Raymond H. Myers, PhD, is Professor Emeritus in the Department of Statistics at Virginia Polytechnic Institute and State University. He has more than 40 years of academic experience in the areas of experimental design and analysis, response surface analysis, and designs for nonlinear models. A Fellow of the American Statistical Association (ASA) and the American Society for Quality (ASQ), Dr. Myers has authored numerous journal articles and books, including Generalized Linear Models: with Applications in Engineering and the Sciences, Second Edition, also published by Wiley.

Douglas C. Montgomery, PhD, is Regents' Professor of Industrial Engineering and Arizona State University Foundation Professor of Engineering. Dr. Montgomery has more than 30 years of academic and consulting experience and his research interest includes the design and analysis of experiments. He is a Fellow of the ASA and the Institute of Industrial Engineers, and an Honorary Member of the ASQ. He has authored numerous journal articles and books, including Design and Analysis of Experiments, Eighth Edition; Generalized Linear Models: with Applications in Engineering and the Sciences, Second Edition; Introduction to Introduction to Linear Regression Analysis, Fifth Edition; and Introduction to Time Series Analysis and Forecasting, Second Edition, all published by Wiley.

Christine M. Anderson-Cook, PhD, is a Research Scientist and Project Leader in the Statistical Sciences Group at the Los Alamos National Laboratory, New Mexico. Dr. Anderson-Cook has over 20 years of academic and consulting experience, and has written numerous journal articles on the topics of design of experiments, response surface methodology and reliability. She is a Fellow of the ASA and the ASQ.

