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Ajit C. Tamhane(auth.)-Statistical Analysis of Designed Experiments_ Theory and .pdf
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Statistical Analysis of
Designed Experiments
Theory and Applications
AJIT C. TAMHANE
Northwestern University
WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Contents
Preface
Abbreviations
1 Introduction
1.1 Observational Studies and Experiments / 1
1.2 Brief Historical Remarks / 4
1.3 Basic Terminology and Concepts of Experimentation / 5
1.4 Basic Principles of Experimentation / 9
1.4.1 How to Minimize Biases and Variability? / 9
1.4.2 Sequential Experimentation / 14
1.5 Chapter Summary / 15
Exercises / 16
2 Review of Elementary Statistics
2.1 Experiments for a Single Treatment / 20
2.1.1 Summary Statistics and Graphical Plots / 21
2.1.2 Confidence Intervals and Hypothesis Tests / 25
2.1.3 Power and Sample Size Calculation / 27
2.2 Experiments for Comparing Two Treatments / 28
2.2.1 Independent Samples Design / 29
2.2.2 Matched Pairs Design / 38
2.3 Linear Regression / 41
2.3.1 Simple Linear Regression / 42
2.3.2 Multiple Linear Regression / 50
2.4 Chapter Summary / 62
Exercises / 62
viii CONTENTS
3 Single Factor Experiments: Completely Randomized Designs 70
3.1 Summary Statistics and Graphical Displays / 71
3.2 Model / 73
3.3 Statistical Analysis / 75
3.3.1 Estimation / 75
3.3.2 Analysis of Variance / 76
3.3.3 Confidence Intervals and Hypothesis Tests / 78
3.4 Model Diagnostics / 79
3.4.1 Checking Homoscedasticity / 80
3.4.2 Checking Normality / 81
3.4.3 Checking Independence / 81
3.4.4 Checking Outliers / 81
3.5 Data Transformations / 85
3.6 Power of F-Test and Sample Size Determination / 87
3.7 Quantitative Treatment Factors / 90
3.8 One-Way Analysis of Covariance / 96
3.8.1 Randomized Block Design versus Analysis of
Covariance / 96
3.8.2 Model / 96
3.8.3 Statistical Analysis / 98
3.9 Chapter Notes / 106
3.9.1 Randomization Distribution of F-Statistic / 106
3.9.2 F-Test for Heteroscedastic Treatment
Variances / 108
3.9.3 Derivations of Formulas for Orthogonal
Polynomials / 110
3.9.4 Derivation of LS Estimators for One-Way Analysis
of Covariance / 112
3.10 Chapter Summary / 113
Exercises / 114
4 Single-Factor Experiments: Multiple Comparison and Selection
Procedures 126
4.1 Basic Concepts of Multiple Comparisons / 127
4.1.1 Family / 127
4.1.2 Family wise Error Rate / 128
4.1.3 Bonferroni Method / 129
4.1.4 Union-Intersection Method / 130
4.1.5 Closure Method / 131
CONTENTS ix
4.2 Pairwise Comparisons / 132
4.2.1 Least Significant Difference and Bonferroni
Procedures / 133
4.2.2 Tukey Procedure for Pairwise Comparisons / 134
4.2.3 Step-Down Procedures for Pairwise Comparisons / 136
4.3 Comparisons with a Control / 139
4.3.1 Dunnett Procedure for Comparisons with a
Control / 139
4.3.2 Step-Down Procedures for Comparisons with a
Control / 142
4.4 General Contrasts / 144
4.4.1 Tukey Procedure for Orthogonal Contrasts / 145
4.4.2 Scheffe Procedure for All Contrasts / 146
4.5 Ranking and Selection Procedures / 148
4.5.1 Indifference-Zone Formulation / 148
4.5.2 Subset Selection Formulation / 154
4.5.3 Multiple Comparisons with the Best / 155
4.5.4 Connection between Multiple Comparisons with
Best and Selection of Best Treatment / 157
4.6 Chapter Summary / 158
Exercises / 159
5 Randomized Block Designs and Extensions 168
5.1 Randomized Block Designs / 169
5.1.1 Model / 169
5.1.2 Statistical Analysis / 171
5.1.3 Randomized Block Designs with Replicates / 177
5.2 Balanced Incomplete Block Designs / 180
5.2.1 Statistical Analysis / 182
5.2.2 Interblock Analysis / 185
5.3 Youden Square Designs / 188
5.3.1 Statistical Analysis / 189
5.4 Latin Square Designs / 192
