review Exercises 78

Methods of Reducing Unexplained Variation 81

5.1 Randomized Complete Block Design 81 5.2 Blocking 81 5.3 Formal Statistical Analysis for the RCBD 87 5.4 Models and Assumptions: Detailed Examination 92 5.5 Statistical Analysis When Data Are Missing in an RCBD 101 5.6 Analysis of Covariance 109 Review Exercises 112 Appendix 5A: Interaction of a Random Block Effect and a Fixed Treatment Effect 116

Latin Squares 118

6.1 Introduction 118 6.2 Formal Structure of Latin Squares 119 6.3 Combining Latin Squares 126 6.4 Graeco-Latin Squares and Orthogonal Latin Squares 134 6.5 Some Special Latin Squares and Variations on Latin Squares 137 6.6 Frequency Squares 140 6.7 Youden Square 145 Review Exercises 150 Appendix 6A: Some Standard Latin Squares 153 Appendix 6B: Mutually Orthogonal Latin Squares 155 Appendix 6C: Possible Youden Squares 157

Split-Plot and Related Designs 158

7.1 Introduction 158 7.2 Background Material 158

CONTENTS vii

7.3 Examples of Situations That Lead to Split-Plots 159 7.4 Statistical Analysis of Split-Plot Experiments 162 7.5 Split-Split-Plot Experiments 175 7.6 Strip-Plot Experiments 176 7.7 Comments on Further Variations 180 7.8 Analysis of Covariance in Split-Plots 181 7.9 Repeated Measures 193 Review Exercises 195

8 Incomplete Block Designs 197

8.1 Introduction 197 8.2 Efficiency of Incomplete Block Designs 207 8.3 Distribution-Free Analysis for Incomplete Block Designs 209 8.4 Balanced Incomplete Block Designs 212 8.5 Lattice Designs 223 8.6 Cyclic Designs 225 8.7 «-Designs 229 8.8 Other Incomplete Block Designs 231 Review Exercises 232 Appendix 8A: Catalog of Incomplete Block Designs 235

9 Repeated Treatments Designs 242

9.1 Introduction 242 9.2 Repeated Treatments Design Model 242 9.3 Construction of Repeated Treatments Designs 245 9.4 Statistical Analysis of Repeated Treatments Design Data 255 9.5 Carryover Design for Two Treatments 260 9.6 Correlated Errors 265 9.7 Designs for More Complex Models 267 Review Exercises 270

10 Factorial Experiments: The 2N System 272

10.1 Introduction 272 10.2 2N Factorials 272 10.3 General Notation for the 2N System 280 10.4 Analysis of Variance for 2N Factorials 284

11 Factorial Experiments: The 3N System 288

11.1 Introduction 288

viii CONTENTS

11.2 3 x 3 Factorial 288 11.3 General System of Notation for the 3N System 295

12 Analysis of Experiments without Designed Error Terms 299

12.1 Introduction 299 12.2 Techniques That Look for Location Parameters 300 12.3 Analysis for Dispersion Effects 304 Review Exercises 308

13 Confounding Effects with Blocks 310

13.1 Introduction 310 13.2 Confounding 23 Factorials 311 13.3 General Confounding Patterns 319 13.4 Double Confounding 325 13.5 3N System 327 13.6 Detailed Numerical Example: 33 Factorial in Blocks of Nine 332 13.7 Confounding Schemes for the 34 System 337 Review Exercises 338

14 Fractional Factorial Experiments 340

14.1 Introduction 340 14.2 Organization of This Chapter 343 14.3 Fractional Replication in the 2N System 344 14.4 Resolution 351 14.5 Constructing Fractional Replicates by Superimposing Factors 356 14.6 Foldover Technique 360 14.7 Franklin-Bailey Algorithm for Constructing Fractions 361 14.8 Irregular Fractions of the 2N System 364 14.9 Fractional Factorials with Compromised Resolution 371 14.10 A Caution About Minimum Aberration 376 14.11 Direct Enumeration Approach to Constructing Plans 378 14.12 Blocking in Small 2N'k Plans 379 14.13 Fractional Replication in the 3^ System 383 Review Exercises 387 Appendix 14A: Minimum Aberration Fractions without Blocking 390 Appendix 14B: Minimum Aberration Fractions with Blocking 393

CONTENTS ■x

15 Response Surface Designs 398

15.1 Introduction 398 15.2 Basic Formulation 399 15.3 Some Basic Designs 400 15.4 Rotatability 407 15.5 Statistical Analyses of Data from Response Surface Designs 408 15.6 Blocking in Response Surface Designs 409 15.7 Mixture Designs 411 15.8 Optimality Criteria and Parametric Modeling 412 15.9 Response Surfaces in Irregular Regions 413 15.10 Searching the Operability Region for an Optimum 416 15.11 Examination of an Experimental Problem 426 Review Exercise 427

16 Plackett-Burman Hadamard Plans 429

16.1 Introduction 429 16.2 Hadamard Matrix 429 16.3 Plackett-Burman Plans 430 16.4 Hadamard Plans and Interactions 436 16.5 Statistical Analyses 439 16.6 Weighing Designs 442 16.7 Projection Properties of Hadamard Plans 445 16.8 Very Large Factorial Experiments 449

