Design and Analysis of Experiments



Volume 1
Introduction to Experimental Design
Second Edition
Klaus Hinkelmann    Oscar Kempthorne


Contents
Preface to the Second Edition
Preface to the First Edition
xvii
xxi
1 The Processes of Science
1.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 . 1 Observations in Science . . . . . . . . . . . . . . . . . . . .
1.1.2 Two Types of Observations . . . . . . . . . . . . . . . . . . .
1.1.3 From Observation to Law . . . . . . . . . . . . . . . . . . .
1.2 DEVELOPMENT OF THEORY . . . . . . . . . . . . . . . . . . . .
1.2.1 The Basic Syllogism . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Induction, Deduction, and Hypothesis . . . . . . . . . . . . .
1.3 THE NATURE AND ROLE OF THEORY IN SCIENCE . . . . . . .
1.3.1 What Is Science? . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Two Types of Science . . . . . . . . . . . . . . . . . . . . .
1.4 VARIETIES OF THEORY . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Two Types of Theory . . . . . . . . . . . . . . . . . . . . . .
1.4.2 What Is a Theory? . . . . . . . . . . . . . . . . . . . . . . .
1.5 THE PROBLEM OF GENERAL SCIENCE . . . . . . . . . . . . . .
1.5.1 Two Problems . . . . . . . . . . . . . . . . . . . . . . . . .
1 S.2 The Role of Data Analysis . . . . . . . . . . . . . . . . . . .
1 S.3 The Problem of Inference . . . . . . . . . . . . . . . . . . .
1.6 CAUSALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Defining Cause. Causation. and Causality . . . . . . . . . . .
1.6.2 The Role of Comparative Experiments . . . . . . . . . . . . .
1.7 THEUPSHOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 WHAT IS AN EXPERIMENT? . . . . . . . . . . . . . . . . . . . .
1.8.1 Absolute and Comparative Experiments . . . . . . . . . . . .
1.8.2 Three Types of Experiments . . . . . . . . . . . . . . . . . .
1.9 STATISTICAL INFERENCE . . . . . . . . . . . . . . . . . . . . . .
1.9.1 Drawing Inference . . . . . . . . . . . . . . . . . . . . . . .
1.9.2 Notions of Probability . . . . . . . . . . . . . . . . . . . . .
1.9.3 Variability and Randomization . . . . . . . . . . . . . . . . .
2 Principles of Experimental Design 29
2.1 CONFIRMATORY AND EXPLORATORY EXPERIMENTS . . . . 29
2.2 STEPS OF DESIGNED INVESTIGATIONS . . . . . . . . . . . . . 30
2.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Subject Matter Model . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Three Aspects of Design . . . . . . . . . . . . . . . . . . . . 33
2.2.4 Modeling the Response . . . . . . . . . . . . . . . . . . . . . 35
2.2.5 Choosing the Response . . . . . . . . . . . . . . . . . . . . . 36
2.2.6 Principles of Analysis . . . . . . . . . . . . . . . . . . . . . 36
2.3 THE LINEAR MODEL . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Three Types of Effects . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Experimental and Observational Units . . . . . . . . . . . . . 38
2.3.3 Outline of a Model . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 The Questions and Hypotheses . . . . . . . . . . . . . . . . . 41
2.4.2 The Experiment and a Model . . . . . . . . . . . . . . . . . . 41
2.4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 ILLUSTRATING INDIVIDUAL STEPS: STUDY 1 . . . . . . . . . 41
2.4.4 Alternative Experimental Setup . . . . . . . . . . . . . . . . 44
2.5 THREE PRINCIPLES OF EXPERIMENTAL DESIGN . . . . . . . . 45
2.6 THE STATISTICAL TRIANGLE: STUDY 2 . . . . . . . . . . . . . 46
2.6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 46
2.6.2 Four Experimental Situations . . . . . . . . . . . . . . . . . 46
2.7 PLANNING THE EXPERIMENT: THINGS TO THINK ABOUT . . 5 1
2.8 COOPERATION BETWEEN SCIENTIST AND STATISTICIAN . . 53
2.9 GENERAL PRINCIPLE OF INFERENCE AND TYPES OF
STATISTICAL ANALYSES . . . . . . . . . . . . . . . . . . . . . . 56
2.9.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9.2 Outline of the ANOVA . . . . . . . . . . . . . . . . . . . . . 56
