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Preface iii

1 Introduction 1

1.1 Strategy of Experimentation 1

1.2 Some Typical Applications of Experimental Design 7

1.3 Basic Principles 11

1.4 Guidelines for Designing Experiments 13

1.5 A Brief History of Statistical Design 19

1.6 Summary: Using Statistical Techniques in Experimentation 20

2 Simple Comparative Experiments 22

2.1 Introduction 22

2.2 Basic Statistical Concepts 23

2.3 Sampling and Sampling Distributions 27

2.4 Inferences About the Differences in Means, Randomized Designs 32

2.4.1 Hypothesis Testing 32

2.4.2 Confidence Intervals 38

2.4.3 Choice of Sample Size 39

2.4.4 The Case Where 𝜎²₁ ≠ 𝜎²₂ 43

2.4.5 The Case Where 𝜎²₁ and 𝜎²₂ Are Known 45

2.4.6 Comparing a Single Mean to a Specified Value 46

2.4.7 Summary 47

2.5 Inferences About the Differences in Means, Paired Comparison Designs 47

2.5.1 The Paired Comparison Problem 47

2.5.2 Advantages of the Paired Comparison Design 50

2.6 Inferences About the Variances of Normal Distributions 52

3 Experiments with a Single Factor: The Analysis of Variance 55

3.1 An Example 55

3.2 The Analysis of Variance 58

3.3 Analysis of the Fixed Effects Model 59

3.3.1 Decomposition of the Total Sum of Squares 60

3.3.2 Statistical Analysis 62

3.3.3 Estimation of the Model Parameters 66

3.3.4 Unbalanced Data 68

3.4 Model Adequacy Checking 68

3.4.1 The Normality Assumption 69

3.4.2 Plot of Residuals in Time Sequence 71

3.4.3 Plot of Residuals Versus Fitted Values 71

3.4.4 Plots of Residuals Versus Other Variables 76

3.5 Practical Interpretation of Results 76

3.5.1 A Regression Model 77

3.5.2 Comparisons Among Treatment Means 78

3.5.3 Graphical Comparisons of Means 78

3.5.4 Contrasts 79

3.5.5 Orthogonal Contrasts 82

3.5.6 Scheffé’s Method for Comparing All Contrasts 83

3.5.7 Comparing Pairs of Treatment Means 85

3.5.8 Comparing Treatment Means with a Control 88

3.6 Sample Computer Output 89

3.7 Determining Sample Size 93

3.7.1 Operating Characteristic and Power Curves 93

3.7.2 Confidence Interval Estimation Method 94

3.8 Other Examples of Single-Factor Experiments 95

3.8.1 Chocolate and Cardiovascular Health 95

3.8.2 A Real Economy Application of a Designed Experiment 97

3.8.3 Discovering Dispersion Effects 99

3.9 The Random Effects Model 101

3.9.1 A Single Random Factor 101

3.9.2 Analysis of Variance for the Random Model 102

3.9.3 Estimating the Model Parameters 103

3.10 The Regression Approach to the Analysis of Variance 109

3.10.1 Least Squares Estimation of the Model Parameters 110

3.10.2 The General Regression Significance Test 111

3.11 Nonparametric Methods in the Analysis of Variance 113

3.11.1 The Kruskal–Wallis Test 113

3.11.2 General Comments on the Rank Transformation 114

4 Randomized Blocks, Latin Squares, and Related Designs 115

4.1 The Randomized Complete Block Design 115

4.1.1 Statistical Analysis of the RCBD 117

4.1.2 Model Adequacy Checking 125

4.1.3 Some Other Aspects of the Randomized Complete Block Design 125

4.1.4 Estimating Model Parameters and the General Regression Significance Test 130

4.2 The Latin Square Design 133

4.3 The Graeco-Latin Square Design 140

4.4 Balanced Incomplete Block Designs 142

4.4.1 Statistical Analysis of the BIBD 143

4.4.2 Least Squares Estimation of the Parameters 147

4.4.3 Recovery of Interblock Information in the BIBD 149

5 Introduction to Factorial Designs 152

5.1 Basic Definitions and Principles 152