5.4.1 Choosing a Latin Square / 192
5.4.2 Model / 195
5.4.3 Statistical Analysis / 195
5.4.4 Crossover Designs / 198
5.4.5 Graeco-Latin Square Designs / 202
5.5 Chapter Notes / 205
X CONTENTS
5.5.1 Restriction Error Model for Randomized Block
Designs / 205
5.5.2 Derivations of Formulas for BIB Design / 206
5.6 Chapter Summary / 211
Exercises / 212
General Factorial Experiments 224
6.1 Factorial versus One-Factor-at-a-Time Experiments / 225
6.2 Balanced Two-Way Layouts / 227
6.2.1 Summary Statistics and Graphical Plots / 227
6.2.2 Model / 230
6.2.3 Statistical Analysis / 231
6.2.4 Model Diagnostics / 235
6.2.5 Tukey's Test for Interaction for Singly Replicated
Two-Way Layouts / 236
6.3 Unbalanced Two-Way Layouts / 240
6.3.1 Statistical Analysis / 240
6.4 Chapter Notes / 245
6.4.1 Derivation of LS Estimators of Parameters for
Balanced Two-Way Layouts / 245
6.4.2 Derivation of ANOVA Sums of Squares and
/''-Tests for Balanced Two-Way Layouts / 246
6.4.3 Three- and Higher Way Layouts / 248
6.5 Chapter Summary / 250
Exercises / 250
Two-Level Factorial Experiments 256
7.1 Estimation of Main Effects and Interactions / 257
7.1.1 2 2 Designs / 257
7.1.2 2 3 Designs / 261
7.1.3 2 p Designs / 266
7.2 Statistical Analysis / 267
7.2.1 Confidence Intervals and Hypothesis Tests / 267
7.2.2 Analysis of Variance / 268
7.2.3 Model Fitting and Diagnostics / 270
7.3 Single-Replicate Case / 272
7.3.1 Normal and Half-Normal Plots of Estimated
Effects / 272
7.3.2 Lenth Method / 278
CONTENTS xi
7.3.3 Augmenting a 2 P Design with Observations at the
Center Point / 279
7.4 2 P Factorial Designs in Incomplete Blocks: Confounding of
Effects / 282
7.4.1 Construction of Designs / 282
7.4.2 Statistical Analysis / 286
7.5 Chapter Notes / 287
7.5.1 Yates Algorithm / 287
7.5.2 Partial Confounding / 288
7.6 Chapter Summary / 289
Exercises / 290
Two-Level Fractional Factorial Experiments 300
8.1 2 p - q Fractional Factorial Designs / 301
8.1.1 2 p ~ l Fractional Factorial Design / 301
8.1.2 General 2 p ~ q Fractional Factorial Designs / 307
8.1.3 Statistical Analysis / 312
8.1.4 Minimum Aberration Designs / 316
8.2 Plackett-Burman Designs / 317
8.3 Hadamard Designs / 323
8.4 Supersaturated Designs / 325
8.4.1 Construction of Supersaturated Designs / 325
8.4.2 Statistical Analysis / 327
8.5 Orthogonal Arrays / 329
8.6 Sequential Assemblies of Fractional Factorials / 333
8.6.1 Foldover of Resolution III Designs / 334
8.6.2 Foldover of Resolution IV Designs / 337
8.7 Chapter Summary / 338
Exercises / 339
Three-Level and Mixed-Level Factorial Experiments 351
9.1 Three-Level Full Factorial Designs / 351
9.1.1 Linear-Quadratic System / 353
9.1.2 Orthogonal Component System / 361
9.2 Three-Level Fractional Factorial Designs / 364
9.3 Mixed-Level Factorial Designs / 372
9.3.1 2 p \ q Designs / 373
9.3.2 2 ρ 7><> Designs / 378
9.4 Chapter Notes / 386
xii CONTENTS
9.4.1 Alternative Derivations of Estimators of Linear and
Quadratic Effects / 386
9.5 Chapter Summary / 388
Exercises / 389
10 Experiments for Response Optimization 395
10.1 Response Surface Methodology / 396
10.1.1 Outline of Response Surface Methodology / 396
10.1.2 First-Order Experimentation Phase / 397
10.1.3 Second-Order Experimentation Phase / 402
10.2 Mixture Experiments / 412
10.2.1 Designs for Mixture Experiments / 414
10.2.2 Analysis of Mixture Experiments / 416
10.3 Taguchi Method of Quality Improvement / 419
10.3.1 Philosophy Underlying Taguchi Method / 422
10.3.2 Implementation of Taguchi Method / 425
10.3.3 Critique of Taguchi Method / 432
10.4 Chapter Summary / 436
Exercises / 437
11 Random and Mixed Crossed-Factors Experiments 448
11.1 One-Way Layouts / 449
11.1.1 Random-Effects Model / 449
11.1.2 Analysis of Variance / 450
11.1.3 Estimation of Variance Components / 452