2.10 OTHER CONSIDERATIONS FOR EXPERIMENTAL DESIGNS . . 58
3 Survey of Designs And Analyses 61
3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 ERROR-CONTROL DESIGNS . . . . . . . . . . . . . . . . . . . . 62
3.3 TREATMENT DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 COMBINING IDEAS FROM ERROR-CONTROL AND
TREATMENT DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 SAMPLING DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 ANALYSIS AND STATISTICAL SOFTWARE . . . . . . . . . . . . 68
3.7 SUMMARY., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Linear Model Theory 71
4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.1 The Concept of a Model . . . . . . . . . . . . . . . . . . . . 71
4.1.2 Comparative and Absolute Experiments . . . . . . . . . . . . 73
4.2 REPRESENTATION OF LINEAR MODELS . . . . . . . . . . . . . 73
4.3 FUNCTIONAL AND CLASSIFICATORY LINEAR MODELS . . . 74


4.3.1 Functional Models . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Classificatory Models . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 THE FITTING OF y = Xp . . . . . . . . . . . . . . . . . . . . . . 76
4.4.1 The Notion of Identifiability . . . . . . . . . . . . . . . . . . 76
4.4.3 The Method of Least Squares . . . . . . . . . . . . . . . . . 77
4.4.4 Theory of Linear Equations . . . . . . . . . . . . . . . . . . 81
4.5 MOORE-PENROSE GENERALIZED INVERSE . . . . . . . . . . . 84
4.6 CONDITIONED LINEAR MODEL . . . . . . . . . . . . . . . . . . 85
4.6.1 Affine Linear Model . . . . . . . . . . . . . . . . . . . . . . 85
4.6.2 Normal Equations for the Conditioned Model . . . . . . . . . 87
4.6.3 Different Types of Conditions . . . . . . . . . . . . . . . . . 88
4.6.4 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 TWO-PART LINEAR MODEL . . . . . . . . . . . . . . . . . . . . 90
4.7.1 Ordered Linear Models . . . . . . . . . . . . . . . . . . . . . 90
4.7.2 Using Orthogonal Projections . . . . . . . . . . . . . . . . . 91
4.7.3 Orthogonal ANOVA . . . . . . . . . . . . . . . . . . . . . . 93
4.8 SPECIAL CASE OF A PARTITIONED MODEL . . . . . . . . . . . 94
4.9 THREE-PART MODELS . . . . . . . . . . . . . . . . . . . . . . . . 94
4.10 TWO-WAY CLASSIFICATION WITHOUT INTERACTION . . . . 95
4.1 1 K-PART LINEAR MODEL . . . . . . . . . . . . . . . . . . . . . . . 97
4.11.1 The General Model and Its Sums of Squares . . . . . . . . . . 97
4.1 1.2 The Means Model . . . . . . . . . . . . . . . . . . . . . . . 99
4.12 BALANCED CLASSIFICATORY STRUCTURES . . . . . . . . . . 100
4.12.1 Factors, Levels, and Partitions . . . . . . . . . . . . . . . . . 101
4.12.2 Nested, Crossed, and Confounded Factors . . . . . . . . . . . 101
4.12.3 The Notion of Balance . . . . . . . . . . . . . . . . . . . . . 102
4.12.4 Balanced One-way Classification . . . . . . . . . . . . . . . 102
4.12.5 Two-way Classification with Equal Numbers . . . . . . . . . 103
4.12.6 Experimental versus Observational Studies . . . . . . . . . . 104
4.12.7 General Classificatory Structure . . . . . . . . . . . . . . . . 106
4.12.8 The Well-Formulated Model . . . . . . . . . . . . . . . . . . 109

4.13.1 Two-Fold Nested Classification . . . . . . . . . . . . . . . . 112
4.13.2 Two-way Cross-Classification . . . . . . . . . . . . . . . . . 113
4.13.3 Two-way Classification without Interaction . . . . . . . . . . 116
4.14 ANALYSIS OF COVARIANCE MODEL . . . . . . . . . . . . . . . 118
4.14.1 The Question of Explaining Data . . . . . . . . . . . . . . . 118
4.14.2 Obtaining the ANOVA Table . . . . . . . . . . . . . . . . . . 120
4.14.3 The Case of One Covariate . . . . . . . . . . . . . . . . . . . 121
4.14.4 The Case of Several Covariates . . . . . . . . . . . . . . . . 121
4.15 FROM DATA ANALYSIS TO STATISTICAL INFERENCE . . . . . 122
4.16 SIMPLE NORMAL STOCHASTIC LINEAR MODEL . . . . . . . . 123
4.16.1 The Notion of Estimability . . . . . . . . . . . . . . . . . . . 123

