1;Preface;6 2;Contents;8 3;Acronyms;12 4;1 The Problem of Simultaneous Inference;14 4.1;1.1 Sources of Multiplicity;16 4.2;1.2 Multiple Hypotheses Testing;17 4.2.1;1.2.1 Measuring and Controlling Errors;17 4.2.2;1.2.2 Structured Systems of Hypotheses;21 4.3;1.3 Relationships to Other Simultaneous Statistical Inference Problems;22 4.4;1.4 Contributions of this Work;24 4.5;References;25 5;Part IGeneral Theory;27 6;2 Some Theory of p-values;28 6.1;2.1 Randomized p-values;31 6.1.1;2.1.1 Randomized p-values in Discrete Models;31 6.1.2;2.1.2 Randomized p-values for Testing Composite Null Hypotheses;32 6.2;2.2 p-value Models;33 6.2.1;2.2.1 The iid.-Uniform Model;33 6.2.2;2.2.2 Dirac-Uniform Configurations;35 6.2.3;2.2.3 Two-Class Mixture Models;36 6.2.4;2.2.4 Copula Models Under Fixed Margins;37 6.2.5;2.2.5 Further Joint Models;37 6.3;References;38 7;3 Classes of Multiple Test Procedures;40 7.1;3.1 Margin-Based Multiple Test Procedures;41 7.1.1;3.1.1 Single-Step Procedures;41 7.1.2;3.1.2 Stepwise Rejective Multiple Tests;43 7.1.3;3.1.3 Data-Adaptive Procedures;46 7.2;3.2 Multivariate Multiple Test Procedures;48 7.2.1;3.2.1 Resampling-Based Methods;48 7.2.2;3.2.2 Methods Based on Central Limit Theorems;49 7.2.3;3.2.3 Copula-Based Methods;49 7.3;3.3 Closed Test Procedures;51 7.4;References;54 8;4 Simultaneous Test Procedures;57 8.1;4.1 Three Important Families of Multivariate Probability Distributions;60 8.1.1;4.1.1 Multivariate Normal Distributions;60 8.1.2;4.1.2 Multivariate t-distributions;61 8.1.3;4.1.3 Multivariate Chi-Square Distributions;61 8.2;4.2 Projection Methods Under Asymptotic Normality;62 8.3;4.3 Probability Bounds and Effective Numbers of Tests;66 8.3.1;4.3.1 Sum-Type Probability Bounds;67 8.3.2;4.3.2 Product-Type Probability Bounds;68 8.3.3;4.3.3 Effective Numbers of Tests;71 8.4;4.4 Simultaneous Test Procedures in Terms of p-value Copulae;72 8.5;4.5 Exploiting the Topological Structure of the Sample Space via Random Field Theory;75 8.6;References;78 9;5 Stepwise Rejective Multiple Tests;80 9.1;5.1 Some Concepts of Dependency;81 9.2;5.2 FWER-Controlling Step-Down Tests;83 9.3;5.3 FWER-Controlling Step-Up Tests;85 9.4;5.4 FDR-Controlling Step-Up Tests;89 9.5;5.5 FDR-Controlling Step-Up-Down Tests;91 9.6;References;98 10;6 Multiple Testing and Binary Classification;100 10.1;6.1 Binary Classification Under Sparsity;102 10.2;6.2 Binary Classification in Non-Sparse Models;105 10.3;6.3 Feature Selection for Binary Classification via Higher Criticism;108 10.4;References;110 11;7 Multiple Testing and Model Selection;111 11.1;7.1 Multiple Testing for Model Selection;112 11.2;7.2 Multiple Testing and Information Criteria;114 11.3;7.3 Multiple Testing After Model Selection;116 11.3.1;7.3.1 Distributions of Regularized Estimators;116 11.3.2;7.3.2 Two-Stage Procedures;119 11.4;7.4 Selective Inference;120 11.5;References;122 12;8 Software Solutions for Multiple Hypotheses Testing;124 12.1;8.1 The R Package multcomp;125 12.2;8.2 The R Package multtest;125 12.3;8.3 The R-based ?TOSS Software;126 12.3.1;8.3.1 The ?TOSS Simulation Tool;127 12.3.2;8.3.2 The ?TOSS Graphical User Interface;129 12.4;References;131 13;Part IIFrom Genotype to Phenotype;133 14;9 Genetic Association Studies;134 14.1;9.1 Statistical Modeling and Test Statistics;135 14.2;9.2 Estimation of the Proportion of Informative Loci;138 14.3;9.3 Effective Numbers of Tests via Linkage Disequilibrium;139 14.4;9.4 Combining Effective Numbers of Tests and Pre-estimation of ?0;142 14.5;9.5 Applicability of Margin-Based Methods;143 14.6;References;144 15;10 Gene Expression Analyses;146 15.1;10.1 Marginal Models and p-values;146 15.2;10.2 Dependency Considerations;148 15.3;10.3 Real Data Examples;151 15.3.1;10.3.1 Application of Generic Multiple Tests to Large-Scale Data;151 15.3.2;10.3.2 Copula Calibration for a Block of Correlated Genes;152 15.4;10.4 LASSO and Statistical Learning Methods;154 15.5;10.5 Gene Set Analyses and Group Structures;155 15.6;References;156 16;11 F