Statistical Analysis of Designed Experiments: Theory and Applications

Preface xv

Abbreviations xxi

1 Introduction 1

1.1 Observational Studies and Experiments 1

1.2 Brief Historical Remarks 4

1.3 Basic Terminology and Concepts of Experimentation 5

1.4 Basic Principles of Experimentation 9

1.4.1 How to Minimize Biases and Variability? 9

1.4.2 Sequential Experimentation 14

1.5 Chapter Summary 15

Exercises 16

2 Review of Elementary Statistics 20

2.1 Experiments for a Single Treatment 20

2.1.1 Summary Statistics and Graphical Plots 21

2.1.2 Confidence Intervals and Hypothesis Tests 25

2.1.3 Power and Sample Size Calculation 27

2.2 Experiments for Comparing Two Treatments 28

2.2.1 Independent Samples Design 29

2.2.2 Matched Pairs Design 38

2.3 Linear Regression 41

2.3.1 Simple Linear Regression 42

2.3.2 Multiple Linear Regression 50

2.4 Chapter Summary 62

Exercises 62

3 Single Factor Experiments: Completely Randomized Designs 70

3.1 Summary Statistics and Graphical Displays 71

3.2 Model 73

3.3 Statistical Analysis 75

3.3.1 Estimation 75

3.3.2 Analysis of Variance 76

3.3.3 Confidence Intervals and Hypothesis Tests 78

3.4 Model Diagnostics 79

3.4.1 Checking Homoscedasticity 80

3.4.2 Checking Normality 81

3.4.3 Checking Independence 81

3.4.4 Checking Outliers 81

3.5 Data Transformations 85

3.6 Power of F-Test and Sample Size Determination 87

3.7 Quantitative Treatment Factors 90

3.8 One-Way Analysis of Covariance 96

3.8.1 Randomized Block Design versus Analysis of Covariance 96

3.8.2 Model 96

3.8.3 Statistical Analysis 98

3.9 Chapter Notes 106

3.9.1 Randomization Distribution of F-Statistic 106

3.9.2 F-Test for Heteroscedastic Treatment Variances 108

3.9.3 Derivations of Formulas for Orthogonal Polynomials 110

3.9.4 Derivation of LS Estimators for One-Way Analysis of Covariance 112

3.10 Chapter Summary 113

Exercises 114

4 Single-Factor Experiments: Multiple Comparison and Selection Procedures 126

4.1 Basic Concepts of Multiple Comparisons 127

4.1.1 Family 127

4.1.2 Familywise Error Rate 128

4.1.3 Bonferroni Method 129

4.1.4 Union–Intersection Method 130

4.1.5 Closure Method 131

4.2 Pairwise Comparisons 132

4.2.1 Least Significant Difference and Bonferroni Procedures 133

4.2.2 Tukey Procedure for Pairwise Comparisons 134

4.2.3 Step-Down Procedures for Pairwise Comparisons 136

4.3 Comparisons with a Control 139

4.3.1 Dunnett Procedure for Comparisons with a Control 139

4.3.2 Step-Down Procedures for Comparisons with a Control 142

4.4 General Contrasts 144

4.4.1 Tukey Procedure for Orthogonal Contrasts 145

4.4.2 Scheff´e Procedure for All Contrasts 146

4.5 Ranking and Selection Procedures 148

4.5.1 Indifference-Zone Formulation 148

4.5.2 Subset Selection Formulation 154

4.5.3 Multiple Comparisons with the Best 155

4.5.4 Connection between Multiple Comparisons with Best and Selection of Best Treatment 157

4.6 Chapter Summary 158

Exercises 159

5 Randomized Block Designs and Extensions 168

5.1 Randomized Block Designs 169

5.1.1 Model 169

5.1.2 Statistical Analysis 171

5.1.3 Randomized Block Designs with Replicates 177

5.2 Balanced Incomplete Block Designs 180

5.2.1 Statistical Analysis 182

5.2.2 Interblock Analysis 185

5.3 Youden Square Designs 188

5.3.1 Statistical Analysis 189

5.4 Latin Square Designs 192

5.4.1 Choosing a Latin Square 192

5.4.2 Model 195

5.4.3 Statistical Analysis 195

5.4.4 Crossover Designs 198

5.4.5 Graeco–Latin Square Designs 202

5.5 Chapter Notes 205