4.16.2 Gauss-Markov Linear Model . . . . . . . . . . . . . . . . . .
4.16.3 Ordinary Least Squares and Best Linear Unbiased Estimators
4.16.4 Expectation of Quadratic Forms . . . . . . . . . . . . . . . .
4.17 DISTRIBUTION THEORY WITH GXNLM
4.17.1 Distributional Properties of X'P . . . . . . . . . . . . . . . .
4.17.2 Distribution of Sums of Squares . . . . . . . . . . . . . . . .
4.17.3 Testing of Hypotheses . . . . . . . . . . . . . . . . . . . . .
4.18 MIXED MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18.1 The Notion of Fixed, Mixed and Random Models . . . . . . .
4.18.2 Aitken-like Model . . . . . . . . . . . . . . . . . . . . . . .
4.18.3 Mixed Models in Experimental Design . . . . . . . . . . . .
4.17 DISTRIBUTION THEORY WITH GXNLM . . . . . . . . . . . . .
5 Randomization
5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Observational versus Intervention Studies . . . . . . . . . . .
5.1.2 Historical Controls versus Repetitions . . . . . . . . . . . . .
5.2 THE TEA TASTING LADY . . . . . . . . . . . . . . . . . . . . . .
5.3 TRIANGULAR EXPERIMENT . . . . . . . . . . . . . . . . . . . .
5.3.1 Medical Example . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Randomization. Probabilities. and Beliefs . . . . . . . . . . .
5.4 SIMPLE ARITHMETICAL EXPERIMENT . . . . . . . . . . . . . .
5.4.1 Noisy Experiments . . . . . . . . . . . . . . . . . . . . . . .
Investigative Experiments and Beliefs . . . . . . . . . . . . .
5.4.3 Randomized Experiments . . . . . . . . . . . . . . . . . . .
5.5 RANDOMIZATION IDEAS . . . . . . . . . . . . . . . . . . . . . .
5.6 EXPERIMENT RANDOMIZATION TEST . . . . . . . . . . . . . .
5.7 INTRODUCTION TO SUBSEQUENT CHAPTERS . . . . . . . . .
5.4.2
6 Completely Randomized Design
6.1 INTRODUCTION AND DEFINITION . . . . . . . . . . . . . . . .
6.2 RANDOMIZATION PROCESS . . . . . . . . . . . . . . . . . . . .
Use of Random Numbers . . . . . . . . . . . . . . . . . . . .
6.2.2 Design Random Variables . . . . . . . . . . . . . . . . . . .
6.3 DERIVED LINEAR MODEL . . . . . . . . . . . . . . . . . . . . .
Conceptual Responses and Observations . . . . . . . . . . . .
6.3.2 Distributional Properties . . . . . . . . . . . . . . . . . . . .
6.3.3 Additivity in the Broad Sense . . . . . . . . . . . . . . . . .
6.3.4 Error Structure . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . .
6.4 ANALYSIS OF VARIANCE . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Deriving the ANOVA Table . . . . . . . . . . . . . . . . . .
6.4.2 Obtaining Expected Mean Squares . . . . . . . . . . . . . . .
6.5 STATISTICAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Enumerating Randomizations . . . . . . . . . . . . . . . . .
