Modern Experimental Design THOMAS P. RYAN Acworth, GA Modern Experimental Design Modern Experimental Design THOMAS P. RYAN Acworth, GA Copyright C 2007 by John Wiley & Sons, Inc. All rights reserved.
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Modern experimental design / by Thomas P. Includes bibliographical references and index. ISBN 978-0-471-21077-1 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents Preface xv 1 Introduction 1 1.1 Experiments All Around Us 2 1.2 Objectives for Experimental Designs 3 1.3 Planned Experimentation versus Use of Observational Data 5 1.4 Basic Design Concepts 6 1.2 Replication versus Repeated Measurements 7 1.4 Size of an Effect That Can be Detected 11 1.6 Steps for the Design of Experiments 13 1.1 Recognition and Statement of the Problem 14 1.2 Selection of Factors and Levels 14 1.1 Choice of Factors 14 1.2 Choice of Levels 15 1.7 Processes Should Ideally be in a State of Statistical Control 18 1.8 Types of Experimental Designs 20 1.9 Analysis of Means 20 1.11 Experimental Designs and Six Sigma 22 1.12 Quasi-Experimental Design 23 1.13 Summary 23 References 23 Exercises 26 v vi contents 2 Completely Randomized Design 31 2.1 Completely Randomized Design 31 2.2 Example: One Factor, Two Levels 33 2.3 Examples: One Factor, More Than Two Levels 35 2.2 Unbalanced and Missing Data 39 2.4 Example Showing the Effect of Unequal Variances 41 2.2 Analysis of Means 42 2.1 ANOM for a Completely Randomized Design 43 2.2 ANOM with Unequal Variances 45 2.4 ANOM for Attributes Data 47 2.3 Software for Experimental Design 48 2.5 Summary 48 Appendix 49 References 49 Exercises 51 3 Designs that Incorporate Extraneous (Blocking) Factors 56 3.1 Randomized Block Design 56 3.2 Blocking an Out-of-Control Process 60 3.3 Efficiency of a Randomized Block Design 61 3.2 Incomplete Block Designs 65 3.1 Balanced Incomplete Block Designs 65 3.2 Recovery of Interblock Information 68 3.2 Partially Balanced Incomplete Block Designs 69 3.3 Nonparametric Analysis for Incomplete Block Designs 70 3.4 Other Incomplete Block Designs 70 3.3 Latin Square Design 71 3.2 Model 74 contents vii 3.4 Efficiency of a Latin Square Design 77 3.5 Using Multiple Latin Squares 77 3.4 Graeco–Latin Square Design 80 3.2 Degrees of Freedom Limitations on the Design Construction 81 3.3 Sets of Graeco–Latin Square Designs 82 3.2 Lists of Youden Designs 86 3.3 Using Replicated Youden Designs 86 3.8 Summary 90 References 91 Exercises 93 4 Full Factorial Designs with Two Levels 101 4.1 The Nature of Factorial Designs 101 4.2 The Deleterious Effects of Interactions 106 4.1 Sample Sizes for Conditional Effects Estimation 113 4.2 Can We “Transform Away” Interactions? 114 4.4 Why Not One-Factor-at-a-Time Designs? 115 4.5 ANOVA Table for Unreplicated Two-Factor Design? 116 4.7 Built-in Replication 122 4.8 Multiple Readings versus Replicates 123 4.9 Reality versus Textbook Examples 124 4.1 Factorial Design but not “Factorial Model” 124 4.10 Bad Data in Factorial Designs 127 4.11 Normal Probability Plot Methods 136 4.12 Missing Data in Factorial Designs 138 4.1 Resulting from Bad Data 139 4.13 Inaccurate Levels in Factorial Designs 140 4.14 Checking for Statistical Control 141 4.15 Blocking 2k Designs 142 viii contents 4.16 The Role of Expected Mean Squares in Experimental Design 144 4.17 Hypothesis Tests with Only Random Factors in 2k Designs? Avoid Them! 146 4.18 Hierarchical versus Nonhierarchical Models 147 4.19 Hard-to-Change Factors 148 4.1 Software for Designs with Hard-to-Change Factors 150 4.20 Factors Not Reset 150 4.21 Detecting Dispersion Effects 150 4.23 Summary 151 Appendix A Derivation of Conditional Main Effects 152 Appendix B Relationship Between Effect Estimates and Regression Coefficients: 153 Appendix C Precision of the Effect Estimates 153 Appendix D Expected Mean Squares for the Replicated 22 Design 153 Appendix E Expected Mean Squares, in General 155 References 157 Exercises 162 5 Fractional Factorial Designs with Two Levels 169 5.2 Effect Estimates and Regression Coefficients 177 5.4 What if I Had Used the Other Fraction? 