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1 INTRODUCTION 1

1．1 The Contribution of Statistics to Experimentation 1

1．2 Initial Steps in the Planning of Experiments 9

References 14

2 METHODS FOR INCREASING THE ACCURACY OF EXPERIMENTS 15

2．1 Introduction 15

2．2 Number of Replications 17

2．3 Other Methods for Increasing Accuracy 31

2．4 The Grouping of Experimental Units 41

References 43

3 NOTES ON THE STATISTICAL ANALYSIS OF THE RESULTS 45

3．1 Introduction 45

3．2 The General Method of Analysis 45

3．3 Accuracy in Computations 58

3．4 Subdivision of the Sum of Squares for Treatments 61

3．5 Calculation of Standard Errors for Comparisons among Treatment Means 70

3．6 Subdivision of the Sum of Squares for Error 77

3．7 Missing Date 80

3．8 The Analysis o f Covaiance 82

3．9 Effects of Errors in the Assumptions Underlying the Analysis of Variance 91

References 92

4 COMPLETELY RANDOMIZED,RANDOMIZED BLOCK,AND LATIN SQUARE DESIGNS 95

4．1 Completely Randomized Designs 95

4．2 Single Grouping:Randomized Blocks 95

4．3 Double Grouping:Latin Squares 117

4．4 Cross-over Designs 127

4．5 Triple Grouping:Graeco-latin Squares 132

4．6a Designs for Estimating Residual Effects When Treatments Are Applied in Sequence 133

References 142

Plans 145

5 FACTORIAL EXPERIMENTS 148

5．1 Description 148

5．2 Calculation of Main Effects and Interactions 153

5．3 Designs for Factorial Experiments 175

References 181

6 CONFOUNDING 183

6．1 The Principle of Confounding 183

6．2 The Use of Confounded Designs 212

6．3 Notes on the Plans and Statistical Analysis 219

References 232

Plans 234

6A FACTORIAL EXPERIMENTS IN FRACTIONAL REPLICATION 244

6A．1 Construction and Properties of Fractionally Replicated Designs 244

6A．2 The Use of Fractional Factorial Designs in Practice 259

6A．3 Designs with Factors at More Than Two Levels 270

References 275

Plans 276

7 FACTORIAL EXPERIMENTS WITH MAIN EFFECTS CONFOUNDED:SPLITPLOT DESIGNS 293

7．1 The Simple Split-plot Design 293

7．2 Repeated Subdivision 304

7．3 Some Variants of the Split-plot Design 305

References 315

8 FACTORIAL EXPERIMENTS CONFOUNDEN IN QUASI-LATIN SQUARES 317

8．1 Introduction 317

8．2 Randomization of Quasi-latin Squares 317

8．3 Notes on the Plans and Statistical Analysis 318

8．4 Other Quasi-latin Squares 322

8．5 Estimation of the Efficiency of Quasi-latin Squares 323

8．6 Treatments Applied to Complete Rows of a Latin Square 324

8．7 Treatments Applied to Complete Rows and Columns of a Latin Square 327

References 327

Plans 328

8A SOME METHODS FOR THE STUDY OF RESPONSE SURFACES 335

8A．1 First Order Designs 335

8A．2 Second Order Designs 342

8A．3 Methods for Determining the Optimum Combination of Factor Levels 354

8A．4 The Single-factor Method 356

8A．5 The Method of Steepest Ascent 357

8A．6 Summary Comments 365

References 369

Plans 370

9 INCOMPLETE BLOCK DESIGNS 376

9．1 Balanced Designs 376

9．2 Partially Balanced Designs 378

9．3 Basis of the Statistical Analysis 380

9．4 Comparison of Incomplete Block and Randomized Block Designs 385

9．5 Comparisons with Other Designs 387

9．6 Choice of Incomplete Block Design 388

References 394

10 LATTICE DESIGNS 396

10．1 Balanced Lattices 396

10．2 Partially Balanced Lattices 403

10．3 Rectangular Lattices 415

10．4 Cubic Lattices 422

References 426

Plans 428

11 BALANCED AND PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS 439

11．1 Balanced Incomplete Blocks 439

11．1a Balanced Incomplete Blocks in Taste and Preference Testing 440

11．2 Comparisons with Other Designs 441

11．3 Arrangement of Experimental Material 442

11．4 Randomization 442

11．5 Statistical Analysis 443

11．6a Partially Balanced Incomplete Block Designs 453

11．7a Chain Block Designs 463

References 468

Plans 469

12 LATTICE SQUARES 483

12．1 Description 483

12．2 Statistical Analysis 485

References 497

Plans 497

13 INCOMPLETE LATIN SQUARES 507

13．1 Description 507

13．2 Statistical Analysis 508

13．3 Other Designs for Small Numbers of Treatments 513

13．4a Partially Balanced Designs 518

References 519

Plans 520

14 ANALYSIS OF THE RESULTS OF A SERIES OF EXPERIMENTS 545

14．1 Initial Steps in the Analysis 545

14．2 Criticisms of the Preliminary Analysis 550

14．3 Experiments of Unequal Size 555

14．4 A Test of the Treatments × Places Interactions 561

14．5 Repetitions in Both Space and Time 565

References 567

15 RANDOM PERMUTATIONS OF 9 AND 16 NUMBERS 569

15．1 Use of the Random Permutations 569

15．2 Construction of the Random Permutations 569

15．3 Randomization of More than 16 Numbers 571

15．4 Tests of Randomness 571

References 576

15．5 Tables of Random Permutations 577

Permutations of 9 577

Permutations of 16 583

SELECTED BIBLIOGRAPHY 597

LIST OF AUTHOR REFERENCES 599

INDEX 603

TABLES OF t AND F 613