Design of Comparative Experiments

by R. A. Bailey

• ISBN13: 9780521865067
• ISBN10: 0521865069
• Format: Hardcover
• Copyright: 2008-06-08
• Publisher: Cambridge University Press

Summary

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.

Author Biography

R. A. Bailey has been Professor of Statistics at Queen Mary, University of London since 1994.

Table of Contents

Preface	p. xi
Forward look	p. 1
Stages in a statistically designed experiment	p. 1
Consultation	p. 1
Statistical design	p. 2
Data collection	p. 2
Data scrutiny	p. 3
Analysis	p. 4
Interpretation	p. 5
The ideal and the reality	p. 5
Purpose of the experiment	p. 5
Replication	p. 5
Local control	p. 6
Constraints	p. 6
Choice	p. 7
An example	p. 7
Defining terms	p. 8
Linear model	p. 14
Summary	p. 15
Questions for discussion	p. 16
Unstructured experiments	p. 19
Completely randomized designs	p. 19
Why and how to randomize	p. 20
The treatment subspace	p. 21
Orthogonal projection	p. 23
Linear model	p. 24
Estimation	p. 24
Comparison with matrix notation	p. 26
Sums of squares	p. 26
Variance	p. 28
Replication: equal or unequal?	p. 30
Allowing for the overall mean	p. 30
Hypothesis testing	p. 33
Sufficient replication for power	p. 35
A more general model	p. 38
Questions for discussion	p. 41
Simple treatment structure	p. 43
Replication of control treatments	p. 43
Comparing new treatments in the presence of a control	p. 44
Other treatment groupings	p. 47
Questions for discussion	p. 52
Blocking	p. 53
Types of block	p. 53
Natural discrete divisions	p. 53
Continuous gradients	p. 55
Choice of blocking for trial management	p. 55
How and when to block	p. 56
Orthogonal block designs	p. 57
Construction and randomization	p. 59
Models for block designs	p. 59
Analysis when blocks have fixed effects	p. 61
Analysis when blocks have random effects	p. 67
Why use blocks?	p. 68
Loss of power with blocking	p. 69
Questions for discussion	p. 71
Factorial treatment structure	p. 75
Treatment factors and their subspaces	p. 75
Interaction	p. 77
Principles of expectation models	p. 84
Decomposing the treatment subspace	p. 87
Analysis	p. 90
Three treatment factors	p. 92
Factorial experiments	p. 97
Construction and randomization of factorial designs	p. 98
Factorial treatments plus control	p. 99
Questions for discussion	p. 99
Row-column designs	p. 105
Double blocking	p. 105
Latin squares	p. 106
Construction and randomization	p. 108
Orthogonal subspaces	p. 110
Fixed row and column effects: model and analysis	p. 110
Random row and column effects: model and analysis	p. 112
Questions for discussion	p. 116
Experiments on people and animals	p. 117
Introduction	p. 117
Historical controls	p. 118
Cross-over trials	p. 118
Matched pairs, matched threes, and so on	p. 119
Completely randomized designs	p. 120
Body parts as experimental units	p. 120
Sequential allocation to an unknown number of patients	p. 121
Safeguards against bias	p. 122
Ethical issues	p. 124
Analysis by intention to treat	p. 126
Questions for discussion	p. 127
Small units inside large units	p. 131
Experimental units bigger than observational units	p. 131
The context	p. 131
Construction and randomization	p. 132
Model and strata	p. 132
Analysis	p. 132
Hypothesis testing	p. 135
Decreasing variance	p. 137
Treatment factors in different strata	p. 138
Split-plot designs	p. 146
Blocking the large units	p. 146
Construction and randomization	p. 147
Model and strata	p. 148
Analysis	p. 149
Evaluation	p. 152
The split-plot principle	p. 152
Questions for discussion	p. 154
More about Latin squares	p. 157
Uses of Latin squares	p. 157
One treatment factor in a square	p. 157