11.2 Two-Way Layouts / 455
11.2.1 Random-Effects Model / 455
11.2.2 Mixed-Effects Model / 459
11.3 Three-Way Layouts / 464
11.3.1 Random- and Mixed-Effects Models / 464
11.3.2 Analysis of Variance / 465
11.3.3 Approximate F-Tests / 468
11.4 Chapter Notes / 472
11.4.1 Maximum Likelihood and Restricted Maximum
Likelihood (REML) Estimation of Variance
Components / 472
11.4.2 Derivations of Results for One- and Two-Way
Random-Effects Designs / 475
11.4.3 Relationship between Unrestricted and Restricted
Models / 478
xiii
11.5 Chapter Summary / 479
Exercises / 480
Nested, Crossed-Nested, and Split-Plot Experiments 487
12.1 Two-Stage Nested Designs / 488
12.1.1 Model / 488
12.1.2 Analysis of Variance / 489
12.2 Three-Stage Nested Designs / 490
12.2.1 Model / 491
12.2.2 Analysis of Variance / 492
12.3 Crossed and Nested Designs / 495
12.3.1 Model / 495
12.3.2 Analysis of Variance / 496
12.4 Split-Plot Designs / 501
12.4.1 Model / 504
12.4.2 Analysis of Variance / 505
12.4.3 Extensions of Split-Plot Designs / 508
12.5 Chapter Notes / 515
12.5.1 Derivations of E(MS) Expressions for Two-Stage
Nested Design of Section 12.1 with Both Factors
Random / 515
12.5.2 Derivations of E(MS) Expressions for Design of
Section 12.3 with Crossed and Nested Factors / 517
12.5.3 Derivations of E(MS) Expressions for Split-Plot
Design / 520
12.6 Chapter Summary / 523
Exercises / 524
Repeated Measures Experiments 536
13.1 Univariate Approach / 536
13.1.1 Model / 537
13.1.2 Univariate Analysis of Variance for RM Designs / 537
13.2 Multivariate Approach / 548
13.2.1 One-Way Multivariate Analysis of Variance / 548
13.2.2 Multivariate Analysis of Variance for RM Designs / 549
13.3 Chapter Notes / 555
13.3.1 Derivations of E(MS) Expressions for Repeated
Measures Design Assuming Compound Symmetry / 555
13.4 Chapter Summary / 558
Exercises / 559
xiv CONTENTS
14.2
14 Theory of Linear Models with Fixed Effects
14.1 Basic Linear Model and Least Squares Estimation / 566
14.1.1 Geometric Interpretation of Least Squares
Estimation / 568
14.1.2 Least Squares Estimation in Singular Case / 570
14.1.3 Least Squares Estimation in Orthogonal Case / 572
Confidence Intervals and Hypothesis Tests / 573
14.2.1 Sampling Distribution of j8 / 573
Sampling Distribution of s 2 I 574
Inferences on Scalar Parameters / 575
Inferences on Vector Parameters / 575
Extra Sum of Squares Method / 577
Analysis of Variance / 579
Power of F-Test / 583
Chapter Notes / 586
14.4.1 Proof of Theorem 14.1 (Gauss-Markov
Theorem) / 586
14.4.2 Proof of Theorem 14.2 / 586
Chapter Summary / 587
Exercises / 588
566
14.2.2
14.2.3
14.2.4
14.2.5
14.2.6
14.3
14.4
14.5
Appendix A Vector-Valued Random Variables and Some
Distribution Theory
A. 1 Mean Vector and Covariance Matrix of Random
Vector / 596
A.2 Covariance Matrix of Linear Transformation of
Random Vector / 597
A.3 Multivariate Normal Distribution / 598
A.4 Chi-Square, F-, and t-Distributions / 599
A.5 Distributions of Quadratic Forms / 601
A.6 Multivariate i-Distribution / 605
A.7 Multivariate Normal Sampling Distribution
Theory / 606
595
Appendix B Case Studies 608
B.l Case Study 1: Effects of Field Strength and Flip
Angle on MRI Contrast / 608
B.l.l Introduction / 608
B.l.2 Design / 609
B.l.3 Data Analysis / 610
CONTENTS XV
B.1.4 Results / 612
B.2 Case Study 2: Growing Stem Cells for Bone
Implants / 613
B.2.1 Introduction / 613
B.2.2 Design / 614
B.2.3 Data Analysis / 614
B.2.4 Results / 614
B.3 Case Study 3: Router Bit Experiment / 619
B.3.1 Introduction / 619
B.3.2 Design / 619
B.3.3 Data Analysis / 623
B.3.4 Results / 624
Appendix C Statistical Tables 627
Answers to Selected Exercises 644
References 664
Index 675
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