6.5.2 Randomization Test . . . . . . . . . . . . . . . . . . . . . .
6.6 APPROXIMATING THE RANDOMIZATION TEST . . . . . . . . .
6.6.1 Moments of the Test Statistic . . . . . . . . . . . . . . . . . . 174
6.6.2 Approximation by the F-Test . . . . . . . . . . . . . . . . . . 177
6.6.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . 177
6.7 CRD WITH UNEQUAL NUMBERS OF REPLICATIONS . . . . . . 179
6.7.1 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.7.2 The Model and ANOVA . . . . . . . . . . . . . . . . . . . . 180
6.7.3 Comparing Randomization Test and F-Test . . . . . . . . . . 180
6.8 NUMBER OF REPLICATIONS . . . . . . . . . . . . . . . . . . . . 180
6.8.1 Power of the F-Test . . . . . . . . . . . . . . . . . . . . . . . 182
6.8.2 Smallest Detectable Difference . . . . . . . . . . . . . . . . . 184
6.8.3 Practical Considerations . . . . . . . . . . . . . . . . . . . . 185
6.9 SUBSAMPLING IN A CRD . . . . . . . . . . . . . . . . . . . . . . 191
6.9.1 Subsampling Model . . . . . . . . . . . . . . . . . . . . . . 191
6.9.2 Inferences with Subsampling . . . . . . . . . . . . . . . . . . 193
6.9.3 Comparison of CRDs without and with Subsampling . . . . . 193
6.10 TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.10.1 Nonadditivity in the General Sense . . . . . . . . . . . . . . 196
6.10.2 Nonconstancy of Variances . . . . . . . . . . . . . . . . . . . 197
6.10.3 Choice of Transformation . . . . . . . . . . . . . . . . . . . 198
6.10.4 Power Transformations . . . . . . . . . . . . . . . . . . . . . 200
6.1 1 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 201
6.12 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7 Comparisons of Treatments 213
7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.2 COMPARISONS FOR QUALITATIVE TREATMENTS . . . . . . . 213
7.2.1 Treatment Contrasts . . . . . . . . . . . . . . . . . . . . . . 214
7.2.2 Orthogonal Contrasts . . . . . . . . . . . . . . . . . . . . . . 214
7.2.3 Partitioning the Treatment Sum of Squares . . . . . . . . . . 215
7.3 ORTHOGONALITY AND ORTHOGONAL COMPARISONS . . . . 218
7.4 COMPARISONSFORQUANTITATIVETREATMENTS . . . . . . 219
7.4.1 Comparisons for Treatments with Equidistant Levels . . . . . 219
7.4.2 Use of Orthogonal Polynomials . . . . . . . . . . . . . . . . 220
7.4.3 Contrast Sums of Squares and the ANOVA . . . . . . . . . . 223
7.5 MULTIPLE COMPARISON PROCEDURES . . . . . . . . . . . . . 224
7.5.1 Multiple Comparisons and Error Rates . . . . . . . . . . . . . 224
7.5.2 Least Significant Difference Test . . . . . . . . . . . . . . . . 225
7.5.3 Bonferroni t-Statistics . . . . . . . . . . . . . . . . . . . . . 225
7.5.4 Studentized Range Procedure . . . . . . . . . . . . . . . . . 226
7.5.5 Duncan’s Multiple Range Test . . . . . . . . . . . . . . . . . 226
7.5.6 Scheffk’s Procedure . . . . . . . . . . . . . . . . . . . . . . 227
7.5.7 Comparisons with a Control . . . . . . . . . . . . . . . . . . 227
7.5.8 Alternatives to Tests Based on Normality . . . . . . . . . . . 228
7.6 GROUPING TREATMENTS . . . . . . . . . . . . . . . . . . . . . . 229
7.7 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 230
7.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236


8 Use of Supplementary Information
8.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 MOTIVATION OF THE PROCEDURE . . . . . . . . . . . . . . . .
8.3 ANALYSIS OF COVARIANCE PROCEDURE . . . . . . . . . . . .
8.3.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Least Squares Analysis . . . . . . . . . . . . . . . . . . . . .
