Design and Analysis of Experiments, Volume 2

The development and introduction of new experimental designs in the last fifty years has been quite staggering, brought about largely by an ever-widening field of applications. Design and Analysis of Experiments, Volume 2: Advanced Experimental Design is the second of a two-volume body of work that builds upon the philosophical foundations of experimental design set forth by Oscar Kempthorne half a century ago and updates it with the latest developments in the field. Designed for advanced-level graduate students and industry professionals, this text includes coverage of incomplete block and row-column designs; symmetrical, asymmetrical, and fractional factorial designs; main effect plans and their construction; supersaturated designs; robust design, or Taguchi experiments; lattice designs; and cross-over designs.

Preface xix

1 General Incomplete Block Design 1

1.1 Introduction and Examples 1

1.2 General Remarks on the Analysis of Incomplete Block Designs 3

1.3 The Intrablock Analysis 4

1.4 Incomplete Designs with Variable Block Size 13

1.5 Disconnected Incomplete Block Designs 14

1.6 Randomization Analysis 16

1.7 Interblock Information in an Incomplete Block Design 23

1.8 Combined Intra- and Interblock Analysis 27

1.9 Relationships Among Intrablock Interblock and Combined Estimation 31

1.10 Estimation of Weights for the Combined Analysis 36

1.11 Maximum-Likelihood Type Estimation 39

1.12 Efficiency Factor of an Incomplete Block Design 43

1.13 Optimal Designs 48

1.14 Computational Procedures 52

2 Balanced Incomplete Block Designs 71

2.1 Introduction 71

2.2 Definition of the BIB Design 71

2.3 Properties of BIB Designs 72

2.4 Analysis of BIB Designs 74

2.5 Estimation of ρ 77

2.6 Significance Tests 79

2.7 Some Special Arrangements 89

2.8 Resistant and Susceptible BIB Designs 98

3 Construction of Balanced Incomplete Block Designs 104

3.1 Introduction 104

3.2 Difference Methods 104

3.3 Other Methods 113

3.4 Listing of Existing BIB Designs 115

4 Partially Balanced Incomplete Block Designs 119

4.1 Introduction 119

4.2 Preliminaries 119

4.3 Definition and Properties of PBIB Designs 123

4.4 Association Schemes and Linear Associative Algebras 127

4.5 Analysis of PBIB Designs 131

4.6 Classification of PBIB Designs 137

4.7 Estimation of ρ for PBIB(2) Designs 155

5 Construction of Partially Balanced Incomplete Block Designs 158

5.1 Group-Divisible PBIB(2) Designs 158

5.2 Construction of Other PBIB(2) Designs 165

5.3 Cyclic PBIB Designs 167

5.4 Kronecker Product Designs 172

5.5 Extended Group-Divisible PBIB Designs 178

5.6 Hypercubic PBIB Designs 187

6 More Block Designs and Blocking Structures 189

6.1 Introduction 189

6.2 Alpha Designs 190

6.3 Generalized Cyclic Incomplete Block Designs 193

6.4 Designs Based on the Successive Diagonalizing Method 194

6.5 Comparing Treatments with a Control 195

6.6 Row–Column Designs 213

7 Two-Level Factorial Designs 241

7.1 Introduction 241

7.2 Case of Two Factors 241

7.3 Case of Three Factors 248

7.4 General Case 253

7.5 Interpretation of Effects and Interactions 260

7.6 Analysis of Factorial Experiments 262

8 Confounding in 2 n Factorial Designs 279

8.1 Introduction 279

8.2 Systems of Confounding 283

8.3 Composition of Blocks for a Particular System of Confounding 289

8.4 Detecting a System of Confounding 291

8.5 Using SAS for Constructing Systems of Confounding 293

8.6 Analysis of Experiments with Confounding 293

8.7 Interblock Information in Confounded Experiments 303

8.8 Numerical Example Using SAS 311

9 Partial Confounding in 2 n Factorial Designs 312

9.1 Introduction 312

9.2 Simple Case of Partial Confounding 312

9.3 Partial Confounding as an Incomplete Block Design 318

9.4 Efficiency of Partial Confounding 323

9.5 Partial Confounding in a 2³ Experiment 324

9.6 Partial Confounding in a 2⁴ Experiment 327

9.7 General Case 329

9.8 Double Confounding 335

9.9 Confounding in Squares 336