5.5.1 Restriction Error Model for Randomized Block Designs 205

5.5.2 Derivations of Formulas for BIB Design 206

5.6 Chapter Summary 211

Exercises 212

6 General Factorial Experiments 224

6.1 Factorial versus One-Factor-at-a-Time Experiments 225

6.2 Balanced Two-Way Layouts 227

6.2.1 Summary Statistics and Graphical Plots 227

6.2.2 Model 230

6.2.3 Statistical Analysis 231

6.2.4 Model Diagnostics 235

6.2.5 Tukey’s Test for Interaction for Singly Replicated Two-Way Layouts 236

6.3 Unbalanced Two-Way Layouts 240

6.3.1 Statistical Analysis 240

6.4 Chapter Notes 245

6.4.1 Derivation of LS Estimators of Parameters for Balanced Two-Way Layouts 245

6.4.2 Derivation of ANOVA Sums of Squares and F-Tests for Balanced Two-Way Layouts 246

6.4.3 Three- and Higher Way Layouts 248

6.5 Chapter Summary 250

Exercises 250

7 Two-Level Factorial Experiments 256

7.1 Estimation of Main Effects and Interactions 257

7.1.1 22 Designs 257

7.1.2 23 Designs 261

7.1.3 2p Designs 266

7.2 Statistical Analysis 267

7.2.1 Confidence Intervals and Hypothesis Tests 267

7.2.2 Analysis of Variance 268

7.2.3 Model Fitting and Diagnostics 270

7.3 Single-Replicate Case 272

7.3.1 Normal and Half-Normal Plots of Estimated Effects 272

7.3.2 Lenth Method 278

7.3.3 Augmenting a 2p Design with Observations at the Center Point 279

7.4 2p Factorial Designs in Incomplete Blocks: Confounding of Effects 282

7.4.1 Construction of Designs 282

7.4.2 Statistical Analysis 286

7.5 Chapter Notes 287

7.5.1 Yates Algorithm 287

7.5.2 Partial Confounding 288

7.6 Chapter Summary 289

Exercises 290

8 Two-Level Fractional Factorial Experiments 300

8.1 2p−q Fractional Factorial Designs 301

8.1.1 2p−1 Fractional Factorial Design 301

8.1.2 General 2p−q Fractional Factorial Designs 307

8.1.3 Statistical Analysis 312

8.1.4 Minimum Aberration Designs 316

8.2 Plackett–Burman Designs 317

8.3 Hadamard Designs 323

8.4 Supersaturated Designs 325

8.4.1 Construction of Supersaturated Designs 325

8.4.2 Statistical Analysis 327

8.5 Orthogonal Arrays 329

8.6 Sequential Assemblies of Fractional Factorials 333

8.6.1 Foldover of Resolution III Designs 334

8.6.2 Foldover of Resolution IV Designs 337

8.7 Chapter Summary 338

Exercises 339

9 Three-Level and Mixed-Level Factorial Experiments 351

9.1 Three-Level Full Factorial Designs 351

9.1.1 Linear–Quadratic System 353

9.1.2 Orthogonal Component System 361

9.2 Three-Level Fractional Factorial Designs 364

9.3 Mixed-Level Factorial Designs 372

9.3.1 2p4q Designs 373

9.3.2 2p3q Designs 378

9.4 Chapter Notes 386

9.4.1 Alternative Derivations of Estimators of Linear and Quadratic Effects 386

9.5 Chapter Summary 388

Exercises 389

10 Experiments for Response Optimization 395

10.1 Response Surface Methodology 396

10.1.1 Outline of Response Surface Methodology 396

10.1.2 First-Order Experimentation Phase 397

10.1.3 Second-Order Experimentation Phase 402

10.2 Mixture Experiments 412

10.2.1 Designs for Mixture Experiments 414

10.2.2 Analysis of Mixture Experiments 416

10.3 Taguchi Method of Quality Improvement 419

10.3.1 Philosophy Underlying Taguchi Method 422

10.3.2 Implementation of Taguchi Method 425

10.3.3 Critique of Taguchi Method 432

10.4 Chapter Summary 436

Exercises 437

11 Random and Mixed Crossed-Factors Experiments 448

11.1 One-Way Layouts 449

11.1.1 Random-Effects Model 449

11.1.2 Analysis of Variance 450

11.1.3 Estimation of Variance Components 452

11.2 Two-Way Layouts 455

11.2.1 Random-Effects Model 455