179 5.1 Normal Probability Plot Methods when k − p = 16 187 5.2 Other Graphical Methods 188 5.4 Utility of Small Fractional Factorials vis-à-vis Normal Probability Plots 188 5.6 Retrieving a Lost Defining Relation 190 5.7 Minimum Aberration Designs and Minimum Confounded Effects Designs 192 5.8 Blocking Factorial Designs 194 5.1 Blocking Fractional Factorial Designs 195 5.1 Blocks of Size 2 200 5.2 Semifolding a 2k−1 Design 210 contents ix 5.4 Semifolding with Software 215 5.11 Projective Properties of 2k− p Designs 219 5.12 Small Fractions and Irregular Designs 220 5.13 An Example of Sequential Experimentation 222 5.1 Critique of Example 224 5.14 Inadvertent Nonorthogonality—Case Study 225 5.15 Fractional Factorial Designs for Natural Subsets of Factors 226 5.16 Relationship Between Fractional Factorials and Latin Squares 228 5.17 Alternatives to Fractional Factorials 229 5.1 Designs Attributed to Genichi Taguchi 229 5.18 Missing and Bad Data 230 5.19 Plackett–Burman Designs 230 5.21 Summary 233 References 234 Exercises 238 6 Designs With More Than Two Levels 248 6.1 Decomposing the A∗ B Interaction 251 6.2 Inference with Unreplicated 3k Designs 252 6.5 Need for Mixed Number of Levels 263 6.6 Replication of 3k− p Designs? 264 6.1 Constructing Mixed Factorials 265 6.5 Mixed Fractional Factorials 274 6.6 Orthogonal Arrays with Mixed Levels 275 6.7 Minimum Aberration Designs and Minimum Confounded Effects Designs 277 6.8 Four or More Levels 278 6.10 Catalog of Designs 284 6.11 Summary 284 References 284 Exercises 286 x contents 7 Nested Designs 291 7.3 Staggered Nested Designs 298 7.4 Nested and Staggered Nested Designs with Factorial Structure 300 7.5 Estimating Variance Components 300 7.6 ANOM for Nested Designs? 302 7.7 Summary 302 References 302 Exercises 304 8 Robust Designs 311 8.2 Identification of Dispersion Effects 314 8.3 Designs with Noise Factors 316 8.4 Product Array, Combined Array, or Compound Array? 318 8.7 Summary 322 References 323 Exercises 326 9 Split-Unit, Split-Lot, and Related Designs 330 9.1 Split-Unit Design 331 9.1 Split-Plot Mirror Image Pairs Designs 336 9.2 Split-Unit Designs in Industry 336 9.3 Split-Unit Designs with Fractional Factorials 340 9.4 Blocking Split-Plot Designs 342 9.5 Split-Unit Plackett-Burman Designs 343 9.6 Examples of Split-Plot Designs for Hard-to-Change Factors 343 9.7 Split-Split-Plot Designs 345 9.2 Split-Lot Design 345 9.1 Strip-Plot Design 346 9.1 Applications of Strip-Block (Strip-Plot) Designs 347 9.3 Commonalities and Differences Between these Designs 349 9.5 Summary 351 References 351 Exercises 354 contents xi 10 Response Surface Designs 360 10.1 Response Surface Experimentation: One Design or More Than One? 362 10.3 Classical Response Surface Designs versus Alternatives 364 10.4 Method of Steepest Ascent (Descent) 370 10.5 Central Composite Designs 373 10.2 Small Composite Designs 377 10.1 Draper–Lin Designs 378 10.6 Properties of Space-Filling Designs 384 10.7 Applications of Uniform Designs 386 10.8 Box–Behnken Designs 386 10.10 Other Response Surface Designs 390 10.2 Uniform Shell Designs 393 10.11 Blocking Response Surface Designs 394 10.1 Blocking Central Composite Designs 394 10.2 Blocking Box–Behnken Designs 396 10.3 Blocking Other Response Surface Designs 396 10.12 Comparison of Designs 397 10.13 Analyzing the Fitted Surface 398 10.1 Characterization of Stationary Points 401 10.2 Confidence Regions on Stationary Points 402 10.1 Ridge Analysis with Noise Factors 404 10.4 Optimum Conditions and Regions of Operability 404 10.14 Response Surface Designs for Computer Simulations 404 10.15 ANOM with Response Surface Designs? 