Nonparametric Hypothesis Testing E-Book

Stefano Bonnini

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 411

Veröffentlichungsjahr: 2014

Beschreibung

A novel presentation of rank and permutation tests, with accessible guidance to applications in R

Nonparametric testing problems are frequently encountered in many scientific disciplines, such as engineering, medicine and the social sciences. This book summarizes traditional rank techniques and more recent developments in permutation testing as robust tools for dealing with complex data with low sample size.

Key Features:

• Examines the most widely used methodologies of nonparametric testing.
• Includes extensive software codes in R featuring worked examples, and uses real case studies from both experimental and observational studies.
• Presents and discusses solutions to the most important and frequently encountered real problems in different fields.

Features a supporting website (www.wiley.com/go/hypothesis_testing) containing all of the data sets examined in the book along with ready to use R software codes.

Nonparametric Hypothesis Testing combines an up to date overview with useful practical guidance to applications in R, and will be a valuable resource for practitioners and researchers working in a wide range of scientific fields including engineering, biostatistics, psychology and medicine.

Leseprobe

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: David J. Balding, Noel A.C. Cressie, Garrett M. Fitzmaurice, Geof H. Givens,

Harvey Goldstein, Geert Molenberghs, David W. Scott, Adrian F.M. Smith,

Ruey S. Tsay, Sanford Weisberg

Editors Emeriti: J. Stuart Hunter, Iain M. Johnstone, Joseph B. Kadane, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

Nonparametric Hypothesis Testing

Rank and Permutation Methods with Applications in R

Stefano Bonnini

University of Ferrara, Italy

Livio Corain

University of Padova, Italy

Marco Marozzi

University of Calabria, Italy

Luigi Salmaso

University of Padova, Italy

This edition first published 2014 © 2014 John Wiley & Sons, Ltd

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

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Library of Congress Cataloging-in-Publication Data

Nonparametric hypothesis testing : rank and permutation methods with applications in R / Stefano Bonnini, Livio Corain, Marco Marozzi, Luigi Salmaso.

pages cm Includes bibliographical references and index. ISBN 978-1-119-95237-4 (cloth) 1. Nonparametric statistics. 2. Statistical hypothesis testing. 3. R (Computer program language) I. Bonnini, Stefano. II. Corain, Livio. III. Marozzi, Marco. IV. Salmaso, Luigi. QA278.8.N64 2014 519.5′4–dc23

2014020574

A catalogue record for this book is available from the British Library.

Cover image: ‘A ship for discovery’ by Serio Salmaso, 1980, Venice

ISBN: 978-1-119-95237-4

The greatest value of a picture is when it forces us to notice what we never expected to see.

                                                          J. Tukey

CONTENTS

Presentation of the book

Preface

Notation and abbreviations

1 One- and two-sample location problems, tests for symmetry and tests on a single distribution

1.1 Introduction

1.2 Nonparametric tests

1.3 Univariate one-sample tests

1.4 Multivariate one-sample tests

1.5 Univariate two-sample tests

1.6 Multivariate two-sample tests

References

2 Comparing variability and distributions

2.1 Introduction

2.2 Comparing variability

2.3 Jointly comparing central tendency and variability

2.4 Comparing distributions

References

3 Comparing more than two samples

3.1 Introduction

3.2 One-way ANOVA layout

3.3 Two-way ANOVA layout

3.4 Pairwise multiple comparisons

3.5 Multivariate multisample tests

References

4 Paired samples and repeated measures

4.1 Introduction

4.2 Two-sample problems with paired data

4.3 Repeated measures tests

References

5 Tests for categorical data

5.1 Introduction

5.2 One-sample tests

5.3 Two-sample tests on proportions or 2 × 2 contingency tables

5.4 Tests for

R

×

C

contingency tables

References

6 Testing for correlation and concordance

6.1 Introduction

6.2 Measuring correlation

6.3 Tests for independence

6.4 Tests for concordance

References

7 Tests for heterogeneity

7.1 Introduction

7.2 Statistical heterogeneity

7.3 Dominance in heterogeneity

7.4 Two-sided and multisample test

References

Appendix A Selected critical values for the null distribution of the peak- known Mack–Wolfe statistic