5.2 The Advantage of Factorials 155

5.3 The Two-Factor Factorial Design 156

5.3.1 An Example 156

5.3.2 Statistical Analysis of the Fixed Effects Model 159

5.3.3 Model Adequacy Checking 164

5.3.4 Estimating the Model Parameters 167

5.3.5 Choice of Sample Size 169

5.3.6 The Assumption of No Interaction in a Two-Factor Model 170

5.3.7 One Observation per Cell 171

5.4 The General Factorial Design 174

5.5 Fitting Response Curves and Surfaces 179

5.6 Blocking in a Factorial Design 188

6 The 2^k Factorial Design 194

6.1 Introduction 194

6.2 The 2² Design 195

6.3 The 2³ Design 203

6.4 The General 2^k Design 215

6.5 A Single Replicate of the 2^k Design 218

6.6 Additional Examples of Unreplicated 2^k Designs 231

6.7 2^k Designs are Optimal Designs 243

6.8 The Addition of Center Points to the 2^k Design 248

6.9 Why We Work with Coded Design Variables 253

7 Blocking and Confounding in the 2^k Factorial Design 256

7.1 Introduction 256

7.2 Blocking a Replicated 2^k Factorial Design 256

7.3 Confounding in the 2^k Factorial Design 259

7.4 Confounding the 2^k Factorial Design in Two Blocks 259

7.5 Another Illustration of Why Blocking is Important 267

7.6 Confounding the 2^k Factorial Design in Four Blocks 268

7.7 Confounding the 2^k Factorial Design in 2^p Blocks 270

7.8 Partial Confounding 271

8 Two-Level Fractional Factorial Designs 274

8.1 Introduction 274

8.2 The One-Half Fraction of the 2^k Design 275

8.2.1 Definitions and Basic Principles 275

8.2.2 Design Resolution 278

8.2.3 Construction and Analysis of the One-Half Fraction 278

8.3 The One-Quarter Fraction of the 2^k Design 290

8.4 The General 2^k−pFractional Factorial Design 297

8.4.1 Choosing a Design 297

8.4.2 Analysis of 2^k−pFractional Factorials 300

8.4.3 Blocking Fractional Factorials 301

8.5 Alias Structures in Fractional Factorials and Other Designs 306

8.6 Resolution III Designs 308

8.6.1 Constructing Resolution III Designs 308

8.6.2 Fold Over of Resolution III Fractions to Separate Aliased Effects 310

8.6.3 Plackett–Burman Designs 313

8.7 Resolution IV and V Designs 322

8.7.1 Resolution IV Designs 322

8.7.2 Sequential Experimentation with Resolution IV Designs 323

8.7.3 Resolution V Designs 329

8.8 Supersaturated Designs 329

8.9 Summary 331

9 Additional Design and Analysis Topics for Factorial and Fractional Factorial Designs 332

9.1 The 3^k Factorial Design 333

9.1.1 Notation and Motivation for the 3^k Design 333

9.1.2 The 3² Design 334

9.1.3 The 3³ Design 335

9.1.4 The General 3^k Design 340

9.2 Confounding in the 3^k Factorial Design 340

9.2.1 The 3^k Factorial Design in Three Blocks 340

9.2.2 The 3^k Factorial Design in Nine Blocks 343

9.2.3 The 3^k Factorial Design in 3^p Blocks 344

9.3 Fractional Replication of the 3^k Factorial Design 345

9.3.1 The One-Third Fraction of the 3^k Factorial Design 345

9.3.2 Other 3^k−pFractional Factorial Designs 348

9.4 Factorials with Mixed Levels 349

9.4.1 Factors at Two and Three Levels 349

9.4.2 Factors at Two and Four Levels 351

9.5 Nonregular Fractional Factorial Designs 352

9.5.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs 354

9.5.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs 362

9.5.3 Analysis of Nonregular Fractional Factorial Designs 368

9.6 Constructing Factorial and Fractional Factorial Designs Using an Optimal Design Tool 369

9.6.1 Design Optimality Criterion 370

9.6.2 Examples of Optimal Designs 370

