Simultaneous Statistical Inference (eBook)

Preface 6
Contents 8
Acronyms 12
1 The Problem of Simultaneous Inference 14
1.1 Sources of Multiplicity 16
1.2 Multiple Hypotheses Testing 17
1.2.1 Measuring and Controlling Errors 17
1.2.2 Structured Systems of Hypotheses 21
1.3 Relationships to Other Simultaneous Statistical Inference Problems 22
1.4 Contributions of this Work 24
References 25
Part IGeneral Theory 27
2 Some Theory of p-values 28
2.1 Randomized p-values 31
2.1.1 Randomized p-values in Discrete Models 31
2.1.2 Randomized p-values for Testing Composite Null Hypotheses 32
2.2 p-value Models 33
2.2.1 The iid.-Uniform Model 33
2.2.2 Dirac-Uniform Configurations 35
2.2.3 Two-Class Mixture Models 36
2.2.4 Copula Models Under Fixed Margins 37
2.2.5 Further Joint Models 37
References 38
3 Classes of Multiple Test Procedures 40
3.1 Margin-Based Multiple Test Procedures 41
3.1.1 Single-Step Procedures 41
3.1.2 Stepwise Rejective Multiple Tests 43
3.1.3 Data-Adaptive Procedures 46
3.2 Multivariate Multiple Test Procedures 48
3.2.1 Resampling-Based Methods 48
3.2.2 Methods Based on Central Limit Theorems 49
3.2.3 Copula-Based Methods 49
3.3 Closed Test Procedures 51
References 54
4 Simultaneous Test Procedures 57
4.1 Three Important Families of Multivariate Probability Distributions 60
4.1.1 Multivariate Normal Distributions 60
4.1.2 Multivariate t-distributions 61
4.1.3 Multivariate Chi-Square Distributions 61
4.2 Projection Methods Under Asymptotic Normality 62
4.3 Probability Bounds and Effective Numbers of Tests 66
4.3.1 Sum-Type Probability Bounds 67
4.3.2 Product-Type Probability Bounds 68
4.3.3 Effective Numbers of Tests 71
4.4 Simultaneous Test Procedures in Terms of p-value Copulae 72
4.5 Exploiting the Topological Structure of the Sample Space via Random Field Theory 75
References 78
5 Stepwise Rejective Multiple Tests 80
5.1 Some Concepts of Dependency 81
5.2 FWER-Controlling Step-Down Tests 83
5.3 FWER-Controlling Step-Up Tests 85
5.4 FDR-Controlling Step-Up Tests 89
5.5 FDR-Controlling Step-Up-Down Tests 91
References 98
6 Multiple Testing and Binary Classification 100
6.1 Binary Classification Under Sparsity 102
6.2 Binary Classification in Non-Sparse Models 105
6.3 Feature Selection for Binary Classification via Higher Criticism 108
References 110
7 Multiple Testing and Model Selection 111
7.1 Multiple Testing for Model Selection 112
7.2 Multiple Testing and Information Criteria 114
7.3 Multiple Testing After Model Selection 116
7.3.1 Distributions of Regularized Estimators 116
7.3.2 Two-Stage Procedures 119
7.4 Selective Inference 120
References 122
8 Software Solutions for Multiple Hypotheses Testing 124
8.1 The R Package multcomp 125
8.2 The R Package multtest 125
8.3 The R-based ?TOSS Software 126
8.3.1 The ?TOSS Simulation Tool 127
8.3.2 The ?TOSS Graphical User Interface 129
References 131
Part IIFrom Genotype to Phenotype 133
9 Genetic Association Studies 134
9.1 Statistical Modeling and Test Statistics 135
9.2 Estimation of the Proportion of Informative Loci 138
9.3 Effective Numbers of Tests via Linkage Disequilibrium 139
9.4 Combining Effective Numbers of Tests and Pre-estimation of ?0 142
9.5 Applicability of Margin-Based Methods 143
References 144
10 Gene Expression Analyses 146
10.1 Marginal Models and p-values 146
10.2 Dependency Considerations 148
10.3 Real Data Examples 151
10.3.1 Application of Generic Multiple Tests to Large-Scale Data 151
10.3.2 Copula Calibration for a Block of Correlated Genes 152
10.4 LASSO and Statistical Learning Methods 154
10.5 Gene Set Analyses and Group Structures 155
References 156
11 Functional Magnetic Resonance Imaging 159
11.1 Spatial Modeling 160
11.2 False Discovery Rate Control for Grouped Hypotheses 161
11.2.1 Clusters of Voxels 161
11.2.2 Multiple Endpoints per Location 163
11.3 Exploiting Topological Structure by Random Field Theory 164
11.4 Spatio-Temporal Models via Multivariate Time Series 165
11.4.1 Which of the Specific Factors have a Non-trivial Autocorrelation Structure? 168
11.4.2 Which of the Common Factors have a Lagged Influence on Which Xi? 169
References 169
Part IIIFurther Applications in the Life Sciences 171
12 Further Life Science Applications 172
12.1 Brain-Computer Interfacing 172
12.2 Gel Electrophoresis-Based Proteome Analysis 175
References 177
Index 179