17 General pN and Nonstandard Factorials 451

17.1 Introduction 451 17.2 Organization of This Chapter 451 17.3 pN System with p Prime 452 17.4 4^ System 456 17.5 4^ System Using Pseudofactors at Two Levels 469 17.6 6^ Factorial System 472 17.7 Asymmetrical Factorials 475 17.8 2N x 3M Plans 478

18 Plans for Which Run Order Is Important 484

18.1 Introduction 484 18.2 Preliminary Concepts 485 18.3 Trend-Resistant Plans 487

X CONTENTS

18.4 Generating Run Orders for Fractions 489 18.5 Extreme Number of Level Changes 492 18.6 Trend-Free Plans from Hadamard Matrices 498 18.7 Extensions to More Than Two Levels 502 18.8 Small One-at-a-Time Plans 503 18.9 Comments 507

19 Sequences of Fractions of Factorials 508

19.1 Introduction 508 19.2 Motivating Example 510 19.3 Foldover Techniques Examined 511 19.4 Augmenting a 24_1 Fraction 516 19.5 Sequence Starting from a Seven-Factor Main Effect Plan 519 19.6 Augmenting a One-Eighth Fraction of 43 520 19.7 Adding Levels of a Factor 523 19.8 Double Semifold 528 19.9 Planned Sequences 530 19.10 Sequential Fractions 537

20 Factorial Experiments with Quantitative Factors: Blocking and Fractions 540 20.1 Introduction 540 20.2 Factors at Three Levels 540 20.3 Factors at Four Levels Based on the 2N System 541 20.4 Pseudofactors and Hadamard Plans 552 20.5 Box-Behnken Plans 553 Review Exercises 555 Appendix 20A: Box-Behnken Plans 556

21 Supersaturated Plans 559

21.1 Introduction 559 21.2 Plans for Small Experiments 560 21.3 Supersaturated Plans That Include an Orthogonal Base 564 21.4 Model-Robust Plans 564

22 Multistage Experiments 569

22.1 Introduction 569 22.2 Factorial Structures in Split-Plot Experiments 570 22.3 Splitting on Interactions 573

CONTENTS xi

22.4 Factorials in Strip-Plot or Strip-Unit Designs 578 22.5 General Comments on Strip-Unit Experiments 590 22.6 Split-Lot Designs 590 Appendix 22A: Fractional Factorial Plans for Split-Plot Designs 595

23 Orthogonal Arrays and Related Structures 598

23.1 Introduction 598 23.2 Orthogonal Arrays and Fractional Factorials 602 23.3 Other Construction Methods 606 23.4 Nearly Orthogonal Arrays 620 23.5 Large Orthogonal Arrays 626 23.6 Summary 630

24 Factorial Plans Derived via Orthogonal Arrays 631

24.1 Introduction 631 24.2 Preliminaries 631 24.3 Product Array Designs 638 24.4 Block Crossed Arrays 641 24.5 Compound Arrays 644

25 Experiments on the Computer 652

25.1 Introduction 652 25.2 Stratified and Latin Hypercube Sampling 653 25.3 Using Orthogonal Arrays for Computer Simulation Studies 656 25.4 Demonstration Simulations 657

References 661

Index 677

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Preface

This is a book based on years of teaching statistics at the graduate level and many more years of consulting with individuals conducting experiments in many disciplines. The object of the book is twofold. It is designed to be used as a textbook for students who have already had a solid course in statistical methods. It is also designed to be a handbook and reference for active researchers and statisticians who serve as consultants in the design of experiments.

We encountered very early in our work that the subject is not linear in the sense that there is a unique order in which topics must be presented—that topics A, B, and C must be presented in that order. All too often, we found that in one sense, A should be before B, and in another, B should be before A. Our aim is to give sufficient detail to show interrelationships, that is, how to reach destinations, yet not hide important points with too much clutter.

We assume that the reader of this book is comfortable with simple sampling, multiple regression, r-tests, confidence intervals, and the analysis of variance at the level found in standard statistical methods textbooks. The demands for mathematics are limited. Simple algebra and matrix manipulation are sufficient.

In our own teaching, we have found that we pick and choose material from various chapters. Chapters 1 to 5 and Chapter 7 provide a review of methods of statistical analysis and establish notation for the remainder of the book. Material selected from the remaining chapters then provides the backbone of an applied course in design of experiments. There is more material than can be covered in one semester. For a graduate level course in design of experiments targeted at a mix of statistics and non-statistics majors, one possible course outline covers Chapter 1, Sections 3.3 and 3.4, Chapter 4, Section 5.5, and Chapters 6 and 8 to 15. If the students have a very strong preparation in statistical methods, we suggest covering selections from Chapters 6 to 16 and 19 to 24, with just the briefest review of Chapters 1 to 5. We have also used material from Chapters 10 to 14, 16, and 19 to 24 for a more specialized course on factorial experiments for students interested in industrial quality control.

In a sense, it would have been much easier to write a book that contained a series of chapters that would serve as a straightforward text. However, our aim is also to provide a handbook for the investigator planning a research program.

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