Experiments: Planning, Analysis, and Optimization
C. F. Jeff Wu and Michael Hamada
Table of Contents
1. Basic
Concepts for Experimental Design and Introductory Regression Analysis 1
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 Simple Linear Regression, 11
1.5 Testing of Hypothesis and Interval Estimation, 14
1.6 Multiple Linear Regression, 20
1.7 Variable Selection in Regression Analysis, 26
1.8 Analysis of Air Pollution Data, 29
1.9 Practical Summary, 34
Exercises, 36
References, 43
2. Experiments
with a Single Factor 45
2.1 One-Way Layout, 45
*2.1.1
Constraint on the Parameters, 50
2.2 Multiple Comparisons, 53
2.3 Quantitative Factors and Orthogonal Polynomials, 57
2.4 Expected Mean Squares and Sample Size Determination, 63
2.5 One-Way Random Effects Model, 70
2.6 Residual Analysis: Assessment of Model Assumptions, 74
2.7 Practical Summary, 79
Exercises, 80
References,
86
3. Experiments
with More Than One Factor 87
3.1 Paired Comparison Designs, 87
3.2 Randomized Block Designs, 90
3.3 Two-Way Layout: Factors with Fixed Levels, 94
3.3.1 Two Qualitative Factors: A Regression Modeling Approach,
97
*3.4 Two-Way Layout: Factors with Random Levels, 99
3.5 Multi-Way Layouts, 108
3.7 Graeco-Latin Square Designs,
114
*3.8 Balanced Incomplete Block Designs, 115
*3.9
3.10 Analysis of Covariance: Incorporating Auxiliary Information,
128
*3.11 Transformation of the Response, 133
3.12 Practical Summary, 137
Exercises, 138
Appendix 3A: Table of Latin Squares, Graeco-Latin
Squares, and Hyper-Graeco-Latin Squares, 150
References,
152
4. Full
Factorial Experiments at Two Levels 155
4.1 An Epitaxial Layer
Growth Experiment, 155
4.2 Full Factorial Designs at Two Levels: A General
Discussion, 157
4.3 Factorial Effects and Plots, 161
4.3.1 Main Effects, 162
4.3.2 Interaction Effects, 164
4.4 Using Regression to Compute Factorial Effects, 169
*4.5 ANOVA Treatment of Factorial Effects, 171
4.6 Fundamental Principles for Factorial Effects:
Effect Hierarchy, Effect Sparsity, and Effect
Heredity, 172
4.7 Comparisons with the “One-Factor-at-a-Time”
Approach, 173
4.8
4.9 Lenth’s Method: Testing
Effect Significance for Experiments without Variance Estimates, 180
4.10 Nominal-the-Best Problem and Quadratic Loss Function,
183
4.11 Use of Log Sample Variance for Dispersion Analysis, 184
4.12 Analysis of Location and Dispersion: Revisiting the
Epitaxial Layer Growth Experiment, 185
*4.13 Test
of Variance Homogeneity and Pooled Estimate of Variance, 188
*4.14 Studentized Maximum Modulus
Test: Testing Effect Significance for Experiments with Variance Estimates, 190
4.15 Blocking and Optimal Arrangement of 2k
Factorial Designs in 2q
Blocks, 193
4.16 Practical Summary, 198
Exercises, 200
Appendix 4A: Table of 2k Factorial Designs in 2q Blocks, 207
References,
208
5. Fractional
Factorial Experiments at Two Levels 211
5.1 A Leaf Spring Experiment, 211
5.2 Fractional Factorial Designs: Effect Aliasing and the
Criteria of Resolution and Minimum Aberration, 213
5.3 Analysis of Fractional Factorial Experiments, 219
5.4 Techniques for Resolving the Ambiguities in Aliased
Effects, 225
5.4.1 Fold-Over Technique for Follow-up Experiments, 225
5.4.2 Optimal Design Approach for Follow-up Experiments, 229
5.5 Selection of 2k−p
Designs Using Minimum Aberration and Related Criteria,
234
5.6 Blocking in Fractional Factorial Designs, 238
5.7 Practical Summary, 240
Exercises, 242
Appendix 5A: Tables of 2k−p
Fractional Factorial Designs, 252
Appendix 5B: Tables of 2k−p
Fractional Factorial Designs in 2q
Blocks, 260
References,
264
6. Full
Factorial and Fractional Factorial Experiments at Three Levels 267
6.1 A Seat-Belt Experiment, 267
6.2 Larger-the-Better and Smaller-the-Better Problems, 268