9.10 Numerical Examples Using SAS 338

10 Designs with Factors at Three Levels 359

10.1 Introduction 359

10.2 Definition of Main Effects and Interactions 359

10.3 Parameterization in Terms of Main Effects and Interactions 365

10.4 Analysis of 3ⁿ Experiments 366

10.5 Confounding in a 3ⁿ Factorial 368

10.6 Useful Systems of Confounding 374

10.7 Analysis of Confounded 3ⁿ Factorials 380

10.8 Numerical Example 387

11 General Symmetrical Factorial Design 393

11.1 Introduction 393

11.2 Representation of Effects and Interactions 395

11.3 Generalized Interactions 396

11.4 Systems of Confounding 398

11.5 Intrablock Subgroup 400

11.6 Enumerating Systems of Confounding 402

11.7 Fisher Plans 403

11.8 Symmetrical Factorials and Finite Geometries 409

11.9 Parameterization of Treatment Responses 410

11.10 Analysis of pⁿ Factorial Experiments 412

11.11 Interblock Analysis 421

11.12 Combined Intra- and Interblock Information 426

11.13 The sⁿ Factorial 431

11.14 General Method of Confounding for the Symmetrical Factorial Experiment 447

11.15 Choice of Initial Block 463

12 Confounding in Asymmetrical Factorial Designs 466

12.1 Introduction 466

12.2 Combining Symmetrical Systems of Confounding 467

12.3 The GC/n Method 477

12.4 Method of Finite Rings 480

12.5 Balanced Factorial Designs (BFD) 491

13 Fractional Factorial Designs 507

13.1 Introduction 507

13.2 Simple Example of Fractional Replication 509

13.3 Fractional Replicates for 2ⁿ Factorial Designs 513

13.4 Fractional Replicates for 3ⁿ Factorial Designs 524

13.5 General Case of Fractional Replication 529

13.6 Characterization of Fractional Factorial Designs of Resolution III IV and V 536

13.7 Fractional Factorials and Combinatorial Arrays 547

13.8 Blocking in Fractional Factorials 549

13.9 Analysis of Unreplicated Factorials 558

14 Main Effect Plans 564

14.1 Introduction 564

14.2 Orthogonal Resolution III Designs for Symmetrical Factorials 564

14.3 Orthogonal Resolution III Designs for Asymmetrical Factorials 582

14.4 Nonorthogonal Resolution III Designs 594

15 Supersaturated Designs 596

15.1 Introduction and Rationale 596

15.2 Random Balance Designs 596

15.3 Definition and Properties of Supersaturated Designs 597

15.4 Construction of Two-Level Supersaturated Designs 598

15.5 Three-Level Supersaturated Designs 603

15.6 Analysis of Supersaturated Experiments 604

16 Search Designs 608

16.1 Introduction and Rationale 608

16.2 Definition of Search Design 608

16.3 Properties of Search Designs 609

16.4 Listing of Search Designs 615

16.5 Analysis of Search Experiments 617

16.6 Search Probabilities 630

17 Robust-Design Experiments 633

17.1 Off-Line Quality Control 633

17.2 Design and Noise Factors 634

17.3 Measuring Loss 635

17.4 Robust-Design Experiments 636

17.5 Modeling of Data 638

18 Lattice Designs 649

18.1 Definition of Quasi-Factorial Designs 649

18.2 Types of Lattice Designs 653

18.3 Construction of One-Restrictional Lattice Designs 655

18.4 General Method of Analysis for One-Restrictional Lattice Designs 657

18.5 Effects of Inaccuracies in the Weights 661

18.6 Analysis of Lattice Designs as Randomized Complete Block Designs 666

18.7 Lattice Designs as Partially Balanced Incomplete Block Designs 669

18.8 Lattice Designs with Blocks of Size K^l 670

18.9 Two-Restrictional Lattices 671

18.10 Lattice Rectangles 678

18.11 Rectangular Lattices 679

18.12 Efficiency Factors 682

19 Crossover Designs 684

19.1 Introduction 684

19.2 Residual Effects 685

19.3 The Model 685

19.4 Properties of Crossover Designs 687

19.5 Construction of Crossover Designs 688

19.6 Optimal Designs 695

19.7 Analysis of Crossover Designs 699

19.8 Comments on Other Models 706

Appendix A Fields and Galois Fields 716

Appendix B Finite Geometries 721

Appendix C Orthogonal and Balanced Arrays 724

Appendix D Selected Asymmetrical Balanced Factorial Designs 728

Appendix E Exercises 736

References 749

Author Index 767