8.3.3 Least Squares Means . . . . . . . . . . . . . . . . . . . . . .
8.3.4 Formulation in Matrix Notation . . . . . . . . . . . . . . . .
8.3.5 ANOVA Table . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 TREATMENT COMPARISONS . . . . . . . . . . . . . . . . . . . .
8.4.1 Preplanned Comparisons . . . . . . . . . . . . . . . . . . . .
8.4.2 Multiple Comparison Procedures . . . . . . . . . . . . . . .
8.5 VIOLATION OF ASSUMPTIONS . . . . . . . . . . . . . . . . . . .
8.5.1 Linear Relationship between x and y . . . . . . . . . . . . . .
8.5.2 Common Slope . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.3 Covariates Affected by Treatments . . . . . . . . . . . . . . .
8.5.4 Normality Assumption . . . . . . . . . . . . . . . . . . . . .
8.6 ANALYSIS OF COVARIANCE WITH
SUBS AMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 CASE OF SEVERAL COVARIATES . . . . . . . . . . . . . . . . .
8.7.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7.2 Two Covariates . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . .
8.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Randomized Block Designs
9.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 RANDOMIZED COMPLETE BLOCK DESIGN . . . . . . . . . . .
9.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Derived Linear Model . . . . . . . . . . . . . . . . . . . . .
9.2.3 Estimation of Treatment Contrasts . . . . . . . . . . . . . . .
9.2.4 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . .
9.2.5 Randomization Test and F-Test . . . . . . . . . . . . . . . .
9.2.6 Additivity in the Broad Sense . . . . . . . . . . . . . . . . .
9.2.7 Subsampling in an RCBD . . . . . . . . . . . . . . . . . . .
9.3 RELATIVE EFFICIENCY OF THE RCBD . . . . . . . . . . . . . .
9.3.1 Question of Effectiveness of Blocking . . . . . . . . . . . . .
9.3.2 Use of Uniformity Trials . . . . . . . . . . . . . . . . . . . .
9.3.3 Interpretation and Use of Relative Efficiency . . . . . . . . .
9.4 ANALYSIS OF COVARIANCE . . . . . . . . . . . . . . . . . . . .
9.4.1 TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Least Squares Analysis . . . . . . . . . . . . . . . . . . . . .
9.4.3 The ANOVA Table . . . . . . . . . . . . . . . . . . . . . . .
9.5 MISSING OBSERVATIONS . . . . . . . . . . . . . . . . . . . . . .
9.5.1 Estimating a Missing Observation . . . . . . . . . . . . . . .
9.5.2 Using the Estimated Missing Observation . . . . . . . . . . .

9.5.3 Several Missing Observations . . . . . . . . . . . . . . . . . 298
9.6 NONADDITIVITY IN THE RCBD . . . . . . . . . . . . . . . . . . 300
9.6.1 The Problem of Nonadditivity . . . . . . . . . . . . . . . . . 300
9.6.2 General Model for Nonadditivity . . . . . . . . . . . . . . . . 300
Nonadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . 302
9.6.4 Testing for Nonadditivity . . . . . . . . . . . . . . . . . . . . 303
9.6.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 305
9.6.7 Several Blocking Factors . . . . . . . . . . . . . . . . . . . . 306
9.6.8 Dealing with Block-Treatment Interaction . . . . . . . . . . . 312
9.7 GENERALIZED RANDOMIZED BLOCK DESIGN . . . . . . . . . 314
9.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.7.2 Derived Linear Model . . . . . . . . . . . . . . . . . . . . . 314
9.7.3 TheANOVATable . . . . . . . . . . . . . . . . . . . . . . . 317
9.7.4 Analyzing Block-Treatment Interaction . . . . . . . . . . . . 319
9.7.5 A More General Formulation . . . . . . . . . . . . . . . . . . 323
9.7.6 Random Block Effects . . . . . . . . . . . . . . . . . . . . . 324
9.7.7 Using Satterthwaite’s Procedure . . . . . . . . . . . . . . . . 326