1 Introduction 1

1.1 Response Surface Methodology, 1

1.1.1 Approximating Response Functions, 2

1.1.2 The Sequential Nature of RSM, 7

1.1.3 Objectives and Typical Applications of RSM, 9

1.1.4 RSM and the Philosophy of Quality Improvement, 11

1.2 Product Design and Formulation (Mixture Problems), 11

1.3 Robust Design and Process Robustness Studies, 12

1.4 Useful References on RSM, 12

2 Building Empirical Models 13

2.1 Linear Regression Models, 13

2.2 Estimation of the Parameters in Linear Regression Models, 14

2.3 Properties of the Least Squares Estimators and Estimation of 𝜎2, 22

2.4 Hypothesis Testing in Multiple Regression, 24

2.4.1 Test for Significance of Regression, 24

2.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients, 27

2.5 Confidence Intervals in Multiple Regression, 31

2.5.1 Confidence Intervals on the Individual Regression Coefficients β, 32

2.5.2 A Joint Confidence Region on the Regression Coefficients β, 32

2.5.3 Confidence Interval on the Mean Response, 33

2.6 Prediction of New Response Observations, 35

2.7 Model Adequacy Checking, 36

2.7.1 Residual Analysis, 36

2.7.2 Scaling Residuals, 38

2.7.3 Influence Diagnostics, 42

2.7.4 Testing for Lack of Fit, 43

2.8 Fitting a Second-Order Model, 47

2.9 Qualitative Regressor Variables, 55

2.10 Transformation of the Response Variable, 61

Exercises, 66

3 Two-Level Factorial Designs 81

3.1 Introduction, 81

3.2 The 22 Design, 82

3.3 The 23 Design, 94

3.4 The General 2k Design, 103

3.5 A Single Replicate of the 2k Design, 108

3.6 2k Designs are Optimal Designs, 125

3.7 The Addition of Center Points to the 2k Design, 130

3.8 Blocking in the 2k Factorial Design, 135

3.8.1 Blocking in the Replicated Design, 135

3.8.2 Confounding in the 2k Design, 137

3.9 Split-Plot Designs, 141

Exercises, 146

4 Two-Level Fractional Factorial Designs 161

4.1 Introduction, 161

4.2 The One-Half Fraction of the 2k Design, 162

4.3 The One-Quarter Fraction of the 2k Design, 174

4.4 The General 2k−p Fractional Factorial Design, 184

4.5 Resolution III Designs, 188

4.6 Resolution IV and V Designs, 197

4.7 Alias Structures in Fractional Factorial and Other Designs, 198

4.8 Nonregular Fractional Factorial Designs, 200

4.8.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs, 203

4.8.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs, 209

4.8.3 Analysis of Nonregular Fractional Factorial Designs, 213

4.9 Fractional Factorial Split-Plot Designs, 216

4.10 Summary, 219

Exercises, 220

5 Process Improvement with Steepest Ascent 233

5.1 Determining the Path of Steepest Ascent, 234

5.1.1 Development of the Procedure, 234

5.1.2 Practical Application of the Method of Steepest Ascent, 237

5.2 Consideration of Interaction and Curvature, 241

5.2.1 What About a Second Phase?, 244

5.2.2 What Happens Following Steepest Ascent?, 244

5.3 Effect of Scale (Choosing Range of Factors), 245

5.4 Confidence Region for Direction of Steepest Ascent, 247

5.5 Steepest Ascent Subject to a Linear Constraint, 250

5.6 Steepest Ascent in a Split-Plot Experiment, 254

Exercises, 262

6 The Analysis of Second-Order Response Surfaces 273

6.1 Second-Order Response Surface, 273

6.2 Second-Order Approximating Function, 274

6.2.1 The Nature of the Second-Order Function and Second-Order Surface, 274

6.2.2 Illustration of Second-Order Response Surfaces, 276

6.3 A Formal Analytical Approach to the Second-Order Model, 277

6.3.1 Location of the Stationary Point, 278

6.3.2 Nature of the Stationary Point (Canonical Analysis), 278

6.3.3 Ridge Systems, 282

6.3.4 Role of Contour Plots, 286

6.4 Ridge Analysis of the Response Surface, 289

6.4.1 Benefits of Ridge Analysis, 290

6.4.2 Mathematical Development of Ridge Analysis, 291

6.5 Sampling Properties of Response Surface Results, 296

6.5.1 Standard Error of Predicted Response, 296

6.5.2 Confidence Region on the Location of the Stationary Point, 299

6.5.3 Use and Computation of the Confidence Region on the Location of the Stationary Point, 300

6.5.4 Confidence Intervals on Eigenvalues in Canonical Analysis, 304

6.6 Further Comments Concerning Response Surface Analysis, 307

Exercises, 307

7 Multiple Response Optimization 325

7.1 Balancing Multiple Objectives, 325

7.2 Strategies for Multiple Response Optimization, 338

7.2.1 Overlaying Contour Plots, 339

7.2.2 Constrained Optimization, 340

7.2.3 Desirability Functions, 341

7.2.4 Pareto Front Optimization, 343

7.2.5 Other Options for Optimization, 349

7.3 A Sequential Process for Optimization—DMRCS, 350

7.4 Incorporating Uncertainty of Response Predictions into Optimization, 352

Exercises, 357

8 Design of Experiments for Fitting Response Surfaces—I 369

8.1 Desirable Properties of Response Surface Designs, 369

8.2 Operability Region, Region of Interest, and Metrics for Desirable Properties, 371

8.2.1 Metrics for Desirable Properties, 372

8.2.2 Model Inadequacy and Model Bias, 373

8.3 Design of Experiments for First-Order Models and First-Order Models with Interactions, 375

8.3.1 The First-Order Orthogonal Design, 376

8.3.2 Orthogonal Designs for Models Containing Interaction, 378

8.3.3 Other First-Order Orthogonal Designs—The Simplex Design, 381

8.3.4 Definitive Screening Designs, 385

8.3.5 Another Variance Property—Prediction Variance, 389

8.4 Designs for Fitting Second-Order Models, 393

8.4.1 The Class of Central Composite Designs, 393

8.4.2 Design Moments and Property of Rotatability, 399

