Design of Comparative Experiments (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 25)

This book should be on the shelf of every practising statistician who designs experiments. Good design considers units and treatments first, and then allocates treatments to units. It does not choose from a menu of named designs. This approach requires a notation for units that does not depend on the treatments applied. Most structure on the set of observational units, or on the set of treatments, can be defined by factors. This book develops a coherent framework for thinking about factors and their relationships, including the use of Hasse diagrams. These are used to elucidate structure, calculate degrees of freedom and allocate treatment subspaces to appropriate strata. Based on a one-term course the author has taught since 1989, the book is ideal for advanced undergraduate and beginning graduate courses. Examples, exercises and discussion questions are drawn from a wide range of real applications: from drug development, to agriculture, to manufacturing. Cover 1 Half-title 3 Serire-title 4 Title 5 Copyright 6 Contents 7 Preface 13 Acknowledgements 15 Chapter 1 Forward look 17 1.1 Stages in a statistically designed experiment 17 1.1.1 Consultation 17 1.1.2 Statistical design 18 1.1.3 Data collection 18 1.1.4 Data scrutiny 19 1.1.5 Analysis 20 1.1.6 Interpretation 21 1.2 The ideal and the reality 21 1.2.1 Purpose of the experiment 21 1.2.2 Replication 21 1.2.3 Local control 22 1.2.4 Constraints 22 1.2.5 Choice 23 1.3 An example 23 1.4 Defining terms 24 1.5 Linear model 30 1.6 Summary 31 Questions for discussion 32 Chapter 2 Unstructured experiments 35 2.1 Completely randomized designs 35 2.2 Why and how to randomize 36 2.3 The treatment subspace 37 2.4 Orthogonal projection 39 2.5 Linear model 40 2.6 Estimation 40 2.7 Comparison with matrix notation 42 2.8 Sums of squares 42 2.9 Variance 44 2.10 Replication: equal or unequal? 46 2.11 Allowing for the overall mean 46 2.12 Hypothesis testing 49 2.13 Sufficient replication for power 51 2.14 A more general model 54 Questions for discussion 57 Chapter 3 Simple treatment structure 59 3.1 Replication of control treatments 59 3.2 Comparing new treatments in the presence of a control 60 3.3 Other treatment groupings 63 Questions for discussion 68 Chapter 4 Blocking 69 4.1 Types of block 69 4.1.1 Natural discrete divisions 69 4.1.2 Continuous gradients 71 4.1.3 Choice of blocking for trial management 71 4.1.4 How and when to block 72 4.2 Orthogonal block designs 73 4.3 Construction and randomization 75 4.4 Models for block designs 75 4.5 Analysis when blocks have fixed effects 77 4.6 Analysis when blocks have random effects 83 4.7 Why use blocks? 84 4.8 Loss of power with blocking 85 Questions for discussion 87 Chapter 5 Factorial treatment structure 91 5.1 Treatment factors and their subspaces 91 5.2 Interaction 93 5.3 Principles of expectation models 100 5.4 Decomposing the treatment subspace 103 5.5 Analysis 106 5.6 Three treatment factors 108 5.7 Factorial experiments 113 5.8 Construction and randomization of factorial designs 114 5.9 Factorial treatments plus control 115 Questions for discussion 115 Chapter 6 Row–column designs 121 6.1 Double blocking 121 6.2 Latin squares 122 6.3 Construction and randomization 124 6.4 Orthogonal subspaces 126 6.5 Fixed row and column effects: model and analysis 126 6.6 Random row and column effects: model and analysis 128 Questions for discussion 132 Chapter 7 Experiments on people and animals 133 7.1 Introduction 133 7.2 Historical controls 134 7.3 Cross-over trials 134 7.4 Matched pairs, matched threes, and so on 135 7.5 Completely randomized designs 136 7.6 Body parts as experimental units 136 7.7 Sequential allocation to an unknown number of patients 137 7.8 Safeguards against