9.8 INCOMPLETE BLOCK DESIGNS . . . . . . . . . . . . . . . . . . 328
Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
9.8.2 Balanced Incomplete Block Designs . . . . . . . . . . . . . . 330
9.8.3 BalancedTreatment IncompleteBlockDesigns . . . . . . . . 333
9.8.4 Partially Balanced Incomplete Block Designs . . . . . . . . . 335
9.8.5 Extended Block Designs . . . . . . . . . . . . . . . . . . . . 337
9.8.6 Some General Remarks . . . . . . . . . . . . . . . . . . . . . 338
9.9 SYSTEMATIC BLOCK DESIGNS . . . . . . . . . . . . . . . . . . 340
9.9.1 Dealing with Trends . . . . . . . . . . . . . . . . . . . . . . 340
9.9.2 Trend-free Designs . . . . . . . . . . . . . . . . . . . . . . . 341
9.10 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 343
9.11 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
9.6.3 One Blocking Factor: A Specific Model for
9.6.5 Tukey’s Test for Nonadditivity . . . . . . . . . . . . . . . . . 303
9.8.1 General Notion of Designs with Incomplete
10 Latin Square Type Designs 373
10.1 INTRODUCTION AND MOTIVATION . . . . . . . . . . . . . . . . 373
10.2 LATIN SQUARE DESIGN . . . . . . . . . . . . . . . . . . . . . . . 374
10.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
10.2.2 Transformation Sets and Randomization . . . . . . . . . . . . 376
10.2.3 Derived Linear Model . . . . . . . . . . . . . . . . . . . . . 377
10.2.4 Estimation of Treatment Contrasts . . . . . . . . . . . . . . . 380
10.2.5 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . 382
The Model under Additivity in the Broad Sense . . . . . . . .
10.2.7 Consequences of Nonadditivity . . . . . . . . . . . . . . . . 386
10.2.8 Investigating Nonadditivity . . . . . . . . . . . . . . . . . . . 387
10.2.9 Miscellaneous Remarks . . . . . . . . . . . . . . . . . . . . 389
10.3 REPLICATED LATIN SQUARES . . . . . . . . . . . . . . . . . . . 390

10.3.1 Different Scenarios for Replication . . . . . . . . . . . . . . 390
10.3.2 Rows and Columns Crossed with
10.3.3 Rows Nested in and Columns Crossed with
10.3.4 Rows and Columns Nested in Replications . . . . . . . . . . 392
10.3.5 Replication x Treatment Interaction . . . . . . . . . . . . . . 392
10.4 LATIN RECTANGLES . . . . . . . . . . . . . . . . . . . . . . . . . 393
10.5 INCOMPLETE LATIN SQUARES . . . . . . . . . . . . . . . . . . 394
10.6 ORTHOGONAL LATIN SQUARES . . . . . . . . . . . . . . . . . . 395
10.6.1 Graco-Latin Squares . . . . . . . . . . . . . . . . . . . . . . 395
10.6.2 Mutually Orthogonal Latin Squares . . . . . . . . . . . . . . 396
10.7.1 Two-Treatment Change-Over Design . . . . . . . . . . . . . 398
10.7.2 Change-Over Designs for More than Two Treatments . . . . . 401
10.7.3 Some Variations and Extensions . . . . . . . . . . . . . . . . 402
10.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
Replications . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Replications . . . . . . . . . . . . . . . . . . . . . . . . . . 391
10.7 CHANGE-OVER DESIGNS . . . . . . . . . . . . . . . . . . . . . . 397
10.8 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 404
11 Factorial Experiments: Basic Ideas 419
11.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
1 1.2 INFERENCES FROM FACTORIAL EXPERIMENTS . . . . . . . . 420
11.3 EXPERIMENTS WITH FACTORS AT TWO LEVELS . . . . . . . . 422
11.3.1 Definition of Main Effects and Interactions . . . . . . . . . . 422
1 1.3.2 Estimation of Main Effects and Interactions . . . . . . . . . . 425
11.3.3 Sums of Squares for Main Effects and Interactions . . . . . . 426
1 1.4 INTERPRETATION OF EFFECTS AND INTERACTIONS . . . . . 426
11.5 INTERACTIONS: A CASE STUDY . . . . . . . . . . . . . . . . . . 428
