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
1Introduction
1.1Observational Studies and Experiments / 1
1.2Brief Historical Remarks / 4
1.3Basic Terminology and Concepts of Experimentation / 5
1.4Basic Principles of Experimentation / 9
1.4.1How to Minimize Biases and Variability? / 9
1.4.2Sequential Experimentation / 14
1.5Chapter Summary / 15
Exercises / 16
2Review of Elementary Statistics
2.1Experiments for a Single Treatment / 20
2.1.1Summary Statistics and Graphical Plots / 21
2.1.2Confidence Intervals and Hypothesis Tests / 25
2.1.3Power and Sample Size Calculation / 27
2.2Experiments for Comparing Two Treatments / 28
2.2.1Independent Samples Design / 29
2.2.2Matched Pairs Design / 38
2.3Linear Regression / 41
2.3.1Simple Linear Regression / 42
2.3.2Multiple Linear Regression / 50
2.4Chapter Summary / 62
Exercises / 62
viiiCONTENTS
3Single Factor Experiments: Completely Randomized Designs70
3.1Summary Statistics and Graphical Displays / 71
3.2Model / 73
3.3Statistical Analysis / 75
3.3.1Estimation / 75
3.3.2Analysis of Variance / 76
3.3.3Confidence Intervals and Hypothesis Tests / 78
3.4Model Diagnostics / 79
3.4.1Checking Homoscedasticity / 80
3.4.2Checking Normality / 81
3.4.3Checking Independence / 81
3.4.4Checking Outliers / 81
3.5Data Transformations / 85
3.6Power of F-Test and Sample Size Determination / 87
3.7Quantitative Treatment Factors / 90
3.8One-Way Analysis of Covariance / 96
3.8.1Randomized Block Design versus Analysis of
Covariance / 96
3.8.2Model / 96
3.8.3Statistical Analysis / 98
3.9Chapter Notes / 106
3.9.1Randomization Distribution of F-Statistic / 106
3.9.2F-Test for Heteroscedastic Treatment
Variances / 108
3.9.3Derivations of Formulas for Orthogonal
Polynomials / 110
3.9.4Derivation of LS Estimators for One-Way Analysis
of Covariance / 112
3.10Chapter Summary / 113
Exercises / 114
4Single-Factor Experiments: Multiple Comparison and Selection
Procedures126
4.1Basic Concepts of Multiple Comparisons / 127
4.1.1Family / 127
4.1.2Family wise Error Rate / 128
4.1.3Bonferroni Method / 129
4.1.4Union-Intersection Method / 130
4.1.5Closure Method / 131
CONTENTSix
4.2Pairwise Comparisons / 132
4.2.1Least Significant Difference and Bonferroni
Procedures / 133
4.2.2Tukey Procedure for Pairwise Comparisons / 134
4.2.3Step-Down Procedures for Pairwise Comparisons / 136
4.3Comparisons with a Control / 139
4.3.1Dunnett Procedure for Comparisons with a
Control / 139
4.3.2Step-Down Procedures for Comparisons with a
Control / 142
4.4General Contrasts / 144
4.4.1Tukey Procedure for Orthogonal Contrasts / 145
4.4.2Scheffe Procedure for All Contrasts / 146
4.5Ranking and Selection Procedures / 148
4.5.1Indifference-Zone Formulation / 148
4.5.2Subset Selection Formulation / 154
4.5.3Multiple Comparisons with the Best / 155
4.5.4Connection between Multiple Comparisons with
Best and Selection of Best Treatment / 157
4.6Chapter Summary / 158
Exercises / 159
5Randomized Block Designs and Extensions168
5.1Randomized Block Designs / 169
5.1.1Model / 169
5.1.2Statistical Analysis / 171
5.1.3Randomized Block Designs with Replicates / 177
5.2Balanced Incomplete Block Designs / 180
5.2.1Statistical Analysis / 182
5.2.2Interblock Analysis / 185
5.3Youden Square Designs / 188
5.3.1Statistical Analysis / 189
5.4Latin Square Designs / 192
