Combinatorial mathematicians and statisticians have made a wide range of contributions to the development of block designs, and this book brings together much of that work. The designs developed for a specific problem are used in a variety of different settings. Applications include controlled sampling, randomized response, validation and valuation studies, intercropping experiments, brand cross-effect designs, lotto and tournaments. The intra- and inter- block, nonparametric and covariance analysis are discussed for general block designs, and the concepts of connectedness, orthogonality, and all types of balances in designs are carefully summarized. Readers are also introduced to the designs currently playing a prominent role in the field: alpha designs, trend-free designs, balanced treatment-control designs, nearest neighbor designs, and nested designs. This book provides the important background results required by researchers in block designs and related areas and prepares them for more complex research on the subject.
Author(s): Damaraju Raghavarao, Lakshmi V. Padgett
Series: Series on Applied Mathematics
Edition: illustrated edition
Publisher: World Scientific Pub Co (
Year: 2005
Language: English
Pages: 225
Contents......Page 9
Preface......Page 13
1.1 The Model......Page 15
1.2 Estimability and Best Linear Unbiased Estimators......Page 16
1.3 Least Squares Estimates......Page 20
1.4 Error Functions......Page 21
1.6 Distributions of Quadratic Forms......Page 23
1.7 Tests of Linear Hypotheses......Page 27
2.1 Preliminaries......Page 30
2.2 Intrablock Analysis of Connected Designs......Page 31
2.3 A Numerical Example......Page 37
2.4 Analysis of Incomplete Block Designs with Recovery of Interblock Information......Page 38
2.5 Nonparametric Analysis......Page 40
2.6 Orthogonality......Page 42
2.7 Variance and Combinatorial Balance......Page 44
2.8 Efficiency Balance......Page 49
2.9 Calinski Patterns......Page 51
2.10 Optimality......Page 52
2.11 Transformations......Page 54
2.12 Covariance Analysis......Page 56
3.1 Analysis with Fixed Block Effects......Page 59
3.2 Analysis with Random Block Effects......Page 63
3.3 Unequal Error Variances......Page 64
3.5 Treatment Block Interactions......Page 65
4.1 Definitions and Basic Results......Page 68
4.2 Intra- and Inter-Block Analysis......Page 70
4.3 Set Structures and Parametric Relations......Page 72
4.4 Related Designs......Page 79
4.5 Construction of BIB Designs from Finite Geometries......Page 85
4.6 Construction by the Method of Differences......Page 86
4.8 Non-Existence Results......Page 91
4.9 Concluding Remarks......Page 93
5.1 Finite Sample Support and Controlled Sampling......Page 100
5.2 Randomized Response Procedure......Page 101
5.4 Group Testing......Page 104
5.5 Fractional Plans to Estimate Main Effects and Two-Factor Interactions Inclusiveof aSpeci.cFactor......Page 107
5.6 Box-Behnken Designs......Page 108
5.7 Intercropping Experiments......Page 110
5.8 Valuation Studies......Page 112
5.9 Tournament and Lotto Designs......Page 118
5.10 Balanced Half-Samples......Page 120
6.1 Introduction......Page 122
6.2 Inequalities on b......Page 123
6.3 Resistance of Variance Balance for the Loss of a Treatment......Page 124
6.4 Constructions......Page 125
6.5 A Cross-Effects Model......Page 128
7.1 Dual Designs—Linked Block Designs......Page 131
7.3 Optimality......Page 133
7.4 Application of LB Designs in Successive Sampling......Page 135
8.1 Definitions and Preliminaries......Page 138
8.2 Some Known Association Schemes......Page 141
8.3 Intra-Block Analysis......Page 149
8.4 Some Combinatorics and Constructions......Page 153
8.5 Partial Geometric Designs......Page 160
8.6 Applications to Group Testing Experiments......Page 161
8.8 Applications in Intercropping......Page 165
8.9 Concluding Remarks......Page 167
9.1 Introduction......Page 169
9.2 Square Lattice Designs......Page 170
9.4 Rectangular Lattice......Page 173
10.1 α-Designs......Page 176
10.2 Trend-Free Designs......Page 178
10.3 Balanced Treatment Incomplete Block Designs......Page 179
10.4 Nested Block Designs......Page 181
10.5 Nearest Neighbor Designs......Page 183
10.6 Augmented Block Designs......Page 184
10.7 Computer Aided Block Designs......Page 185
10.8 Design for Identifying Differentially Expressed Genes......Page 186
10.9 Symmetrical Factorial Experiments with Correlated Observations in Blocks......Page 189
References and Selected Bibliography......Page 193
Author Index......Page 217
Subject Index......Page 221
