Experiments Planning, Analysis, and Parameter Design Optimization

C. F. Jeff Wu Michael Hamada

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Weinheim • Brisbane • Singapore • Toronto

Contents

Preface

Suggestions of Topics for Instructors

List of Experiments and Data Sets

Basic Principles and Experiments with a Single Factor

1.1 Introduction and Historical Perspective, 1 1.2 A Systematic Approach to the Planning and

Implementation of Experiments, 4 1.3 Fundamental Principles: Replication, Randomization, and

Blocking, 8 1.4 The General Linear Model, 11 1.5 Variable Selection in Regression Analysis, 17 1.6 One-Way Layout, 19 1.7 Multiple Comparisons, 26 1.8 Quantitative Factors and Orthogonal Polynomials, 30 1.9 Residual Analysis: Assessment of Model

Assumptions, 35 1.10 Practical Summary, 40 Exercises, 41 References, 47

Experiments With More Than One Factor

2.1 Paired Comparison Design, 48 2.2 Randomized Block Design, 51 2.3 Two-Way Layout, 55

2.3.1 Two Qualitative Factors, 59 2.3.2 One Qualitative Factor and One Quantitative

Factor, 61

viii CONTENTS

2.3.3 Two Quantitative Factors, 62 2.4 Multi-Way Layout, 63 2.5 Transformation of the Response, 66 2.6 Latin Square Design: Two Blocking Variables, 70 2.7 Graeco-Latin Square Design, 74 2.8 Balanced Incomplete Block Design, 76 2.9 Analysis of Covariance: Incorporating Auxiliary

Information, 80 2.10 Practical Summary, 83 Exercises, 84 Appendix 2A: Table of Latin Squares, Graeco-Latin Squares,

and Hyper-Graeco-Latin Squares, 93 References, 94

3 Füll Factorial Experiments at Two Levels 96

3.1 An Epitaxial Layer Growth Experiment, 96 3.2 Nominal-the-Best Problem and Quadratic Loss

Function, 98 3.3 Füll Factorial Designs at Two Levels: A General

Discussion, 99 3.4 Factorial Effects and Plots, 103

3.4.1 Main Effects, 104 3.4.2 Interaction Effects, 106

3.5 Fundamental Principles for Factorial Effects: Hierarchical Ordering, Effect Sparsity, and Effect Heredity, 111

3.6 Using Regression and the Model Matrix to Compute Factorial Effects, 113

3.7 Comparisons with the "One-Factor-at-a-Time" Approach, 114

3.8 Log Transformation of the Sample Variances, 117 3.9 Effect of Simultaneous Testing, 118 3.10 Normal and Half-Normal Plots, 120 3.11 Analysis of Location and Dispersion: Revisiting the

Epitaxial Layer Growth Experiment, 123 3.12 Blocking and Optimal Arrangement of 2* Factorial

Designs in 2q blocks, 126 3.13 A Formal Test of Effect Significance for Unreplicated

Experiments: Without s2, 131

CONTENTS ix

3.14 Testing Variance Homogeneity, 135 3.15 A Formal Test of Effect Significance for yt: With s2, 136 3.16 A Formal Test of Effect Significance for Ins?, 139 3.17 A Summary of Analysis Strategies, 140 3.18 Practical Summary, 141 Exercises, 143 Appendix 3A: Table of 2* Factorial Designs in 2q Blocks, 150 References, 151

Fractional Factorial Experiments at Two Levels 153

4.1 A Leaf Spring Experiment, 153 4.2 Fractional Factorial Designs: Effect Aliasing and the

Criteria of Resolution and Minimum Aberration, 154 4.3 Analysis, 161 4.4 Techniques for Resolving the Ambiguities in Aliased

Effects, 167 4.4.1 Method of Adding Orthogonal Runs, 167 4.4.2 Optimal Design Approach for Follow-up

Experiments, 169 4.4.3 Fold-Over Technique for Follow-up Experiments,

172 4.5 Selection of 2k~p Designs Using Minimum Aberration

and Related Criteria, 175 4.6 Blocking in Fractional Factorial Designs, 179 4.7 Practical Summary, 181 Exercises, 183 Appendix 4A: Tables of 2k~p Fractional Factorial Designs,

193 Appendix 4B: Tables of 2k~p Fractional Factorial Designs in 2q

Blocks, 199 References, 203

Füll Factorial and Fractional Factorial Experiments at Three Levels 205

5.1 A Seat-Belt Experiment, 205 5.2 Larger-the-Better and Smaller-the-Better Problems, 207 5.3 3* Füll Factorial Designs, 208 5.4 3k~p Fractional Factorial Designs, 214 5.5 Simple Analysis Methods: Plots and Analysis of