Subject Index 771

"…a massively impressive work of scholarship…" (Short Book Reviews, December 2006)

"...a broad and in-depth book...covers not only classic but also up-to-date results and references, making it convenient for researchers. It is one of the very few advanced textbooks on experimental design..." (Technometrics, November 2006)

"I suspect this excellent book will be used most often by specialists in design...the book's importance is largely as a reference for experts...or as an independent learning tool…" (Journal of the American Statistical Association, June 2006)

"I would expect HK to attain essentially the same stature and appeal to virtually the same markets as the 1952 edition." (Journal of Quality Technology, January 2006)

"…the authors have done a commendable job in putting together the vast amount of literature that is available on the topics…of great value to students, and also to teachers and researchers." (Mathematical Reviews, 2006b)

KLAUS HINKELMANN, PHD, is Emeritus Professor of Statistics at Virginia Polytechnic Institute and State University Department of Statistics, where he also served as both graduate administrator and department head. In addition to being a Fellow of both the American Statistical Association and the American Association for the Advancement of Science, Professor Hinkelmann is a member of the International Statistical Institute, and has served as a council member of the International Biometric Society. He was editor of Biometrics and the Current Index to Statistics.

OSCAR KEMPTHORNE, SCD, was Emeritus Professor of Statistics and Emeritus Distinguished Professor of Liberal Arts and Sciences at Iowa State University. He was a Fellow of the American Statistical Association, the Institute of Mathematical Statistics, and the American Association for the Advancement of Science, as well as an Honorary Fellow of the Royal Statistical Society and a member of the International Statistical Institute. In addition, Dr. Kempthorne was a past president of the Eastern North American Region (ENAR) of the International Biometric Society, a former chairman of statistics within the American Association for the Advancement of Science, and a past president of the Institute of Mathematical Statistics.

A comprehensive overview of experimental design at the advanced level

The development and introduction of new experimental designs in the last fifty years has been quite staggering and was brought about largely by an ever-widening field of applications. Design and Analysis of Experiments, Volume 2: Advanced Experimental Design is the second of a two-volume body of work that builds upon the philosophical foundations of experimental design set forth half a century ago by Oscar Kempthorne, and features the latest developments in the field.

Volume 1: An Introduction to Experimental Design introduced students at the MS level to the principles of experimental design, including the groundbreaking work of R. A. Fisher and Frank Yates, and Kempthorne's work in randomization theory with the development of derived linear models. Design and Analysis of Experiments, Volume 2 provides more detail about aspects of error control and treatment design, with emphasis on their historical development and practical significance, and the connections between them. Designed for advanced-level graduate students and industry professionals, this text includes coverage of:

• Incomplete block and row-column designs
• Symmetrical and asymmetrical factorial designs
• Systems of confounding
• Fractional factorial designs, including main effect plans
• Supersaturated designs
• Robust design or Taguchi experiments
• Lattice designs
• Crossover designs

In order to facilitate the application of text material to a broad range of fields, the authors take a general approach to their discussions. To aid in the construction and analysis of designs, many procedures are illustrated using Statistical Analysis System (SAS(r)) software.