Design and Analysis of Experiments, Advanced Experimental Design

813 Pages·2005·3.39 MB·English

by  Klaus Hinkelmann

Design and Analysis
of Experiments
Volume 2
Advanced Experimental Design
KLAUS HINKELMANN
Virginia Polytechnic Institute and State University
Department of Statistics
Blacksburg, VA
OSCAR KEMPTHORNE
Iowa State University
Department of Statistics
Ames, IA
A JOHN WILEY & SONS, INC., PUBLICATION

Design and Analysis
of Experiments

Design and Analysis
of Experiments
Volume 2
Advanced Experimental Design
KLAUS HINKELMANN
Virginia Polytechnic Institute and State University
Department of Statistics
Blacksburg, VA
OSCAR KEMPTHORNE
Iowa State University
Department of Statistics
Ames, IA
A JOHN WILEY & SONS, INC., PUBLICATION
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Contents
Preface xix
1 General Incomplete Block Design 1
1.1 Introduction and Examples, 1
1.2 General Remarks on the Analysis of Incomplete Block
Designs, 3
1.3 The Intrablock Analysis, 4
1.3.1 Notation and Model, 4
1.3.2 Normal and Reduced Normal Equations, 5
1.3.3 The C Matrix and Estimable Functions, 7
1.3.4 Solving the Reduced Normal Equations, 8
1.3.5 Estimable Functions of Treatment Effects, 10
1.3.6 Analyses of Variance, 12
1.4 Incomplete Designs with Variable Block Size, 13
1.5 Disconnected Incomplete Block Designs, 14
1.6 Randomization Analysis, 16
1.6.1 Derived Linear Model, 16
1.6.2 Randomization Analysis of ANOVA Tables, 18
1.7 Interblock Information in an Incomplete Block Design, 23
1.7.1 Introduction and Rationale, 23
1.7.2 Interblock Normal Equations, 23
1.7.3 Nonavailability of Interblock Information, 27
1.8 Combined Intra- and Interblock Analysis, 27
1.8.1 Combining Intra- and Interblock Information, 27
1.8.2 Linear Model, 27
1.8.3 Normal Equations, 28
1.8.4 Some Special Cases, 31
v
vi CONTENTS
1.9 Relationships Among Intrablock, Interblock, and Combined
Estimation, 31
1.9.1 General Case, 32
1.9.2 Case of Proper, Equireplicate Designs, 34
1.10 Estimation of Weights for the Combined Analysis, 36
1.10.1 Yates Procedure, 37
1.10.2 Properties of Combined Estimators, 38
1.11 Maximum-Likelihood Type Estimation, 39
1.11.1 Maximum-Likelihood Estimation, 39
1.11.2 Restricted Maximum-Likelihood
Estimation, 40
1.12 Efficiency Factor of an Incomplete Block Design, 43
1.12.1 Average Variance for Treatment Comparisons for
an IBD, 43
1.12.2 Definition of Efficiency Factor, 45
1.12.3 Upper Bound for the Efficiency Factor, 47
1.13 Optimal Designs, 48
1.13.1 Information Function, 48
1.13.2 Optimality Criteria, 49
1.13.3 Optimal Symmetric Designs, 50
1.13.4 Optimality and Research, 50
1.14 Computational Procedures, 52
1.14.1 Intrablock Analysis Using SAS PROC GLM, 52
1.14.2 Intrablock Analysis Using the Absorb Option in
SAS PROC GLM, 58
1.14.3 CombinedIntra-andInterblockAnalysisUsingthe
Yates Procedure, 61
1.14.4 Combined Intra- and Interblock Analysis Using
SAS PROC MIXED, 63
1.14.5 Comparison of Estimation Procedures, 63
1.14.6 Testing of Hypotheses, 66
2 Balanced Incomplete Block Designs 71
2.1 Introduction, 71
2.2 Definition of the BIB Design, 71
2.3 Properties of BIB Designs, 72
2.4 Analysis of BIB Designs, 74
2.4.1 Intrablock Analysis, 74
2.4.2 Combined Analysis, 76
2.5 Estimation of ρ, 77
CONTENTS vii
2.6 Significance Tests, 79
2.7 Some Special Arrangements, 89
2.7.1 Replication Groups Across Blocks, 89
2.7.2 Grouped Blocks, 91
2.7.3 α-Resolvable BIB Designs with Replication
Groups Across Blocks, 96
2.8 Resistant and Susceptible BIB Designs, 98
2.8.1 Variance-Balanced Designs, 98
2.8.2 Definition of Resistant Designs, 99
2.8.3 Characterization of Resistant Designs, 100
2.8.4 Robustness and Connectedness, 103
3 Construction of Balanced Incomplete Block Designs 104
3.1 Introduction, 104
3.2 Difference Methods, 104
3.2.1 Cyclic Development of Difference Sets, 104
3.2.2 Method of Symmetrically Repeated
Differences, 107
3.2.3 FormulationinTermsofGaloisFieldTheory, 112
3.3 Other Methods, 113
3.3.1 Irreducible BIB Designs, 113
3.3.2 Complement of BIB Designs, 113
3.3.3 Residual BIB Designs, 114
3.3.4 Orthogonal Series, 114
3.4 Listing of Existing BIB Designs, 115
4 Partially Balanced Incomplete Block Designs 119
4.1 Introduction, 119
4.2 Preliminaries, 119
4.2.1 Association Scheme, 120
4.2.2 Association Matrices, 120
4.2.3 Solving the RNE, 121
4.2.4 Parameters of the Second Kind, 122
4.3 Definition and Properties of PBIB Designs, 123
4.3.1 Definition of PBIB Designs, 123
4.3.2 Relationships Among Parameters of a PBIB
Design, 125
4.4 Association Schemes and Linear Associative Algebras, 127
4.4.1 Linear Associative Algebra of Association
Matrices, 127