Permutation Tests for Complex Data - Fortunato Pesarin

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Résumé :

Preface xvii

Acknowledgments xxi

Acronyms xxiii

Notation xxv

About the Companion Website xxix

1 Introduction 1

1.1 On Permutation Analysis 1

1.2 Basic Notation 6

1.3 The Permutation - Conditional - Testing Principle 8

1.3.1 Nonparametric Family of Distributions 9

1.3.2 The Permutation - Conditional - Testing Principle 10

1.4 Permutation Approaches 13

1.5 When and Why Conditioning Is Appropriate 15

1.6 Randomization and Permutation 18

1.7 Computational Aspects 19

1.8 A Problem with Paired Observations 20

1.8.1 Modelling Responses 21

1.8.2 Symmetry Induced by Exchangeability 23

1.8.3 Further Aspects 24

1.8.4 Student's t-Paired Solution 24

1.8.5 The Signed Rank Solution 26

1.8.6 The McNemar Solution 27

1.9 The Permutation Solution 28

1.9.1 General Aspects 28

1.9.2 The Permutation Sample Space 28

1.9.3 The Conditional Monte Carlo Method 29

1.9.3.1 An Algorithm for Inspecting Permutation Sample Spaces 29

1.9.3.2 Approximating the Permutation Distribution 31

1.9.4 Problems and Exercises 32

1.10 A Two-Sample Problem 33

1.10.1 Modelling Responses 34

1.10.2 The Student t Solution 36

1.10.3 The Permutation Solution 36

1.10.4 Rank Solutions 38

1.10.5 Fisher's Exact Probability Solution 39

1.10.6 Problems and Exercises 39

1.11 One-Way ANOVA 40

1.11.1 Modelling Responses 41

1.11.2 Permutation Solutions 42

1.11.3 Problems and Exercises 43

2 Theory of One-Dimensional Permutation Tests 45

2.1 Introduction 45

2.1.1 Notation and Basic Assumptions 45

2.1.2 The Conditional Reference Space 49

2.1.3 Conditioning on a Set of Sufficient Statistics 55

2.2 Definition of Permutation Tests 59

2.2.1 General Aspects 59

2.2.2 Randomized Permutation Tests 60

2.2.3 Non-Randomized Permutation Tests 61

2.2.4 Unbiasedness of Permutation Tests 63

2.3 Some Useful Test Statistics 65

2.4 Equivalence of Permutation Statistics 69

2.4.1 Two Examples 70

2.4.2 Problems and Exercises 72

2.5 Further Relevant Properties 74

2.5.1 The p-Value-Like Statistic 74

2.5.2 Unbiasedness of p-Value-Like Test Statistics 75

2.5.3 Weak Consistency of Permutation Tests 77

2.5.4 A CMC Algorithm for Estimating the p-Value-Like Statistic 80

2.6 Some Arguments for Choosing Permutation Tests 82

2.7 Power Functions of Permutation Tests 87

2.7.1 Definition and Algorithm for the Conditional Power 87

2.7.2 The Conditional ROC Curve 90

2.7.3 Definition and Algorithm for the Unconditional Power: Fixed Effects 90

2.7.4 Unconditional Power: Random Effects 92

2.7.5 Comments on Power Functions 92

2.8 Permutation Confidence Interval for ? 93

2.8.1 Problems and Exercises 98

2.9 Generalizing Inference from Conditional to Unconditional 99

2.10 Optimal Properties 103

2.10.1 Problems and Exercises 105

2.11 Some Further Asymptotic Properties 107

2.11.1 Introduction 107

2.11.2 Two Basic Theorems 107

2.12 Permutation Central Limit Theorems 109

2.12.1 Basic Notions 109

2.12.2 Permutation Central Limit Theorems 110

2.12.3 Problems and Exercises 113

3 Some Examples of Unidimensional Permutation Tests 117

3.1 Examples of One-Sample Problems 117

3.1.1 A Problem with Repeated Observations 125

3.1.1.1 Friedman's Rank Test 126

3.1.1.2 A Permutation Solution 126

3.1.1.3 An Example 127

3.1.2 Problems and Exercises 128

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Biographie:

TIMELY UPDATE OF A POPULAR EDITION ON PERMUTATION TESTING WITH NUMEROUS CASE STUDIES INCLUDED THROUGHOUT

The newly revised and updated Second Edition of Permutation Tests for Complex Data describes permutation tests from the point of view of experimental design, with methodological details and illustrating the process of devising an appropriate permutation test through case studies. In addition to the text, this book includes two open source packages for permutation tests in Python and R which include a comprehensive code base to implement common permutation tests as well as code to implement each of the book's case studies.

The focus of this book is the permutation approach to a variety of univariate and multivariate problems of hypothesis testing in a typical nonparametric framework. The book examines the most up-to-date methodologies of univariate and multivariate permutation testing, includes real case studies from both experimental and observational studies, and presents and discusses solutions to the most important and frequently encountered real problems in multivariate analyses.

Written by two highly qualified authors in the field of nonparametrics and applied statistics, Permutation Tests for Complex Data includes information on sample topics including:

• Theory of one-dimensional and multi-dimensional permutation tests, covering test statistics, arguments for selecting permutation tests, and examples of one-sample and multi-sample problems
• Multiplicity control and closed testing, covering raw and adjusted p-values, the MinP Bonferroni-Holm procedure, and weighted methods for controlling FWE and FDR
• Multivariate categorical variables, covering stochastic ordering, tests on moments for ordered variables, and heterogeneity comparisons
• NPC tests for survival analysis and shape analysis, covering analysis of PERC data and analysis with correlated landmarks

Presenting a thorough overview of permutation testing with both formal description and proofs, Permutation Tests for Complex Data is an excellent introduction to permutation tests for graduate-level statistics or data science courses and will be ideal as a handbook for researchers hoping to use the open source code....

Sommaire:

FORTUNATO PESARIN is Emeritus Professor at the Department of Statistical Sciences, University of Padova, Italy.

LUIGI SALMASO is Professor of Statistics at the Department of Management and Engineering, University of Padova, Italy....