More general row-column designs	p. 158
Two treatment factors in a block design	p. 159
Three treatment factors in an unblocked design	p. 161
Graeco-Latin squares	p. 162
Uses of Graeco-Latin squares	p. 166
Superimposed design in a square	p. 166
Two treatment factors in a square	p. 166
Three treatment factors in a block design	p. 166
Four treatment factors in an unblocked design	p. 167
Questions for discussion	p. 167
The calculus of factors	p. 169
Introduction	p. 169
Relations on factors	p. 169
Factors and their classes	p. 169
Aliasing	p. 170
One factor finer than another	p. 171
Two special factors	p. 171
Operations on factors	p. 171
The infimum of two factors	p. 171
The supremum of two factors	p. 172
Uniform factors	p. 175
Hasse diagrams	p. 175
Subspaces defined by factors	p. 178
One subspace per factor	p. 178
Fitted values and crude sums of squares	p. 178
Relations between subspaces	p. 178
Orthogonal factors	p. 178
Definition of orthogonality	p. 178
Projection matrices commute	p. 179
Proportional meeting	p. 180
How replication can affect orthogonality	p. 181
A chain of factors	p. 181
Orthogonal decomposition	p. 182
A second subspace for each factor	p. 182
Effects and sums of squares	p. 184
Calculations on the Hasse diagram	p. 185
Degrees of freedom	p. 185
Sums of squares	p. 187
Orthogonal treatment structures	p. 189
Conditions on treatment factors	p. 189
Collections of expectation models	p. 190
Orthogonal plot structures	p. 193
Conditions on plot factors	p. 193
Variance and covariance	p. 194
Matrix formulation	p. 195
Strata	p. 196
Randomization	p. 196
Orthogonal designs	p. 197
Desirable properties	p. 197
General definition	p. 198
Locating treatment subspaces	p. 198
Analysis of variance	p. 200
Further examples	p. 202
Questions for discussion	p. 215
Incomplete-block designs	p. 219
Introduction	p. 219
Balance	p. 219
Lattice designs	p. 221
Randomization	p. 223
Analysis of balanced incomplete-block designs	p. 226
Efficiency	p. 229
Analysis of lattice designs	p. 230
Optimality	p. 233
Supplemented balance	p. 234
Row-column designs with incomplete columns	p. 235
Questions for discussion	p. 238
Factorial designs in incomplete blocks	p. 241
Confounding	p. 241
Decomposing interactions	p. 242
Constructing designs with specified confounding	p. 245
Confounding more than one character	p. 249
Pseudofactors for mixed numbers of levels	p. 251
Analysis of single-replicate designs	p. 253
Several replicates	p. 257
Questions for discussion	p. 258
Fractional factorial designs	p. 259
Fractional replicates	p. 259
Choice of defining contrasts	p. 260
Weight	p. 262
Resolution	p. 265
Analysis of fractional replicates	p. 266
Questions for discussion	p. 270
Backward look	p. 271
Randomization	p. 271
Random sampling	p. 271
Random permutations of the plots	p. 272
Random choice of plan	p. 273
Randomizing treatment labels	p. 273
Randomizing instances of each treatment	p. 275
Random allocation to position	p. 275
Restricted randomization	p. 278
Factors such as time, sex, age and breed	p. 279
Writing a protocol	p. 282
What is the purpose of the experiment?	p. 282
What are the treatments?	p. 282
Methods	p. 283
What are the experimental units?	p. 283
What are the observational units?	p. 283
What measurements are to be recorded?	p. 283
What is the design?	p. 283
Justification for the design	p. 284
Randomization used	p. 284
Plan	p. 284
Proposed statistical analysis	p. 284
The eight stages	p. 285
A story	p. 286
Questions for discussion	p. 290
Exercises	p. 291
Sources of examples, questions and exercises	p. 313
Further reading	p. 319
References	p. 321
Index	p. 327
Table of Contents provided by Ingram. All Rights Reserved.