5.4.1Choosing a Latin Square / 192
5.4.2Model / 195
5.4.3Statistical Analysis / 195
5.4.4Crossover Designs / 198
5.4.5Graeco-Latin Square Designs / 202
5.5Chapter Notes / 205
XCONTENTS
5.5.1Restriction Error Model for Randomized Block
Designs / 205
5.5.2Derivations of Formulas for BIB Design / 206
5.6Chapter Summary / 211
Exercises / 212
General Factorial Experiments224
6.1Factorial versus One-Factor-at-a-Time Experiments / 225
6.2Balanced Two-Way Layouts / 227
6.2.1Summary Statistics and Graphical Plots / 227
6.2.2Model / 230
6.2.3Statistical Analysis / 231
6.2.4Model Diagnostics / 235
6.2.5Tukey's Test for Interaction for Singly Replicated
Two-Way Layouts / 236
6.3Unbalanced Two-Way Layouts / 240
6.3.1Statistical Analysis / 240
6.4Chapter Notes / 245
6.4.1Derivation of LS Estimators of Parameters for
Balanced Two-Way Layouts / 245
6.4.2Derivation of ANOVA Sums of Squares and
/''-Tests for Balanced Two-Way Layouts / 246
6.4.3Three- and Higher Way Layouts / 248
6.5Chapter Summary / 250
Exercises / 250
Two-Level Factorial Experiments256
7.1Estimation of Main Effects and Interactions / 257
7.1.12 2 Designs / 257
7.1.22 3 Designs / 261
7.1.32 p Designs / 266
7.2Statistical Analysis / 267
7.2.1Confidence Intervals and Hypothesis Tests / 267
7.2.2Analysis of Variance / 268
7.2.3Model Fitting and Diagnostics / 270
7.3Single-Replicate Case / 272
7.3.1Normal and Half-Normal Plots of Estimated
Effects / 272
7.3.2Lenth Method / 278
CONTENTSxi
7.3.3Augmenting a 2 P Design with Observations at the
Center Point / 279
7.42 P Factorial Designs in Incomplete Blocks: Confounding of
Effects / 282
7.4.1Construction of Designs / 282
7.4.2Statistical Analysis / 286
7.5Chapter Notes / 287
7.5.1Yates Algorithm / 287
7.5.2Partial Confounding / 288
7.6Chapter Summary / 289
Exercises / 290
Two-Level Fractional Factorial Experiments300
8.12 p - q Fractional Factorial Designs / 301
8.1.12 p ~ l Fractional Factorial Design / 301
8.1.2General 2 p ~ q Fractional Factorial Designs / 307
8.1.3Statistical Analysis / 312
8.1.4Minimum Aberration Designs / 316
8.2Plackett-Burman Designs / 317
8.3Hadamard Designs / 323
8.4Supersaturated Designs / 325
8.4.1Construction of Supersaturated Designs / 325
8.4.2Statistical Analysis / 327
8.5Orthogonal Arrays / 329
8.6Sequential Assemblies of Fractional Factorials / 333
8.6.1Foldover of Resolution III Designs / 334
8.6.2Foldover of Resolution IV Designs / 337
8.7Chapter Summary / 338
Exercises / 339
Three-Level and Mixed-Level Factorial Experiments351
9.1Three-Level Full Factorial Designs / 351
9.1.1Linear-Quadratic System / 353
9.1.2Orthogonal Component System / 361
9.2Three-Level Fractional Factorial Designs / 364
9.3Mixed-Level Factorial Designs / 372
9.3.12 p \ q Designs / 373
9.3.2 2 ρ 7><> Designs / 378
9.4Chapter Notes / 386
xiiCONTENTS
9.4.1Alternative Derivations of Estimators of Linear and
Quadratic Effects / 386
9.5Chapter Summary / 388
Exercises / 389
10 Experiments for Response Optimization395
10.1Response Surface Methodology / 396
10.1.1Outline of Response Surface Methodology / 396
10.1.2First-Order Experimentation Phase / 397
10.1.3Second-Order Experimentation Phase / 402
10.2 Mixture Experiments / 412
10.2.1Designs for Mixture Experiments / 414
10.2.2Analysis of Mixture Experiments / 416
10.3 Taguchi Method of Quality Improvement / 419