9.6.3 Extensions of the Optimal Design Approach 378

10 Fitting Regression Models 382

10.1 Introduction 382

10.2 Linear Regression Models 383

10.3 Estimation of the Parameters in Linear Regression Models 384

10.4 Hypothesis Testing in Multiple Regression 395

10.4.1 Test for Significance of Regression 395

10.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients 397

10.5 Confidence Intervals in Multiple Regression 399

10.5.1 Confidence Intervals on the Individual Regression Coefficients 400

10.5.2 Confidence Interval on the Mean Response 400

10.6 Prediction of New Response Observations 401

10.7 Regression Model Diagnostics 402

10.7.1 Scaled Residuals and PRESS 402

10.7.2 Influence Diagnostics 405

10.8 Testing for Lack of Fit 405

11 Response Surface Methods and Designs 408

11.1 Introduction to Response Surface Methodology 408

11.2 The Method of Steepest Ascent 411

11.3 Analysis of a Second-Order Response Surface 416

11.3.1 Location of the Stationary Point 416

11.3.2 Characterizing the Response Surface 418

11.3.3 Ridge Systems 424

11.3.4 Multiple Responses 425

11.4 Experimental Designs for Fitting Response Surfaces 430

11.4.1 Designs for Fitting the First-Order Model 430

11.4.2 Designs for Fitting the Second-Order Model 430

11.4.3 Blocking in Response Surface Designs 437

11.4.4 Optimal Designs for Response Surfaces 440

11.5 Experiments with Computer Models 454

11.6 Mixture Experiments 461

11.7 Evolutionary Operation 472

12 Robust Parameter Design and Process Robustness Studies 477

12.1 Introduction 477

12.2 Crossed Array Designs 479

12.3 Analysis of the Crossed Array Design 481

12.4 Combined Array Designs and the Response Model Approach 484

12.5 Choice of Designs 490

13 Experiments with Random Factors 493

13.1 Random Effects Models 493

13.2 The Two-Factor Factorial with Random Factors 494

13.3 The Two-Factor Mixed Model 500

13.4 Rules for Expected Mean Squares 505

13.5 Approximate F-Tests 508

13.6 Some Additional Topics on Estimation of Variance Components 512

13.6.1 Approximate Confidence Intervals on Variance Components 512

13.6.2 The Modified Large-Sample Method 516

14 Nested and Split-Plot Designs 518

14.1 The Two-Stage Nested Design 518

14.1.1 Statistical Analysis 519

14.1.2 Diagnostic Checking 524

14.1.3 Variance Components 526

14.1.4 Staggered Nested Designs 526

14.2 The General m-Stage Nested Design 528

14.3 Designs with Both Nested and Factorial Factors 530

14.4 The Split-Plot Design 534

14.5 Other Variations of the Split-Plot Design 540

14.5.1 Split-Plot Designs with More Than Two Factors 540

14.5.2 The Split-Split-Plot Design 545

14.5.3 The Strip-Split-Plot Design 549

15 Other Design and Analysis Topics (Available in e-text for students) W-1

Problems P-1

Appendix A-1

Table I. Cumulative Standard Normal Distribution A-2

Table II. Percentage Points of the t Distribution A-4

Table III. Percentage Points of the 𝜒 ² Distribution A-5

Table IV. Percentage Points of the F Distribution A-6

Table V. Percentage Points of the Studentized Range Statistic A-11

Table VI. Critical Values for Dunnett’s Test for Comparing Treatments with a Control A-13

Table VII. Coefficients of Orthogonal Polynomials A-15

Table VIII. Alias Relationships for 2^k−pFractional Factorial Designs with k ≤ 15 and n ≤ 64 A-16

OC Bibliography (Available in e-text for students) B-1

Index I-1

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