Variance, 218

X CONTENTS

5.6 An Alternative Analysis Method, 226 5.7 Analysis Strategies for Multiple Responses I: Out-of-Spec

Probabilities, 232 5.8 Biocking in 3* and 3k~p Designs, 240 5.9 Practical Summary, 242 Exercises, 244 Appendix 5A: Tables of 3k~p Fractional Factorial Designs,

250 Appendix 5B: Tables of 3k~p Fractional Factorial Designs in 3q

Blocks, 251 References, 255

Other Design and Analysis Techniques for Experiments at More Than Two Levels 256

6.1 A Router Bit Experiment Based on a Mixed Two-Level and Four-Level Design, 256

6.2 Method of Replacement and Construction of 2m4" Designs, 258

6.3 Minimum Aberration 2m4" Designs with n = 1,2, 262 6.4 An Analysis Strategy For 2m4" Experiments, 265 6.5 Analysis of the Router Bit Experiment, 267 6.6 A Paint Experiment Based on a Mixed Two-Level and

Three-Level Design, 271 6.7 Design and Analysis of 36-Run Experiments at Two

and Three Levels, 273 6.8 rk~p Fractional Factorial Designs for Any Prime

Number r, 278 6.8.1 25-Run Fractional Factorial Designs at Five

Levels, 279 , 6.8.2 49-Run Fractional Factorial Designs at Seven

Levels, 282 6.8.3 General Construction, 282

6.9 Related Factors: Method of Sliding Levels and Nested Effects Analysis, 283 6.9.1 Analysis of Light Bulb Experiment, 287

6.10 Practical Summary, 290 Exercises, 291 Appendix 6A: Tables of 2m41 Minimum Aberration Designs,

298 Appendix 6B: Tables of 2m42 Minimum Aberration Designs,

300

CONTENTS xi

Appendix 6C: OA(25,56), 302 Appendix 6D: OL4(49,78), 302 References, 304

7 Nonregular Designs: Construction and Properties 305

7.1 Two Experiments: Weld-Repaired Castings and Blood Glucose Testing, 305

7.2 Some Advantages of Nonregular Designs over the 2k~p

and 3k~p Series of Designs, 307 7.3 A Lemma on Orthogonal Arrays, 308 7.4 Plackett-Burman Designs and Hall's Designs, 309 7.5 A Collection of Useful Mixed-Level Orthogonal

Arrays, 313 7.6 Construction of Mixed-Level Orthogonal Arrays Based

on Difference Matrices, 315 7.7 Construction of Mixed-Level Orthogonal Arrays through

the Method of Replacement, 318 7.8 Orthogonal Main-Effect Plans through Collapsing

Factors, 320 7.9 Practical Summary, 324 Exercises, 326 Appendix 7A: Plackett-Burman Designs OA(N,2N~1) with

12 < N < 48 and N = Ak But Not a Power of2, 330

Appendix 7B: Hall's 16-Run Orthogonal Arrays of Types II toV, 333

Appendix 7C: Some Useful Mixed-Level Orthogonal Arrays, 335

Appendix 7D: Some Useful Difference Matrices, 345 Appendix 7E: Some Useful Orthogonal Main-Effect

Plans, 347 References, 348

8 Experiments with Complex Aliasing 350

8.1 Partial Aliasing of Effects and the Alias Matrix, 350 8.2 Traditional Analysis Strategy: Screening Design and Main

Effect Analysis, 353 8.3 Simplification of Complex Aliasing via Effect Sparsity,

354 8.4 An Analysis Strategy for Designs with Complex Aliasing,

356

xii CONTENTS

8.4.1 Some Limitations, 361 8.5 A Bayesian Variable Selection Strategy for Designs with

Complex Aliasing, 363 8.5.1 Bayesian Model Priors, 364 8.5.2 Gibbs Sampling, 366 8.5.3 Choice of Prior Tuning Constants, 367 8.5.4 Blood Glucose Experiment Revisited, 369 8.5.5 Other Applications, 371

8.6 Supersaturated Designs: Design Construction and Analysis, 371

8.8 Practical Summary, 375 Exercises, 376 Appendix 8A: Further Details for the Füll Conditional

Distributions, 382 References, 385

9 Response Surface Methodology 387

9.1 A Ranitidine Separation Experiment, 387 9.2 Sequential Nature of Response Surface Methodology,

389 9.3 From First-Order Experiments to Second-Order

Experiments: Steepest Ascent Search and Rectangular Grid Search, 392 9.3.1 Curvature Check, 392 9.3.2 Steepest Ascent Search, 394 9.3.3 Rectangular Grid Search, 399

