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Design and Analysis of Experiments, 9th Edition

Douglas C. Montgomery

ISBN: 978-1-119-11347-8 May 2017 640 Pages

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Design and Analysis of Experiments, 9th Edition
 
continues to help senior and graduate students in engineering, business, and statistics-as well as working practitioners-to design and analyze experiments for improving the quality, efficiency and performance of working systems.

This bestselling text maintains its comprehensive coverage by including: new examples, exercises, and problems (including in the areas of biochemistry and biotechnology); new topics and problems in the area of response surface; new topics in nested and split-plot design; and the residual maximum likelihood method is now emphasized throughout the book.

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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

1.7 Problems 21

2 Simple Comparative Experiments 23

2.1 Introduction 24

2.2 Basic Statistical Concepts 25

2.3 Sampling and Sampling Distributions 28

2.4 Inferences About the Differences in Means, Randomized Designs 33

2.4.1 Hypothesis Testing 33

2.4.2 Confidence Intervals 39

2.4.3 Choice of Sample Size 41

2.4.4 The Case Where 𝜎21 ≠ 𝜎22 44

2.4.5 The Case Where 𝜎21 and 𝜎22 Are Known 47

2.4.6 Comparing a Single Mean to a Specified Value 47

2.4.7 Summary 48

2.5 Inferences About the Differences in Means, Paired Comparison Designs 50

2.5.1 The Paired Comparison Problem 50

2.5.2 Advantages of the Paired Comparison Design 52

2.6 Inferences About the Variances of Normal Distributions 53

2.7 Problems 55

3 Experiments with a Single Factor: The Analysis of Variance 64

3.1 An Example 65

3.2 The Analysis of Variance 67

3.3 Analysis of the Fixed Effects Model 69

3.3.1 Decomposition of the Total Sum of Squares 69

3.3.2 Statistical Analysis 72

3.3.3 Estimation of the Model Parameters 76

3.3.4 Unbalanced Data 78

3.4 Model Adequacy Checking 78

3.4.1 The Normality Assumption 79

3.4.2 Plot of Residuals in Time Sequence 81

3.4.3 Plot of Residuals Versus Fitted Values 81

3.4.4 Plots of Residuals Versus Other Variables 86

3.5 Practical Interpretation of Results 86

3.5.1 A Regression Model 87

3.5.2 Comparisons Among Treatment Means 88

3.5.3 Graphical Comparisons of Means 88

3.5.4 Contrasts 89

3.5.5 Orthogonal Contrasts 92

3.5.6 Scheffé’s Method for Comparing All Contrasts 93

3.5.7 Comparing Pairs of Treatment Means 95

3.5.8 Comparing Treatment Means with a Control 98

3.6 Sample Computer Output 99

3.7 Determining Sample Size 103

3.7.1 Operating Characteristic and Power Curves 103

3.7.2 Confidence Interval Estimation Method 104

3.8 Other Examples of Single-Factor Experiments 105

3.8.1 Chocolate and Cardiovascular Health 105

3.8.2 A Real Economy Application of a Designed Experiment 107

3.8.3 Discovering Dispersion Effects 109

3.9 The Random Effects Model 111

3.9.1 A Single Random Factor 111

3.9.2 Analysis of Variance for the Random Model 112

3.9.3 Estimating the Model Parameters 113

3.10 The Regression Approach to the Analysis of Variance 119

3.10.1 Least Squares Estimation of the Model Parameters 120

3.10.2 The General Regression Significance Test 121

3.11 Nonparametric Methods in the Analysis of Variance 123

3.11.1 The Kruskal–Wallis Test 123

3.11.2 General Comments on the Rank Transformation 124

3.12 Problems 125

4 Randomized Blocks, Latin Squares, and Related Designs 135

4.1 The Randomized Complete Block Design 135

4.1.1 Statistical Analysis of the RCBD 137

4.1.2 Model Adequacy Checking 145

4.1.3 Some Other Aspects of the Randomized Complete Block Design 145

4.1.4 Estimating Model Parameters and the General Regression Significance Test 150

4.2 The Latin Square Design 153

4.3 The Graeco-Latin Square Design 160

4.4 Balanced Incomplete Block Designs 162

4.4.1 Statistical Analysis of the BIBD 163

4.4.2 Least Squares Estimation of the Parameters 167

4.4.3 Recovery of Interblock Information in the BIBD 169

4.5 Problems 171

5 Introduction to Factorial Designs 179

5.1 Basic Definitions and Principles 179

5.2 The Advantage of Factorials 182

5.3 The Two-Factor Factorial Design 183

5.3.1 An Example 183

5.3.2 Statistical Analysis of the Fixed Effects Model 186

5.3.3 Model Adequacy Checking 191

5.3.4 Estimating the Model Parameters 194

5.3.5 Choice of Sample Size 196

5.3.6 The Assumption of No Interaction in a Two-Factor Model 197

