Advanced Analysis of Variance

Chihiro Hirotsu is a Senior Researcher at the Collaborative Research Center, Meisei University, and Professor Emeritus at the University of Tokyo. He is a fellow of the American Statistical Association, an elected member of the International Statistical Institute, and he has been awarded the Japan Statistical Society Prize (2005) and the Ouchi Prize (2006). His work has been published in Biometrika, Biometrics, and Computational Statistics & Data Analysis, among other premier research journals.

Preface xi

Notation and Abbreviations xvii

1 Introduction to Design and Analysis of Experiments 1

1.1 Why Simultaneous Experiments? 1

1.2 Interaction Effects 2

1.3 Choice of Factors and Their Levels 4

1.4 Classification of Factors 5

1.5 Fixed or Random Effects Model? 5

1.6 Fisher’s Three Principles of Experiments vs. Noise Factor 6

1.7 Generalized Interaction 7

1.8 Immanent Problems in the Analysis of Interaction Effects 7

1.9 Classification of Factors in the Analysis of Interaction Effects 8

1.10 Pseudo Interaction Effects (Simpson’s Paradox) in Categorical Data 8

1.11 Upper Bias by Statistical Optimization 9

1.12 Stage of Experiments: Exploratory, Explanatory or Confirmatory? 10

2 Basic Estimation Theory 11

2.1 Best Linear Unbiased Estimator 11

2.2 General Minimum Variance Unbiased Estimator 12

2.3 Efficiency of Unbiased Estimator 14

2.4 Linear Model 18

2.5 Least Squares Method 19

2.6 Maximum Likelihood Estimator 31

2.7 Sufficient Statistics 34

3 Basic Test Theory 41

3.1 Normal Mean 41

3.2 Normal Variance 53

3.3 Confidence Interval 56

3.4 Test Theory in the Linear Model 58

3.5 Likelihood Ratio Test and Efficient Score Test 62

4 Multiple Decision Processes and an Accompanying Confidence Region 71

4.1 Introduction 71

4.2 Determining the Sign of a Normal Mean – Unification of One- and Two-Sided Tests 71

4.3 An Improved Confidence Region 73

5 Two-Sample Problem 75

5.1 Normal Theory 75

5.2 Non-parametric Tests 84

5.3 Unifying Approach to Non-inferiority, Equivalence and Superiority Tests 92

6 One-Way Layout, Normal Model 113

6.1 Analysis of Variance (Overall F-Test) 113

6.2 Testing the Equality of Variances 115

6.3 Linear Score Test (Non-parametric Test) 118

6.4 Multiple Comparisons 121

6.5 Directional Tests 128

7 One-Way Layout, Binomial Populations 165

7.1 Introduction 165

7.2 Multiple Comparisons 166

7.3 Directional Tests 167

8 Poisson Process 193

8.1 Max acc. t1 for the Monotone and Step Change-Point Hypotheses 193

8.2 Max acc. t2 for the Convex and Slope Change-Point Hypotheses 197

9 Block Experiments 201

9.1 Complete Randomized Blocks 201

9.2 Balanced Incomplete Blocks 205

9.3 Non-parametric Method in Block Experiments 211

10 Two-Way Layout, Normal Model 237

10.1 Introduction 237

10.2 Overall ANOVA of Two-Way Data 238

10.3 Row-wise Multiple Comparisons 244

10.4 Directional Inference 256

10.5 Easy Method for Unbalanced Data 260

11 Analysis of Two-Way Categorical Data 273

11.1 Introduction 273

11.2 Overall Goodness-of-Fit Chi-Square 275

11.3 Row-wise Multiple Comparisons 276

11.4 Directional Inference in the Case of Natural Ordering Only in Columns 281

11.5 Analysis of Ordered Rows and Columns 291

12 Mixed and Random Effects Model 299

12.1 One-Way Random Effects Model 299

12.2 Two-Way Random Effects Model 306

12.3 Two-Way Mixed Effects Model 314

12.4 General Linear Mixed Effects Model 322

13 Profile Analysis of Repeated Measurements 329

13.1 Comparing Treatments Based on Upward or Downward Profiles 329

13.2 Profile Analysis of 24-Hour Measurements of Blood Pressure 338

14 Analysis of Three-Way Categorical Data 347

14.1 Analysis of Three-Way Response Data 348

14.2 One-Way Experiment with Two-Way Categorical Responses 361

14.3 Two-Way Experiment with One-Way Categorical Responses 375

15 Design and Analysis of Experiments by Orthogonal Arrays 383

15.1 Experiments by Orthogonal Array 383

15.2 Ordered Categorical Responses in a Highly Fractional Experiment 393

15.3 Optimality of an Orthogonal Array 397

References 399

Appendix 401

Index 407