6.3 3k Full Factorial Designs, 270
6.4 3k−p
Fractional Factorial Designs, 275
6.5 Simple Analysis Methods: Plots and Analysis of Variance,
279
6.6 An Alternative Analysis Method, 287
6.7 Analysis Strategies for Multiple Responses I: Out-of-Spec
Probabilities, 293
6.8 Blocking in 3k and 3k−p
Designs, 302
6.9 Practical Summary, 303
Exercises, 305
Appendix 6A: Tables of 3k−p
Fractional Factorial Designs, 312
Appendix 6B: Tables of 3k−p
Fractional Factorial Designs in 3q
Blocks, 313
References, 317
7. Other
Design and Analysis Techniques for Experiments at More Than Two Levels 319
7.1 A Router Bit
Experiment Based on a Mixed Two-Level and Four-Level Design, 319
7.2 Method of Replacement and Construction of 2m4n Designs, 322
7.3 Minimum Aberration 2m4n Designs with n = 1,
2, 325
7.4 An Analysis Strategy for 2m4n Experiments, 328
7.5 Analysis of the Router Bit Experiment, 330
7.6 A Paint Experiment Based on a Mixed Two-Level and
Three-Level Design, 334
7.7 Design and Analysis of 36-Run Experiments at Two and
Three Levels, 334
7.8 rk−p
Fractional Factorial Designs for any Prime Number
r, 341
7.8.1 25-Run Fractional Factorial Designs at Five Levels, 342
7.8.2 49-Run Fractional Factorial Designs at Seven Levels,
345
7.8.3 General Construction, 345
*7.9 Related Factors: Method of Sliding Levels, Nested
Effects Analysis, and Response Surface Modeling, 346
7.9.1 Nested Effects Modeling, 348
7.9.2 Analysis of Light Bulb Experiment, 350
7.9.3 Response Surface Modeling, 353
7.9.4 Symmetric and Asymmetric Relationships Between Related Factors, 355
7.10 Practical Summary, 356
Exercises, 357
Appendix 7A: Tables of 2m41 Minimum Aberration Designs, 364
Appendix 7B: Tables of 2m42 Minimum Aberration Designs, 366
Appendix 7C: OA(25, 56), 368
Appendix 7D: OA(49, 78), 368
References, 370
8. Nonregular
Designs: Construction and Properties 371
8.1 Two Experiments: Weld-Repaired Castings and
Blood Glucose Testing, 371
8.2 Some Advantages of Nonregular
Designs Over the 2k−p
and 3k−p
Series of Designs, 373
8.3 A Lemma on Orthogonal Arrays, 374
8.4 Plackett–Burman Designs and Hall’s Designs, 375
8.5 A Collection of Useful Mixed-Level Orthogonal Arrays, 379
*8.6 Construction of Mixed-Level Orthogonal Arrays Based on Difference
Matrices, 381
8.6.1 General Method for Constructing Asymmetrical Orthogonal
Arrays, 382
*8.7 Construction of Mixed-Level Orthogonal Arrays Through the Method of Replacement, 384
8.8 Orthogonal Main-Effect Plans Through
Collapsing Factors, 386
8.9 Practical Summary, 390
Exercises, 391
Appendix 8A: Plackett–Burman Designs OA(N,
2N−1) with 12 ≤ N ≤ 48 and N = 4k
But Not a Power of 2, 397
Appendix 8B: Hall’s 16-Run Orthogonal Arrays of Types
II to V, 401
Appendix 8C: Some Useful Mixed-Level Orthogonal Arrays, 405
Appendix 8D: Some Useful Difference Matrices, 416
Appendix 8E: Some Useful Orthogonal Main-Effect Plans, 418
References,
419
9. Experiments
with Complex Aliasing 421
9.1 Partial Aliasing of Effects and the Alias
Matrix, 421
9.2 Traditional Analysis Strategy: Screening Design and Main
Effect Analysis, 424
9.3
Simplification of Complex Aliasing via Effect Sparsity,
424
9.4 An Analysis Strategy for Designs with Complex Aliasing,
426
9.4.1 Some Limitations, 432
*9.5 A Bayesian Variable Selection Strategy for Designs with
Complex Aliasing, 433
9.5.1 Bayesian Model Priors, 435
9.5.2 Gibbs Sampling, 437
9.5.3 Choice of Prior Tuning Constants, 438
9.5.4 Blood Glucose Experiment Revisited, 439
9.5.5 Other Applications, 441
*9.6 Supersaturated Designs: Design Construction and
Analysis, 442
9.7 Practical Summary, 445
Exercises, 446
Appendix 9A: Further Details for the Full Conditional
Distributions, 454
References,
456
10.