10.3.1Philosophy Underlying Taguchi Method / 422
10.3.2Implementation 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 Experiments448
11.1One-Way Layouts / 449
11.1.1Random-Effects Model / 449
11.1.2Analysis of Variance / 450
11.1.3 Estimation of Variance Components / 452
11.2 Two-Way Layouts / 455
11.2.1Random-Effects Model / 455
11.2.2Mixed-Effects Model / 459
11.3 Three-Way Layouts / 464
11.3.1Random- and Mixed-Effects Models / 464
11.3.2Analysis of Variance / 465
11.3.3Approximate F-Tests / 468
11.4 Chapter Notes / 472
11.4.1 Maximum Likelihood and Restricted Maximum
Likelihood (REML) Estimation of Variance
Components / 472
11.4.2Derivations 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 Experiments487
12.1 Two-Stage Nested Designs / 488
12.1.1Model / 488
12.1.2Analysis of Variance / 489
12.2 Three-Stage Nested Designs / 490
12.2.1Model / 491
12.2.2Analysis of Variance / 492
12.3Crossed and Nested Designs / 495
12.3.1Model / 495
12.3.2Analysis of Variance / 496
12.4Split-Plot Designs / 501
12.4.1Model / 504
12.4.2Analysis of Variance / 505
12.4.3 Extensions of Split-Plot Designs / 508
12.5 Chapter Notes / 515
12.5.1Derivations of E(MS) Expressions for Two-Stage
Nested Design of Section 12.1 with Both Factors
Random / 515
12.5.2Derivations 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 Experiments536
13.1Univariate Approach / 536
13.1.1Model / 537
13.1.2Univariate Analysis of Variance for RM Designs / 537
13.2 Multivariate Approach / 548
13.2.1One-Way Multivariate Analysis of Variance / 548
13.2.2Multivariate Analysis of Variance for RM Designs / 549
13.3 Chapter Notes / 555
13.3.1Derivations of E(MS) Expressions for Repeated
Measures Design Assuming Compound Symmetry / 555
13.4 Chapter Summary / 558
Exercises / 559
xivCONTENTS
14.2
14 Theory of Linear Models with Fixed Effects
14.1 Basic Linear Model and Least Squares Estimation / 566
14.1.1Geometric Interpretation of Least Squares
Estimation / 568
14.1.2Least Squares Estimation in Singular Case / 570
14.1.3 Least Squares Estimation in Orthogonal Case / 572
Confidence Intervals and Hypothesis Tests / 573
14.2.1Sampling 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
14.4.1Proof of Theorem 14.1 (Gauss-Markov
Theorem) / 586
14.4.2Proof 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 AVector-Valued Random Variables and Some
Distribution Theory
A. 1 Mean Vector and Covariance Matrix of Random
Vector / 596
A.2Covariance Matrix of Linear Transformation of
Random Vector / 597
A.3 Multivariate Normal Distribution / 598
A.4Chi-Square, F-, and t-Distributions / 599
A.5Distributions of Quadratic Forms / 601
A.6Multivariate i-Distribution / 605
A.7Multivariate Normal Sampling Distribution
Theory / 606
595
Appendix BCase Studies608
B.lCase Study 1: Effects of Field Strength and Flip
Angle on MRI Contrast / 608
B.l.lIntroduction / 608
B.l.2Design / 609
B.l.3Data Analysis / 610
CONTENTSXV
B.1.4Results / 612
B.2Case Study 2: Growing Stem Cells for Bone
Implants / 613
B.2.1Introduction / 613
B.2.2Design / 614
B.2.3Data Analysis / 614
B.2.4Results / 614
B.3Case Study 3: Router Bit Experiment / 619
B.3.1Introduction / 619
B.3.2Design / 619
B.3.3Data Analysis / 623
B.3.4Results / 624
Appendix C Statistical Tables627
Answers to Selected Exercises644
References664
Index675