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 ¿ 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 ¿ 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 Kl 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