11.2.2 Mixed-Effects Model 459

11.3 Three-Way Layouts 464

11.3.1 Random- and Mixed-Effects Models 464

11.3.2 Analysis of Variance 465

11.3.3 Approximate F-Tests 468

11.4 Chapter Notes 472

11.4.1 Maximum Likelihood and Restricted Maximum Likelihood (REML) Estimation of Variance Components 472

11.4.2 Derivations of Results for One- and Two-Way Random-Effects Designs 475

11.4.3 Relationship between Unrestricted and Restricted Models 478

11.5 Chapter Summary 479

Exercises 480

12 Nested, Crossed–Nested, and Split-Plot Experiments 487

12.1 Two-Stage Nested Designs 488

12.1.1 Model 488

12.1.2 Analysis of Variance 489

12.2 Three-Stage Nested Designs 490

12.2.1 Model 491

12.2.2 Analysis of Variance 492

12.3 Crossed and Nested Designs 495

12.3.1 Model 495

12.3.2 Analysis of Variance 496

12.4 Split-Plot Designs 501

12.4.1 Model 504

12.4.2 Analysis of Variance 505

12.4.3 Extensions of Split-Plot Designs 508

12.5 Chapter Notes 515

12.5.1 Derivations of E(MS) Expressions for Two-Stage Nested Design of Section 12.1 with Both Factors
Random 515

12.5.2 Derivations of E(MS) Expressions for Design of Section 12.3 with Crossed and Nested Factors 517

12.5.3 Derivations of E(MS) Expressions for Split-Plot Design 520

12.6 Chapter Summary 523

Exercises 524

13 Repeated Measures Experiments 536

13.1 Univariate Approach 536

13.1.1 Model 537

13.1.2 Univariate Analysis of Variance for RM Designs 537

13.2 Multivariate Approach 548

13.2.1 One-Way Multivariate Analysis of Variance 548

13.2.2 Multivariate Analysis of Variance for RM Designs 549

13.3 Chapter Notes 555

13.3.1 Derivations of E(MS) Expressions for Repeated Measures Design Assuming Compound Symmetry 555

13.4 Chapter Summary 558

Exercises 559

14 Theory of Linear Models with Fixed Effects 566

14.1 Basic Linear Model and Least Squares Estimation 566

14.1.1 Geometric Interpretation of Least Squares Estimation 568

14.1.2 Least Squares Estimation in Singular Case 570

14.1.3 Least Squares Estimation in Orthogonal Case 572

14.2 Confidence Intervals and Hypothesis Tests 573

14.2.1 Sampling Distribution of_β 573

14.2.2 Sampling Distribution of s2 574

14.2.3 Inferences on Scalar Parameters 575

14.2.4 Inferences on Vector Parameters 575

14.2.5 Extra Sum of Squares Method 577

14.2.6 Analysis of Variance 579

14.3 Power of F-Test 583

14.4 Chapter Notes 586

14.4.1 Proof of Theorem 14.1 (Gauss–Markov Theorem) 586

14.4.2 Proof of Theorem 14.2 586

14.5 Chapter Summary 587

Exercises 588

Appendix A Vector-Valued Random Variables and Some Distribution Theory 595

A.1 Mean Vector and Covariance Matrix of Random Vector 596

A.2 Covariance Matrix of Linear Transformation of Random Vector 597

A.3 Multivariate Normal Distribution 598

A.4 Chi-Square, F-, and t-Distributions 599

A.5 Distributions of Quadratic Forms 601

A.6 Multivariate t-Distribution 605

A.7 Multivariate Normal Sampling Distribution Theory 606

Appendix B Case Studies 608

B.1 Case Study 1: Effects of Field Strength and Flip Angle on MRI Contrast 608

B.1.1 Introduction 608

B.1.2 Design 609

B.1.3 Data Analysis 610

B.1.4 Results 612

B.2 Case Study 2: Growing Stem Cells for Bone Implants 613

B.2.1 Introduction 613

B.2.2 Design 614

B.2.3 Data Analysis 614

B.2.4 Results 614

B.3 Case Study 3: Router Bit Experiment 619

B.3.1 Introduction 619

B.3.2 Design 619

B.3.3 Data Analysis 623

B.3.4 Results 624

Appendix C Statistical Tables 627

Answers to Selected Exercises 644

References 664

Index 675