bias 138 7.9 Ethical issues 140 7.10 Analysis by intention to treat 142 Questions for discussion 143 Chapter 8 Small units inside large units 147 8.1 Experimental units bigger than observational units 147 8.1.1 The context 147 8.1.2 Construction and randomization 148 8.1.3 Model and strata 148 8.1.4 Analysis 148 8.1.5 Hypothesis testing 151 8.1.6 Decreasing variance 153 8.2 Treatment factors in different strata 154 8.3 Split-plot designs 162 8.3.1 Blocking the large units 162 8.3.2 Construction and randomization 163 8.3.3 Model and strata 164 8.3.4 Analysis 165 8.3.5 Evaluation 168 8.4 The split-plot principle 168 Questions for discussion 170 Chapter 9 More about Latin squares 173 9.1 Uses of Latin squares 173 9.1.1 One treatment factor in a square 173 9.1.2 More general row–column designs 174 9.1.3 Two treatment factors in a block design 175 9.1.4 Three treatment factors in an unblocked design 177 9.2 Graeco-Latin squares 178 Prime numbers If n = p, a prime number, label the rows and columns of the square by 179 Finite fields If n is a power of a prime but not itself prime, use a similar construction using 179 Product method If S1 and T1 are mutually orthogonal Latin squares of order n1 and S2 181 9.3 Uses of Graeco-Latin squares 182 9.3.1 Superimposed design in a square 182 9.3.2 Two treatment factors in a square 182 9.3.3 Three treatment factors in a block design 182 9.3.4 Four treatment factors in an unblocked design 183 Questions for discussion 183 Chapter 10 The calculus of factors 185 10.1 Introduction 185 10.2 Relations on factors 185 10.2.1 Factors and their classes 185 10.2.2 Aliasing 186 10.2.3 One factor finer than another 187 10.2.4 Two special factors 187 10.3 Operations on factors 187 10.3.1 The infimum of two factors 187 10.3.2 The supremum of two factors 188 10.3.3 Uniform factors 191 10.4 Hasse diagrams 191 10.5 Subspaces defined by factors 194 10.5.1 One subspace per factor 194 10.5.2 Fitted values and crude sums of squares 194 10.5.3 Relations between subspaces 194 10.6 Orthogonal factors 194 10.6.1 Definition of orthogonality 194 10.6.2 Projection matrices commute 195 10.6.3 Proportional meeting 196 10.6.4 How replication can affect orthogonality 197 10.6.5 A chain of factors 197 10.7 Orthogonal decomposition 198 10.7.1 A second subspace for each factor 198 10.7.2 Effects and sums of squares 200 10.8 Calculations on the Hasse diagram 201 10.8.1 Degrees of freedom 201 10.8.2 Sums of squares 203 10.9 Orthogonal treatment structures 205 10.9.1 Conditions on treatment factors 205 10.9.2 Collections of expectation models 206 10.10 Orthogonal plot structures 209 10.10.1 Conditions on plot factors 209 10.10.2 Variance and covariance 210 10.10.3 Matrix formulation 211 10.10.4 Strata 212 10.11 Randomization 212 10.12 Orthogonal designs 213 10.12.1 Desirable properties 213 10.12.2 General definition 214 10.12.3 Locating treatment subspaces 214 10.12.4 Analysis of variance 216 10.13 Further examples 218 Questions for discussion 231 Chapter 11 Incomplete-block designs 235 11.1 Introduction 235 11.2 Balance 235 11.3 Lattice designs 237 11.4 Randomization 239 11.5 Analysis of balanced incomplete-block designs 242 11.6 Efficiency 245 11.7 Analysis of lattice designs 246 11.8 Optimality 249 11.9 Supplemented balance 250 11.10 Row–column designs with incomplete columns 251 Questions for discussion 254 Chapter 12 Factorial designs in incomplete blocks 257 12.1 Confounding 257 12.2 Decomposing interactions 258 12.3 Constructing designs with specified confounding 261 12.4 Confounding more than one character 265 12.5 Pseudofactors for mixed numbers of levels 267 12.6 Analysis of single-replicate designs 269 12.7 Several replicates 273 Questions for discussion 274 Chapter 13 Fractional factorial designs 275 13.1 Fractional replicates 275 13.2 Choice