5.3.7 One Observation per Cell 198

5.4 The General Factorial Design 201

5.5 Fitting Response Curves and Surfaces 206

5.6 Blocking in a Factorial Design 215

5.7 Problems 220

6 The 2k Factorial Design 230

6.1 Introduction 230

6.2 The 22 Design 231

6.3 The 23 Design 240

6.4 The General 2k Design 252

6.5 A Single Replicate of the 2k Design 254

6.6 Additional Examples of Unreplicated 2k Designs 268

6.7 2k Designs are Optimal Designs 280

6.8 The Addition of Center Points to the 2k Design 285

6.9 Why We Work with Coded Design Variables 290

6.10 Problems 292

7 Blocking and Confounding in the 2k Factorial Design 308

7.1 Introduction 308

7.2 Blocking a Replicated 2k Factorial Design 309

7.3 Confounding in the 2k Factorial Design 311

7.4 Confounding the 2k Factorial Design in Two Blocks 311

7.5 Another Illustration of Why Blocking Is Important 319

7.6 Confounding the 2k Factorial Design in Four Blocks 320

7.7 Confounding the 2k Factorial Design in 2p Blocks 322

7.8 Partial Confounding 323

7.9 Problems 325

8 Two-Level Fractional Factorial Designs 328

8.1 Introduction 329

8.2 The One-Half Fraction of the 2k Design 329

8.2.1 Definitions and Basic Principles 329

8.2.2 Design Resolution 332

8.2.3 Construction and Analysis of the One-Half Fraction 332

8.3 The One-Quarter Fraction of the 2k Design 344

8.4 The General 2k−p Fractional Factorial Design 351

8.4.1 Choosing a Design 351

8.4.2 Analysis of 2k−p Fractional Factorials 354

8.4.3 Blocking Fractional Factorials 355

8.5 Alias Structures in Fractional Factorials and Other Designs 360

8.6 Resolution III Designs 362

8.6.1 Constructing Resolution III Designs 362

8.6.2 Fold Over of Resolution III Fractions to Separate Aliased Effects 364

8.6.3 Plackett–Burman Designs 367

8.7 Resolution IV and V Designs 376

8.7.1 Resolution IV Designs 376

8.7.2 Sequential Experimentation with Resolution IV Designs 377

8.7.3 Resolution V Designs 383

8.8 Supersaturated Designs 384

8.9 Summary 385

8.10 Problems 386

9 Additional Design and Analysis Topics for Factorial and Fractional Factorial Designs 405

9.1 The 3k Factorial Design 406

9.1.1 Notation and Motivation for the 3k Design 406

9.1.2 The 32 Design 407

9.1.3 The 33 Design 408

9.1.4 The General 3k Design 413

9.2 Confounding in the 3k Factorial Design 413

9.2.1 The 3k Factorial Design in Three Blocks 413

9.2.2 The 3k Factorial Design in Nine Blocks 416

9.2.3 The 3k Factorial Design in 3p Blocks 417

9.3 Fractional Replication of the 3k Factorial Design 418

9.3.1 The One-Third Fraction of the 3k Factorial Design 418

9.3.2 Other 3k−p Fractional Factorial Designs 421

9.4 Factorials with Mixed Levels 422

9.4.1 Factors at Two and Three Levels 422

9.4.2 Factors at Two and Four Levels 424

9.5 Nonregular Fractional Factorial Designs 425

9.5.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs 427

9.5.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs 436

9.5.3 Analysis of Nonregular Fractional Factorial Designs 441

9.6 Constructing Factorial and Fractional Factorial Designs Using an Optimal Design Tool 442

9.6.1 Design Optimality Criterion 443

9.6.2 Examples of Optimal Designs 443

9.6.3 Extensions of the Optimal Design Approach 453

9.7 Problems 454

10 Fitting Regression Models (online at www.wiley.com/college/montgomery) 460

10.1 Introduction 461

10.2 Linear Regression Models 461

10.3 Estimation of the Parameters in Linear Regression Models 462

10.4 Hypothesis Testing in Multiple Regression 473

10.4.1 Test for Significance of Regression 473

10.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients 475

10.5 Confidence Intervals in Multiple Regression 478

10.5.1 Confidence Intervals on the Individual Regression Coefficients 478

10.5.2 Confidence Interval on the Mean Response 478

10.6 Prediction of New Response Observations 479

10.7 Regression Model Diagnostics 480

10.7.1 Scaled Residuals and PRESS 480

10.7.2 Influence Diagnostics 483

10.8 Testing for Lack of Fit 483

10.9 Problems 485

11 Response Surface Methods and Designs 489

11.1 Introduction to Response Surface Methodology 490

11.2 The Method of Steepest Ascent 492

11.3 Analysis of a Second-Order Response Surface 497

11.3.1 Location of the Stationary Point 497

11.3.2 Characterizing the Response Surface 499

11.3.3 Ridge Systems 505

11.3.4 Multiple Responses 506

11.4 Experimental Designs for Fitting Response Surfaces 511

11.4.1 Designs for Fitting the First-Order Model 511

11.4.2 Designs for Fitting the Second-Order Model 511