405 10.17 The Present and Future Direction of Response Surface Designs 406 10.19 Catalogs of Designs 408 10.20 Summary 408 References 409 Exercises 414 xii contents 11 Repeated Measures Designs 425 11.1 The Example in Section 2.2 More Than One Factor 428 11.4 Designs for Carryover Effects 432 11.5 How Many Repeated Measures? 437 11.8 Summary 439 References 439 Exercises 444 12 Multiple Responses 447 12.1 Overlaying Contour Plots 448 12.2 Seeking Multiple Response Optimization with Desirability Functions 449 12.1 Weight and Importance 451 12.3 Dual Response Optimization 452 12.4 Designs Used with Multiple Responses 452 12.6 Multiple Response Optimization Variations 463 12.7 The Importance of Analysis 469 12.9 Summary 471 References 472 Exercises 474 13 Miscellaneous Design Topics 483 13.1 One-Factor-at-a-Time Designs 483 13.1 Plackett–Burman Designs 489 13.1 Projection Properties of Plackett–Burman Designs 493 13.3 Lesser-Known Screening Designs 500 13.5 Design of Experiments for Analytic Studies 500 13.1 One Factor, Two Levels 502 13.2 Are Commonly Used Designs Equileverage? 502 contents xiii 13.2 Applications of Optimal Designs 507 13.8 Designs for Restricted Regions of Operability 508 13.9 Space-Filling Designs 514 13.1 From Raw Form to Coded Form 518 13.2 Sphere-Packing Designs 518 13.3 Latin Hypercube Design 519 13.10 Trend-Free Designs 521 13.11 Cost-Minimizing Designs 522 13.1 Optimal Mixture Designs or Not? 523 13.13 Design of Measurement Capability Studies 523 13.14 Design of Computer Experiments 523 13.15 Design of Experiments for Categorical Response Variables 524 13.16 Weighing Designs and Calibration Designs 524 13.17 Designs for Assessing the Capability of a System 528 13.18 Designs for Nonlinear Models 528 13.19 Model-Robust Designs 528 13.20 Designs and Analyses for Non-normal Responses 529 13.21 Design of Microarray Experiments 529 13.22 Multi-Vari Plot 530 13.25 Summary 532 References 533 Exercises 542 14 Tying It All Together 544 14.1 Training for Experimental Design Use 544 References 545 Exercises 546 Answers to Selected Exercises 551 Appendix: Statistical Tables 565 Author Index 575 Subject Index 587 Preface Although there is a moderate amount of data analysis, especially in certain chapters, the emphasis in this book is on the statistical design of experiments. Such emphasis is justified by the widely held view that data from a well-designed experiment are easy to analyze.
Certain types of designs are not simple, however, such as those covered in Chapters 7, 8, and 11, and the problem is compounded by the fact that some popular statistical software packages have quite limited capability for those designs. The book would be suitable for an undergraduate one-semester course in design of experiments. For a course taught to nonstatistics majors, an instructor may wish to cover Chapters 1–4, part of Chapter 5, and then pick and choose from the other chapters in accordance with the needs of the students. The selection might include either or both of Chapters 10 and 12 and then cover sections of interest in Chapter 13.
