Multiple Testing Procedures with Applications to Genomics (eBook)

Preface 7
Intended readership 8
Overview 8
Supplements 14
Acknowledgments 14
Contents 16
List of Figures 26
List of Tables 30
1 Multiple Hypothesis Testing 33
1.1 Introduction 33
1.2 Multiple hypothesis testing framework 41
2 Test Statistics Null Distribution 80
2.1 Introduction 80
2.2 Type I error control and choice of a test statistics null distribution 83
2.3 Null shift and scale-transformed test statistics null distribution 91
2.4 Null quantile-transformed test statistics null distribution 100
2.5 Null distribution for transformations of the test statistics 106
2.6 Testing single-parameter null hypotheses based on t- statistics 110
2.7 Testing multiple-parameter null hypotheses based on F- statistics 118
2.8 Weak and strong Type I error control and subset pivotality 125
2.9 Test statistics null distributions based on bootstrap and permutation data generating distributions 129
3 Overview of Multiple Testing Procedures 140
3.1 Introduction 140
3.2 Multiple testing procedures for controlling the number of Type I errors: FWER 143
3.3 Multiple testing procedures for controlling the number of Type I errors: gFWER 165
3.4 Multiple testing procedures for controlling the proportion of Type I errors among the rejected hypotheses: FDR 176
3.5 Multiple testing procedures for controlling the proportion of Type I errors among the rejected hypotheses: TPPFP 180
4 Single-Step Multiple Testing Procedures for Controlling General Type I Error Rates, T( FVn ) 192
4.1 Introduction 192
4.2 T( FVn )- controlling single- step procedures 194
4.3 Adjusted p-values for T( FVn )- controlling single- step procedures 200
4.4 T( FVn )- controlling bootstrap- based single- step procedures 205
4.5 T( FVn )- controlling two- sided single- step procedures 218
4.6 Multiple hypothesis testing and confidence regions 222
4.7 Optimal multiple testing procedures 228
5 Step-Down Multiple Testing Procedures for Controlling the Family- Wise Error Rate 230
5.1 Introduction 230
5.2 FWER-controlling step-down common-cut-off procedure based on maxima of test statistics 233
5.3 FWER-controlling step-down common-quantile procedure based on minima of unadjusted p- values 243
5.4 FWER-controlling step-up common-cut-off and common- quantile procedures 255
5.5 FWER-controlling bootstrap-based step-down procedures 258
6 Augmentation Multiple Testing Procedures for Controlling Generalized Tail Probability Error Rates 265
6.1 Introduction 265
6.2 Augmentation multiple testing procedures for controlling the generalized family- wise error rate, gFWER( k) = Pr( Vn > k)
6.3 Augmentation multiple testing procedures for controlling the tail probability for the proportion of false positives, TPPFP( q) = Pr( Vn/ Rn > q)
6.4 TPPFP-based multiple testing procedures for controlling the false discovery rate, FDR = E[ Vn/Rn] 281
6.5 General results on augmentation multiple testing procedures 286
6.6 gTP-based multiple testing procedures for controlling the generalized expected value, gEV ( g) = E[ g( Vn, Rn)] 299
6.7 Initial FWER- and gFWER-controlling multiple testing procedures 302
6.8 Discussion 303
7 Resampling-Based Empirical Bayes Multiple Testing Procedures for Controlling Generalized Tail Probability Error Rates 318
7.1 Introduction 318
7.2 gTP-controlling resampling-based empirical Bayes procedures 320
7.3 Adjusted p-values for gTP-controlling resampling- based empirical Bayes procedures 329
7.4 Finite sample rationale for gTP control by resampling- based empirical Bayes procedures 332
7.5 Formal asymptotic gTP control results for resampling- based empirical Bayes procedures 335
7.6 gTP-controlling resampling-based weighted empirical Bayes procedures 341
7.7 FDR-controlling empirical Bayes procedures 342
7.8 Discussion 347
Color Plates 349
8 Simulation Studies: Assessment of Test Statistics Null Distributions 373
8.1 Introduction 373
8.2 Bootstrap-based multiple testing procedures 376
8.3 Simulation Study 1: Tests for regression coefficients in linear models with dependent covariates and error terms 379
8.4 Simulation Study 2: Tests for correlation coefficients 388
9 Identification of Differentially Expressed and Co- Expressed Genes in High- Throughput Gene Expression Experiments 395
9.1 Introduction 395
9.2 Apolipoprotein AI experiment of Callow et al. (2000) 396
9.3 Cancer microRNA study of Lu et al. (2005) 430
10 Multiple Tests of Association with Biological Annotation Metadata 441
10.1 Introduction 441
10.2 Statistical framework for multiple tests of association with biological annotation metadata 445
10.3 The Gene Ontology 453
10.4 Tests of association between GO annotation and differential gene expression in ALL 467
10.5 Discussion 481
11 HIV-1 Sequence Variation and Viral Replication Capacity 505
11.1 Introduction 505
11.2 HIV-1 dataset of Segal et al. (2004) 505
11.3 Multiple testing procedures 507
11.4 Software implementation in SAS 509
11.5 Results 510
11.6 Discussion 512
12 Genetic Mapping of Complex Human Traits Using Single Nucleotide Polymorphisms: The ObeLinks Project 516
12.1 Introduction 516
12.2 The ObeLinks Project 518
12.3 Multiple testing procedures 522
12.4 Results 524
12.5 Discussion 528
13 Software Implementation 545
13.1 R package multtest 545
13.2 SAS macros 555
A Summary of Multiple Testing Procedures 558
B Miscellaneous Mathematical and Statistical Results 575
B.1 Probability inequalities 575
B.2 Convergence results 576
B.3 Properties of floor and ceiling functions 577
C SAS Code 579
References 584
Author Index 598
Subject Index 601