Response Surface Methodology 459
10.1 A Ranitidine Separation Experiment, 459
10.2 Sequential Nature of Response Surface Methodology, 461
10.3 From First-Order Experiments to Second-Order
Experiments: Steepest Ascent Search and Rectangular Grid Search, 464
10.3.1 Curvature Check, 465
10.3.2 Steepest Ascent Search, 466
10.3.3 Rectangular Grid Search, 470
10.4 Analysis of Second-Order Response Surfaces, 473
10.4.1 Ridge Systems, 475
10.5 Analysis of the Ranitidine Experiment, 477
10.6 Analysis Strategies for Multiple Responses II: Contour
Plots and the Use of Desirability Functions, 481
10.7 Central Composite Designs, 484
10.8 Box–Behnken Designs and
Uniform Shell Designs, 489
10.9 Practical Summary, 492
Exercises, 494
Appendix 10A: Table of Central Composite Designs, 505
Appendix 10B: Table of Box–Behnken
Designs, 507
Appendix 10C: Table of Uniform Shell Designs, 508
References, 509
11. Introduction
to Robust Parameter Design 511
11.1 A Robust Parameter Design Perspective of the
Layer Growth and Leaf Spring Experiments, 511
11.1.1 Layer Growth Experiment Revisited, 511
11.1.2 Leaf Spring Experiment Revisited, 512
11.2 Strategies for Reducing Variation, 514
11.3 Noise (Hard-to-Control) Factors, 516
11.4 Variation Reduction Through Robust
Parameter Design, 518
11.5 Experimentation and Modeling Strategies I: Cross Array,
520
11.5.1 Location and Dispersion Modeling, 521
11.5.2 Response Modeling, 526
11.6 Experimentation and Modeling Strategies II: Single Array
and Response Modeling, 532
11.7 Cross Arrays: Estimation Capacity and Optimal Selection,
535
11.8 Choosing Between Cross Arrays and Single Arrays, 538
*11.8.1 Compound Noise Factor, 542
11.9 Signal-to-Noise Ratio and Its
Limitations for Parameter Design Optimization, 543
11.9.1 SN Ratio Analysis of Layer Growth Experiment, 546
*11.10 Further Topics, 547
11.11 Practical Summary, 548
Exercises, 550
References,
560
12. Robust
Parameter Design for Signal–Response Systems 563
12.1 An Injection
Molding Experiment, 563
12.2 Signal–Response Systems
and Their Classification, 565
12.2.1 Calibration of Measurement Systems, 570
12.3 Performance Measures for Parameter Design Optimization,
571
12.4 Modeling and Analysis Strategies, 575
12.5 Analysis of the Injection Molding Experiment, 577
12.5.1 PMM Analysis, 580
12.5.2 RFM Analysis, 581
*12.6 Choice
of Experimental Plans, 584
12.7 Practical Summary, 587
Exercises, 588
References,
596
13. Experiments
for Improving Reliability 599
13.1 Experiments with Failure Time Data, 599
13.1.1 Light Experiment, 599
13.1.2 Thermostat Experiment, 600
13.1.3 Drill Bit Experiment, 600
13.2 Regression Model for Failure Time Data, 604
13.3 A Likelihood Approach for Handling Failure Time Data
with Censoring, 605
13.3.1 Estimability Problem with MLEs, 608
13.4 Design-Dependent Model Selection Strategies, 609
13.5 A Bayesian Approach to Estimation and Model Selection
for Failure Time Data, 610
13.6 Analysis of Reliability Experiments with Failure Time
Data, 613
13.6.1 Analysis of Light Experiment, 613
13.6.2 Analysis of Thermostat Experiment, 614
13.6.3 Analysis of Drill Bit Experiment, 615
13.7 Other Types of Reliability Data, 617
13.8 Practical Summary, 618
Exercises, 619
References, 623
14.
Analysis of Experiments with Nonnormal
Data 625
14.1 A Wave Soldering Experiment with Count Data, 625
14.2 Generalized Linear Models, 627
14.2.1 The Distribution of the Response, 627
14.2.2 The Form of the Systematic Effects, 629
14.2.3 GLM versus Transforming the Response, 630
14.3 Likelihood-Based Analysis of Generalized Linear Models,
631
14.4 Likelihood-Based Analysis of the Wave Soldering Experiment,
634
14.5 Bayesian Analysis of Generalized Linear Models, 635
14.6 Bayesian Analysis of the Wave Soldering Experiment, 637
14.7 Other Uses and Extensions of Generalized Linear Models
and Regression Models for Nonnormal Data, 639
*14.8 Modeling and Analysis for Ordinal Data, 639
14.8.1 The Gibbs Sampler for Ordinal Data, 642
*14.9 Analysis of Foam Molding Experiment, 644
14.10
Scoring: A Simple Method for Analyzing Ordinal Data, 647
14.11 Practical Summary, 649
Exercises, 649
References, 661
Appendices. Statistical Tables 663-703