8.4.3 Rotatability and the CCD, 403

8.4.4 More on Prediction Variance—Scaled, Unscaled, and Estimated, 406

8.4.5 The Face-Centered Cube in Cuboidal Regions, 408

8.4.6 Choosing between Spherical and Cuboidal Regions, 411

8.4.7 The Box–Behnken Design, 413

8.4.8 Definitive Screening Designs for Fitting Second-Order Models, 417

8.4.9 Orthogonal Blocking in Second-Order Designs, 422

Exercises, 434

9 Experimental Designs for Fitting Response Surfaces—II 451

9.1 Designs that Require a Relatively Small Run Size, 452

9.1.1 The Hoke Designs, 452

9.1.2 Koshal Design, 454

9.1.3 Hybrid Designs, 455

9.1.4 The Small Composite Design, 458

9.1.5 Some Saturated or Near-Saturated Cuboidal Designs, 462

9.1.6 Equiradial Designs, 463

9.2 General Criteria for Constructing, Evaluating, and Comparing Designed Experiments, 465

9.2.1 Practical Design Optimality, 467

9.2.2 Use of Design Efficiencies for Comparison of Standard Second-Order Designs, 474

9.2.3 Graphical Procedure for Evaluating the Prediction Capability of an RSM Design, 477

9.3 Computer-Generated Designs in RSM, 488

9.3.1 Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality, 491

9.3.2 Algorithms for Computer-Generated Designs, 494

9.3.3 Comparison of D-, G-, and I-Optimal Designs, 497

9.3.4 Illustrations Involving Computer-Generated Design, 499

9.3.5 Computer-Generated Designs Involving Qualitative Variables, 508

9.4 Multiple Objective Computer-Generated Designs for RSM, 517

9.4.1 Pareto Front Optimization for Selecting a Design, 518

9.4.2 Pareto Aggregating Point Exchange Algorithm, 519

9.4.3 Using DMRCS for Design Optimization, 520

9.5 Some Final Comments Concerning Design Optimality and Computer-Generated Design, 525

Exercises, 527

10 Advanced Topics in Response Surface Methodology 543

10.1 Effects of Model Bias on the Fitted Model and Design, 543

10.2 A Design Criterion Involving Bias and Variance, 547

10.2.1 The Case of a First-Order Fitted Model and Cuboidal Region, 550

10.2.2 Minimum Bias Designs for a Spherical Region of Interest, 556

10.2.3 Simultaneous Consideration of Bias and Variance, 558

10.2.4 How Important Is Bias?, 558

10.3 Errors in Control of Design Levels, 560

10.4 Experiments with Computer Models, 563

10.4.1 Design for Computer Experiments, 567

10.4.2 Analysis for Computer Experiments, 570

10.4.3 Combining Information from Physical and Computer Experiments, 574

10.5 Minimum Bias Estimation of Response Surface Models, 575

10.6 Neural Networks, 579

10.7 Split-Plot Designs for Second-Order Models, 581

10.8 RSM for Non-Normal Responses—Generalized Linear Models, 591

10.8.1 Model Framework: The Link Function, 592

10.8.2 The Canonical Link Function, 593

10.8.3 Estimation of Model Coefficients, 593

10.8.4 Properties of Model Coefficients, 595

10.8.5 Model Deviance, 595

10.8.6 Overdispersion, 597

10.8.7 Examples, 598

10.8.8 Diagnostic Plots and Other Aspects of the GLM, 605

Exercises, 609

11 Robust Parameter Design and Process Robustness Studies 619

11.1 Introduction, 619

11.2 What is Parameter Design?, 619

11.2.1 Examples of Noise Variables, 620

11.2.2 An Example of Robust Product Design, 621

11.3 The Taguchi Approach, 622

11.3.1 Crossed Array Designs and Signal-to-Noise Ratios, 622

11.3.2 Analysis Methods, 625

11.3.3 Further Comments, 630

11.4 The Response Surface Approach, 631

11.4.1 The Role of the Control × Noise Interaction, 631

11.4.2 A Model Containing Both Control and Noise Variables, 635

11.4.3 Generalization of Mean and Variance Modeling, 638

11.4.4 Analysis Procedures Associated with the Two Response Surfaces, 642

11.4.5 Estimation of the Process Variance, 651

11.4.6 Direct Variance Modeling, 655

11.4.7 Use of Generalized Linear Models, 657

11.5 Experimental Designs For RPD and Process Robustness Studies, 661

11.5.1 Combined Array Designs, 661

11.5.2 Second-Order Designs, 663

11.5.3 Other Aspects of Design, 665

11.6 Dispersion Effects in Highly Fractionated Designs, 672

11.6.1 The Use of Residuals, 673

11.6.2 Further Diagnostic Information from Residuals, 674

11.6.3 Further Comments Concerning Variance Modeling, 680

Exercises, 684

12 Experiments with Mixtures 693

12.1 Introduction, 693

12.2 Simplex Designs and Canonical Mixture Polynomials, 696

12.2.1 Simplex Lattice Designs, 696

12.2.2 The Simplex-Centroid Design and Its Associated Polynomial, 704

12.2.3 Augmentation of Simplex Designs with Axial Runs, 707

12.3 Response Trace Plots, 716

12.4 Reparameterizing Canonical Mixture Models to Contain A Constant Term (𝛽0), 716

Exercises, 720

13 Other Mixture Design and Analysis Techniques 731

13.1 Constraints on the Component Proportions, 731

13.1.1 Lower-Bound Constraints on the Component Proportions, 732

13.1.2 Upper-Bound Constraints on the Component Proportions, 743

13.1.3 Active Upper- and Lower-Bound Constraints, 747

13.1.4 Multicomponent Constraints, 758

13.2 Mixture Experiments Using Ratios of Components, 759

13.3 Process Variables in Mixture Experiments, 763

13.3.1 Mixture-Process Model and Design Basics, 763

13.3.2 Split-Plot Designs for Mixture-Process Experiments, 767

13.3.3 Robust Parameter Designs for Mixture-Process Experiments, 778

13.4 Screening Mixture Components, 783

Exercises, 785

Appendix 1 Moment Matrix of a Rotatable Design 797

Appendix 2 Rotatability of a Second-Order Equiradial Design 803

References 807

Index 821

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