1 1.5.1 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 428
1 1.5.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
1 1 S.3 The Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 430
11.5.4 Separate Analyses . . . . . . . . . . . . . . . . . . . . . . . 439
1 1 S.5 Blocking by Intrinsic Factor Only . . . . . . . . . . . . . . . 440
1 1 S.6 Using the Half-normal Plot Technique . . . . . . . . . . . . . 441
1 1 S.7 The Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 443
11.5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
1 1.6 2n FACTORIALS IN INCOMPLETE BLOCKS . . . . . . . . . . . . 446
11.6.1 23 Factorial in Blocks of Size 4 . . . . . . . . . . . . . . . . 446
11.6.2 23 Factorial in Blocks of Size 2 . . . . . . . . . . . . . . . . 447
1 1.6.3 Partial Confounding . . . . . . . . . . . . . . . . . . . . . . 449
1 1.7 FRACTIONS OF 2n FACTORIALS . . . . . . . . . . . . . . . . . . 451
11.7.1 Rationale for Fractional Replication . . . . . . . . . . . . . . 451
1 1.7.2 1/2 Fraction of the 23 Factorial . . . . . . . . . . . . . . . . . 454
11.7.3 The Alias Structure . . . . . . . . . . . . . . . . . . . . . . . 454
11.7.4 1/4 Fraction of the 28 Factorial . . . . . . . . . . . . . . . . . 456
11.7.5 Systems of Confounding for Fractional Factorials . . . . . . . 457

11.8 ORTHOGONAL MAIN EFFECT PLANS FOR 2n FACTORIALS . . 462
11.9 EXPERIMENTS WITH FACTORS AT THREE LEVELS . . . . . . 464
11.9.1 The 3' Factorial . . . . . . . . . . . . . . . . . . . . . . . . 465
11.9.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
11.9.4 Systems of Confounding for the 3" Factorial . . . . . . . . . 470
1 1.9.5 Fractions of 3" Factorials . . . . . . . . . . . . . . . . . . . 472
11.9.6 Highly Fractionated 3" Factorials . . . . . . . . . . . . . . . 475
11.9.3 Formal Definition of Main Effects and Interactions . . . . . . 468
11.9.7 Systems of Confounding for Fractions of 3n
Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
476
11.10.1 Asymmetrical Factorial Experiments . . . . . . . . . . . . . 476
11 . 10.2 Confounding in 2" x 3n Factorials . . . . . . . . . . . . . . 477
Blocks of Size 18: . . . . . . . . . . . . . . . . . . . . . . . 478
Blocks of Size 12: . . . . . . . . . . . . . . . . . . . . . . . 478
Blocks of Size 9: . . . . . . . . . . . . . . . . . . . . . . . . 478
Blocks of Size 6: . . . . . . . . . . . . . . . . . . . . . . . . 478
Blocks of Size 4: . . . . . . . . . . . . . . . . . . . . . . . . 478
479
11.1 1 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 481
11.12 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
11.10 FACTORS AT TWO AND THREE LEVELS . . . . . . . . . . . . .
11.10.3 Fractions of 2m x 3n Factorials . . . . . . . . . . . . . . . .
12 Response Surface Designs 497
12.2 FORMULATION OF THE PROBLEM . . . . . . . . . . . . . . . . 498
12.3 FIRST-ORDER MODELS AND DESIGNS . . . . . . . . . . . . . . 500
12.3.1 First-Order Regression Model . . . . . . . . . . . . . . . . . 500
12.3.2 Least Squares Analysis . . . . . . . . . . . . . . . . . . . . .
12.3.3 Alternative Designs . . . . . . . . . . . . . . . . . . . . . . . 503
12.4 SECOND-ORDER MODELS AND DESIGNS . . . . . . . . . . . . 504
12.4.1 Second-Order Linear Regression . . . . . . . . . . . . . . . . 504
12.4.2 Possible Designs . . . . . . . . . . . . . . . . . . . . . . . . 505
12.4.3 Central Composite Designs . . . . . . . . . . . . . . . . . . 506
Blocking in Central Composite Designs . . . . . . . . . . . .
12.4.5 Box-Behnken Designs . . . . . . . . . . . . . . . . . . . . . 509
Hard-to-Change versus Easy-to-Change Factors . . . . . . . .