of defining contrasts 276 13.3 Weight 278 13.4 Resolution 281 13.5 Analysis of fractional replicates 282 Questions for discussion 286 Chapter 14 Backward look 287 14.1 Randomization 287 14.1.1 Random sampling 287 14.1.2 Random permutations of the plots 288 14.1.3 Random choice of plan 289 14.1.4 Randomizing treatment labels 289 14.1.5 Randomizing instances of each treatment 291 14.1.6 Random allocation to position 291 14.1.7 Restricted randomization 294 14.2 Factors such as time, sex, age and breed 295 14.3 Writing a protocol 298 14.3.1 What is the purpose of the experiment? 298 14.3.2 What are the treatments? 298 14.3.3 Methods 299 14.3.4 What are the experimental units? 299 14.3.5 What are the observational units? 299 14.3.6 What measurements are to be recorded? 299 14.3.7 What is the design? 299 14.3.8 Justification for the design 300 14.3.9 Randomization used 300 14.3.10 Plan 300 14.3.11 Proposed statistical analysis 300 14.4 The eight stages 301 14.5 A story 302 Questions for discussion 306 Exercises 307 Sources of examples, questions and exercises 329 Further reading 335 References 337 Index 343 Good Design Considers Units And Treatments First, And Then Allocates Treatments To Units. It Does Not Choose From A Menu Of Named Designs. This Approach Requires A Notation For Units That Does Not Depend On The Treatments Applied. Most Structure On The Set Of Observational Units, Or On The Set Of Treatments, Can Be Defined By Factors. This Book Develops A Coherent Framework For Thinking About Factors And Their Relationships, Including The Use Of Hasse Diagrams. These Are Used To Elucidate Structure, Calculate Degrees Of Freedom And Allocate Treatment Subspaces To Appropriate Strata. Based On A One-term Course The Author Has Taught Since 1989, The Book Is Ideal For Advanced Undergraduate And Beginning Graduate Courses. Examples, Exercises And Discussion Questions Are Drawn From A Wide Range Of Real Applications: From Drug Development, To Agriculture, To Manufacturing.--jacket. Forward Look -- Unstructured Experiments -- Simple Treatment Structure -- Blocking -- Factorial Treatment Structure -- Row-column Designs -- Experiments On People And Animals -- Small Units Inside Large Units -- More About Latin Squares -- The Calculus Of Factors -- Incomplete-block Designs -- Factorial Designs In Incomplete Blocks -- Fractional Factorial Designs -- Backward Look. R.a. Bailey. Includes Bibliographical References (p. 321-326) And Index. Content: Cover; Half-title; Serire-title; Title; Copyright; Contents; Preface; Chapter 1 Forward look; Chapter 2 Unstructured experiments; Chapter 3 Simple treatment structure; Chapter 4 Blocking; Chapter 5 Factorial treatment structure; Chapter 6 Row-column designs; Chapter 7 Experiments on people and animals; Chapter 8 Small units inside large units; Chapter 9 More about Latin squares; Chapter 10 The calculus of factors; Chapter 11 Incomplete-block designs; Chapter 12 Factorial designs in incomplete blocks; Chapter 13 Fractional factorial designs; Chapter 14 Backward look; Exercises Abstract: The coherent framework behind good practice; for working statisticians, advanced undergraduates, beginning graduate students Design of Comparative Experiments develops a coherent framework for thinking about factors that affect experiments and their relationships, including the use of Hasse diagrams. These diagrams are used to elucidate structure, calculate degrees of freedom and allocate treatment sub-spaces to appropriate strata. Good design considers units and treatments first, and then allocates treatments to units. Based on a one-term course the author has taught since 1989, the book is ideal for advanced undergraduate and beginning graduate courses. This book should be on the shelf of every practicing statistician who designs experiments.