11.4.3 Blocking in Response Surface Designs 518

11.4.4 Optimal Designs for Response Surfaces 521

11.5 Experiments with Computer Models 535

11.6 Mixture Experiments 542

11.7 Evolutionary Operation 553

11.8 Problems 558

12 Robust Parameter Design and Process Robustness Studies (online at www.wiley.com/college/montgomery) 569

12.1 Introduction 569

12.2 Crossed Array Designs 571

12.3 Analysis of the Crossed Array Design 573

12.4 Combined Array Designs and the Response Model Approach 576

12.5 Choice of Designs 582

12.6 Problems 585

13 Experiments with Random Factors 589

13.1 Random Effects Models 589

13.2 The Two-Factor Factorial with Random Factors 590

13.3 The Two-Factor Mixed Model 597

13.4 Rules for Expected Mean Squares 602

13.5 Approximate F-Tests 605

13.6 Some Additional Topics on Estimation of Variance Components 609

13.6.1 Approximate Confidence Intervals on Variance Components 609

13.6.2 The Modified Large-Sample Method 613

13.7 Problems 615

14 Nested and Split-Plot Designs 618

14.1 The Two-Stage Nested Design 619

14.1.1 Statistical Analysis 619

14.1.2 Diagnostic Checking 624

14.1.3 Variance Components 626

14.1.4 Staggered Nested Designs 626

14.2 The General m-Stage Nested Design 628

14.3 Designs with Both Nested and Factorial Factors 630

14.4 The Split-Plot Design 634

14.5 Other Variations of the Split-Plot Design 640

14.5.1 Split-Plot Designs with More Than Two Factors 640

14.5.2 The Split-Split-Plot Design 645

14.5.3 The Strip-Split-Plot Design 649

14.6 Problems 650

15 Other Design and Analysis Topics (online at www.wiley.com/college/montgomery) 656

15.1 Nonnormal Responses and Transformations 657

15.1.1 Selecting a Transformation: The Box–Cox Method 657

15.1.2 The Generalized Linear Model 659

15.2 Unbalanced Data in a Factorial Design 666

15.2.1 Proportional Data: An Easy Case 667

15.2.2 Approximate Methods 668

15.2.3 The Exact Method 670

15.3 The Analysis of Covariance 670

15.3.1 Description of the Procedure 671

15.3.2 Computer Solution 679

15.3.3 Development by the General Regression Significance Test 680

15.3.4 Factorial Experiments with Covariates 682

15.4 Repeated Measures 692

15.5 Problems 694

Appendix (online at www.wiley.com/college/montgomery) 697

Table I. Cumulative Standard Normal Distribution 698

Table II. Percentage Points of the t Distribution 700

Table III. Percentage Points of the 𝜒 2 Distribution 701

Table IV. Percentage Points of the F Distribution 702

Table V. Percentage Points of the Studentized Range Statistic 707

Table VI. Critical Values for Dunnett’s Test for Comparing Treatments with a Control 709

Table VII. Coefficients of Orthogonal Polynomials 711

Table VIII. Alias Relationships for 2k−p Fractional Factorial Designs with k ≤ 15 and n ≤ 64 712

Bibliography (online at www.wiley.com/college/montgomery) 724

Index 731

  • 83 new homework problems (including in the areas of biochemistry and biotechnology).
  • Additional examples of single-factor experiments, such as a study involving chocolate consumption and cardiovascular health (Chapter 3)
  • New section on the random effects model (Chapter 3)
  • New material on nonregular fractions as alternatives to traditional abberation fractions in 16 runs and analysis methods for those designs discussed and illustrated.
  • New material on constructing factorial and fractional factorial designs using an optimal design tool (Chapter 9).
  • New topics and problems in the area of response surface, including designs that combine screening and optimization and use optimal designs (Chapter 11).
  • New topics in nested and split-plot design
  • Includes software examples taken from the four most dominant programs in the field: Design-Expert, Minitab, JMP, and SAS.
  • Focuses on the connection between the experiment and the model that the experimenter can develop from the results of the experiement.
  • Stresses the importance of experimental design as a tool for engineers and scientists to use for product design and development as well as process development and improvement. The use of experiemental design in developing products that are robust to environmental factors and other sources of variability is illustrated. The use of experimental design early in the product cycle can subsatantially reduce development lead time and cost, leading to processes and products that perform better in the field and have higher reliability than those developed using other approaches.