12.5 INTEGRATED MEAN SQUARED ERROR DESIGNS . . . . . . . .
12.5.1 Variance and Bias for the One-Factor Case . . . . . . . . . . 514
12.5.2 Choice of Design . . . . . . . . . . . . . . . . . . . . . . . .
12.6 SEARCHING FOR AN OPTIMUM . . . . . . . . . . . . . . . . . . 518
12.7 EXPERIMENTS WITH MIXTURES . . . . . . . . . . . . . . . . . 519
12.7.1 Defining the Problem . . . . . . . . . . . . . . . . . . . . . . 519
12.7.2 Simplex-Lattice Designs . . . . . . . . . . . . . . . . . . . . 520
12.7.3 Simplex-Centroid Designs . . . . . . . . . . . . . . . . . . . 521
12.7.4 Axial Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 521
12.7.5 Canonical Polynomials . . . . . . . . . . . . . . . . . . . . . 521

12.7.6 Including Process Variables . . . . . . . . . . . . . . . . . . 523
12.8 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 523
12.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
13 Split-Plot Type Designs 533
13.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
13.2 SIMPLE SPLIT-PLOT DESIGN . . . . . . . . . . . . . . . . . . . . 534
13.2.1 Superimposing Two Randomized Complete Block Designs . . 534
13.2.2 Derived Linear Model . . . . . . . . . . . . . . . . . . . . . 537
13.2.3 Testing of Hypotheses . . . . . . . . . . . . . . . . . . . . . 538
13.2.4 Estimating Treatment Contrasts . . . . . . . . . . . . . . . . 539
13.2.5 Testing Hypotheses about Treatment Contrasts . . . . . . . . 542
13.3 RELATIVEEFFICIENCY OFSPLIT-PLOTDESIGN . . . . . . . . 543
13.4 OTHER FORMS OF SPLIT-PLOT DESIGNS . . . . . . . . . . . . . 544
13.4.2 Split-Plot Design in Time . . . . . . . . . . . . . . . . . . . 545
13.4.4 SPD(LSD, RCBD) . . . . . . . . . . . . . . . . . . . . . . . 548
13.4.5 SPD(CRD, IBD) . . . . . . . . . . . . . . . . . . . . . . . . 549
13.4.6 SPD(GRBD, RCBD) . . . . . . . . . . . . . . . . . . . . . . 550
13.4.7 SPD(GRBD, IBD) . . . . . . . . . . . . . . . . . . . . . . . 552
13.4.1 SPD(CRD, RCBD) . . . . . . . . . . . . . . . . . . . . . . . 545
13.4.3 SPD(CRD, LSD) . . . . . . . . . . . . . . . . . . . . . . . . 547
13.4.8 SPD(IBD, RCBD) . . . . . . . . . . . . . . . . . . . . . . . 553
13.4.9 SPD(RCBD, GRBD) . . . . . . . . . . . . . . . . . . . . . . 554
13.4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
13.5.1 The Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
13.5.2 Linear Model and ANOVA . . . . . . . . . . . . . . . . . . . 557
13.5.3 Estimating Treatment Contrasts . . . . . . . . . . . . . . . . 557
13.7 EXAMPLES USING SAS@ . . . . . . . . . . . . . . . . . . . . . . 562
13.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
13.5 SPLIT-BLOCK DESIGN . . . . . . . . . . . . . . . . . . . . . . . . 555
13.6 SPLIT-SPLIT-PLOT DESIGN . . . . . . . . . . . . . . . . . . . . . 560
14 Designs with Repeated Measures 573
14.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
14.2 METHODS FOR ANALYZING REPEATED MEASURES DATA . . 574
14.2.1 Comparisons at Separate Time Points . . . . . . . . . . . . . 574
14.2.2 Use of Summary Measures . . . . . . . . . . . . . . . . . . . 575
14.2.3 Trend Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 575
14.2.4 The ANOVA Method . . . . . . . . . . . . . . . . . . . . . . 577
14.2.5 Mixed Model Analysis . . . . . . . . . . . . . . . . . . . . . 578
14.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593