9.4 Analysis of Second-Order Response Surfaces, 401 9.4.1 Ridge Systems, 404

9.5 Analysis of the Ranitidine Experiment, 405 9.6 Analysis Strategies for Multiple Responses II: Contour

Plots and the Use of Desirability Functions, 409 9.7 Central Composite Designs, 412 9.8 Box-Behnken Designs and Uniform Shell Designs, 417 9.9 Practical Summary, 420 Exercises, 422 Appendix 9A: Table of Central Composite Designs, 431 Appendix 9B: Table of Box-Behnken Designs, 433 Appendix 9C: Table of Uniform Shell Designs, 434 References, 435

CONTENTS xiii

10 Introduction to Robust Parameter Design 436

10.1 A Robust Parameter Design Perspective of the Layer Growth and Leaf Spring Experiments, 436 10.1.1 Layer Growth Experiment Revisited, 436 10.1.2 Leaf Spring Experiment Revisited, 438

10.2 Strategies for Reducing Variation, 439 10.3 Noise (Hard-to-Control) Factors, 440 10.4 Variation Reduction through Robust Parameter

Design, 442 10.5 Experimentation and Modeling Strategies I: Cross

Array, 444 10.5.1 Location and Dispersion Modeling, 445 10.5.2 Response Modeling, 451

10.6 Experimentation and Modeling Strategies II: Single Array and Response Modeling, 456

10.7 Cross Arrays: Estimation Capacity and Optimal Selection, 459

10.8 Choosing between Cross Arrays and Single Arrays, 461 10.9 Optimal Selection of Single Arrays, 466 10.10 Signal-to-Noise Ratio and Its Limitations for Parameter

Design Optimization, 468 10.10.1 SN Ratio Analysis of Layer Growth Experiment,

470 10.11 Further Topics, 472 10.12 Practical Summary, 473 Exercises, 474 Appendix 10A: Tables of Single Arrays Based on 2k~p

Designs, 484 References, 492

11 Robust Parameter Design for Signal-Response Systems 495

11.1 An Injection Molding Experiment, 495 11.2 Signal-Response Systems and Their Classification, 499

11.2.1 Calibration of Measurement Systems, 501 11.3 Performance Measures for Parameter Design

Optimization, 503 11.4 Modeling and Analysis Strategies, 507 11.5 Analysis of the Injection Molding Experiment, 510

11.5.1 PMM Analysis, 512 11.5.2 RFM Analysis, 513

xiv CONTENTS

11.6 Choice of Experimental Plans, 516 11.7 Practical Summary, 519 Exercises, 520 References, 528

12 Experiments for Improving Reliability 529

12.1 Experiments with Failure Time Data, 529 12.1.1 Light Experiment, 530 12.1.2 Thermostat Experiment, 531 12.1.3 Drill Bit Experiment, 532

12.2 Regression Model for Failure Time Data, 533 12.3 A Likelihood Approach for Handling Failure Time Data

with Censoring, 535 12.3.1 Estimability Problem with MLEs, 538

12.4 Design-Dependent Model Selection Strategies, 539 12.5 A Bayesian Approach to Estimation and Model

Selection for Failure Time Data, 540 12.6 Analysis of Reliability Experiments with Failure Time

Data, 543 12.6.1 Analysis of Light Experiment, 543 12.6.2 Analysis of Thermostat Experiment, 545 12.6.3 Analysis of Drill Bit Experiment, 546

12.7 Reliability Experiments with Degradation Data, 548 12.8 A Simple Analysis for Degradation Data, 552 12.9 Practical Summary, 555 Exercises, 556 References, 561

13 Experiments With Nonnormal Data 563

13.1 A Wave Soldering Experiment with Count Data, 563 13.2 Generalized Linear Models, 564

13.2.1 The Distribution of the Response, 565 13.2.2 The Form of the Systematic Effects, 567 13.2.3 GLM versus Transforming the Response, 568

13.3 Analysis of Generalized Linear Models, 569 13.4 Analysis of the Wave Soldering Experiment, 572 13.5 Other Uses and Extensions of Generalized Linear

Models, 574 13.6 A Foam Molding Experiment with Ordinal Data, 575

CONTENTS XV

13.7 Model and Analysis for Ordinal Data, 576 13.7.1 The Gibbs Sampler for Ordinal Data, 577

13.8 Analysis of Foam Molding Experiment, 580 13.9 Scoring: A Simple Method for Analyzing Ordinal

Data, 583 13.10 Practical Summary, 584 Exercises, 585 References, 594

A Upper Tail Probabilities of the Standard Normal Distribution 596

B Upper Percentiles of the t Distribution 598

C Upper Percentiles of the x2 Distribution 600

D Upper Percentiles of the F Distribution 602

E Upper Percentiles of the Studentized Range Distribution 606

F Upper Percentiles of the Studentized Maximum Modulus

Distribution 611

G Coefficients of Orthogonal Contrast Vectors 618

H Critical Values for Lenth's Method 619

Author Index 621

Subject Index 624