Design of Comparative Experiments

Cover......Page 1
Half-title......Page 3
Serire-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
Acknowledgements......Page 15
1.1.1 Consultation......Page 17
1.1.3 Data collection......Page 18
1.1.4 Data scrutiny......Page 19
1.1.5 Analysis......Page 20
1.2.2 Replication......Page 21
1.2.4 Constraints......Page 22
1.3 An example......Page 23
1.4 Defining terms......Page 24
1.5 Linear model......Page 30
1.6 Summary......Page 31
Questions for discussion......Page 32
2.1 Completely randomized designs......Page 35
2.2 Why and how to randomize......Page 36
2.3 The treatment subspace......Page 37
2.4 Orthogonal projection......Page 39
2.6 Estimation......Page 40
2.8 Sums of squares......Page 42
2.9 Variance......Page 44
2.11 Allowing for the overall mean......Page 46
2.12 Hypothesis testing......Page 49
2.13 Sufficient replication for power......Page 51
2.14 A more general model......Page 54
Questions for discussion......Page 57
3.1 Replication of control treatments......Page 59
3.2 Comparing new treatments in the presence of a control......Page 60
3.3 Other treatment groupings......Page 63
Questions for discussion......Page 68
4.1.1 Natural discrete divisions......Page 69
4.1.3 Choice of blocking for trial management......Page 71
4.1.4 How and when to block......Page 72
4.2 Orthogonal block designs......Page 73
4.4 Models for block designs......Page 75
4.5 Analysis when blocks have fixed effects......Page 77
4.6 Analysis when blocks have random effects......Page 83
4.7 Why use blocks?......Page 84
4.8 Loss of power with blocking......Page 85
Questions for discussion......Page 87
5.1 Treatment factors and their subspaces......Page 91
5.2 Interaction......Page 93
5.3 Principles of expectation models......Page 100
5.4 Decomposing the treatment subspace......Page 103
5.5 Analysis......Page 106
5.6 Three treatment factors......Page 108
5.7 Factorial experiments......Page 113
5.8 Construction and randomization of factorial designs......Page 114
Questions for discussion......Page 115
6.1 Double blocking......Page 121
6.2 Latin squares......Page 122
6.3 Construction and randomization......Page 124
6.5 Fixed row and column effects: model and analysis......Page 126
6.6 Random row and column effects: model and analysis......Page 128
Questions for discussion......Page 132
7.1 Introduction......Page 133
7.3 Cross-over trials......Page 134
7.4 Matched pairs, matched threes, and so on......Page 135
7.6 Body parts as experimental units......Page 136
7.7 Sequential allocation to an unknown number of patients......Page 137
7.8 Safeguards against bias......Page 138
7.9 Ethical issues......Page 140
7.10 Analysis by intention to treat......Page 142
Questions for discussion......Page 143
8.1.1 The context......Page 147
8.1.4 Analysis......Page 148
8.1.5 Hypothesis testing......Page 151
8.1.6 Decreasing variance......Page 153
8.2 Treatment factors in different strata......Page 154
8.3.1 Blocking the large units......Page 162
8.3.2 Construction and randomization......Page 163
8.3.3 Model and strata......Page 164
8.3.4 Analysis......Page 165
8.4 The split-plot principle......Page 168
Questions for discussion......Page 170
9.1.1 One treatment factor in a square......Page 173
9.1.2 More general row–column designs......Page 174
9.1.3 Two treatment factors in a block design......Page 175
9.1.4 Three treatment factors in an unblocked design......Page 177
9.2 Graeco-Latin squares......Page 178
Finite fields If n is a power of a prime but not itself prime, use a similar construction using......Page 179
Product method If S1 and T1 are mutually orthogonal Latin squares of order n1 and S2......Page 181
9.3.3 Three treatment factors in a block design......Page 182
Questions for discussion......Page 183
10.2.1 Factors and their classes......Page 185
10.2.2 Aliasing......Page 186
10.3.1 The infimum of two factors......Page 187
10.3.2 The supremum of two factors......Page 188
10.4 Hasse diagrams......Page 191
10.6.1 Definition of orthogonality......Page 194
10.6.2 Projection matrices commute......Page 195
10.6.3 Proportional meeting......Page 196
10.6.5 A chain of factors......Page 197
10.7.1 A second subspace for each factor......Page 198
10.7.2 Effects and sums of squares......Page 200
10.8.1 Degrees of freedom......Page 201
10.8.2 Sums of squares......Page 203
10.9.1 Conditions on treatment factors......Page 205
10.9.2 Collections of expectation models......Page 206
10.10.1 Conditions on plot factors......Page 209
10.10.2 Variance and covariance......Page 210
10.10.3 Matrix formulation......Page 211
10.11 Randomization......Page 212
10.12.1 Desirable properties......Page 213
10.12.3 Locating treatment subspaces......Page 214
10.12.4 Analysis of variance......Page 216
10.13 Further examples......Page 218
Questions for discussion......Page 231
11.2 Balance......Page 235
11.3 Lattice designs......Page 237
11.4 Randomization......Page 239
11.5 Analysis of balanced incomplete-block designs......Page 242
11.6 Efficiency......Page 245
11.7 Analysis of lattice designs......Page 246
11.8 Optimality......Page 249
11.9 Supplemented balance......Page 250
11.10 Row–column designs with incomplete columns......Page 251
Questions for discussion......Page 254
12.1 Confounding......Page 257
12.2 Decomposing interactions......Page 258
12.3 Constructing designs with specified confounding......Page 261
12.4 Confounding more than one character......Page 265
12.5 Pseudofactors for mixed numbers of levels......Page 267
12.6 Analysis of single-replicate designs......Page 269
12.7 Several replicates......Page 273
Questions for discussion......Page 274
13.1 Fractional replicates......Page 275
13.2 Choice of defining contrasts......Page 276
13.3 Weight......Page 278
13.4 Resolution......Page 281
13.5 Analysis of fractional replicates......Page 282
Questions for discussion......Page 286
14.1.1 Random sampling......Page 287
14.1.2 Random permutations of the plots......Page 288
14.1.4 Randomizing treatment labels......Page 289
14.1.6 Random allocation to position......Page 291
14.1.7 Restricted randomization......Page 294
14.2 Factors such as time, sex, age and breed......Page 295
14.3.2 What are the treatments?......Page 298
14.3.7 What is the design?......Page 299
14.3.11 Proposed statistical analysis......Page 300
14.4 The eight stages......Page 301
14.5 A story......Page 302
Questions for discussion......Page 306
Exercises......Page 307
Sources of examples, questions and exercises......Page 329
Further reading......Page 335
References......Page 337
Index......Page 343