Appendix B Selected critical values for the null distribution of the peak- unknown Mack–Wolfe statistic

Appendix C Selected upper-tail probabilities for the null distribution of the Page

L

statistic

Appendix D

R

functions and codes

Index

Series

End User License Agreement

List of Tables

Chapter 1

Table 1.1

Table 1.2

Table 1.3

Table 1.4

Table 1.5

Table 1.6

Table 1.7

Table 1.8

Table 1.9

Table 1.10

Chapter 2

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Chapter 3

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 3.6

Table 3.7

Table 3.8

Chapter 4

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Chapter 5

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Table 5.7

Table 5.8

Table 5.9

Table 5.10

Table 5.11

Table 5.12

Table 5.13

Table 5.14

Table 5.15

Chapter 6

Table 6.1

Table 6.2

Table 6.3

Chapter 7

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 7.5

Table 7.6

Table 7.7

Table 7.8

Appendix A

Table A.1

Appendix B

Table B.1

Appendix C

Table C.1

Guide

Cover

Table of Contents

Preface

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Presentation of the book

The importance and usefulness of nonparametric methods for testing statistical hypotheses has been growing in recent years mainly due to their flexibility, their efficiency and their ease of application to several different types of problems, including most important and frequently encountered multivariate cases. By also taking account that with respect to parametric counterparts they are much less demanding in terms of required assumptions, these peculiarities of nonparametric methods are making them quite popular and widely used even by non-statisticians.

The growing availability of adequate hardware and software tools for their practical application, and in particular of free access to software environments for statistical computing like R, represents one more reason for the great success of these methods.

The recognized simplicity and good power behavior of rank and permutation tests often make them preferable to the classical parametric procedures based on the assumption of normality or other distribution laws. In particular, permutation tests are generally asymptotically as powerful as their parametric counterparts in the conditions for the latter. Moreover, when data exchangeability with respect to samples is satisfied in the null hypothesis, permutation tests are always exact in the sense that their null distributions are known for any given dataset of any sample size. On the other hand, those of parametric counterparts are often known only asymptotically. Thus for most sample sizes of practical interest, the related lack of efficiency of unidimensional permutation solutions may sometimes be compensated by the lack of approximation of parametric asymptotic competitors. For multivariate cases, especially when the number of processed variables is large in comparison with sample sizes, permutation solutions in most situations are more powerful than their parametric counterparts.

For these reasons in the specialized literature a book dedicated to rank and permutation tests, problem oriented with exhaustive but simple and easy to understand theoretical explanations, a practical guide for the application of the methods to most frequently encountered scientific problems, including related R codes and with many clearly discussed examples from several different disciplines, was lacking.

The present book fully satisfies these objectives and can be considered a practical and complete handbook for the application of the most important rank and permutation tests. The presentation style is simple and comprehensible also for non-statisticians with elementary education in statistical inference, but at the same time precise and formally rigorous in the theoretical explanations of the methods.

Fortunato Pesarin Department of StatisticsUniversity of Padova

Preface

This book deals with nonparametric statistical solutions for hypotheses testing problems and codes for the software environment R for the application of these solutions. In particular rank based and permutation procedures are presented and discussed, also considering real-world application problems related to engineering, economics, educational sciences, biology, medicine and several other scientific disciplines. Since the importance of nonparametric methods in modern statistics continues to grow, the goal of the book consists of providing effective, simple and user friendly instruments for applying these methods.

The statistical techniques are described mainly highlighting properties and applicability of the methods in relation to application problems, with the intention of providing methodological solutions to a wide range of problems. Hence this book presents a practical approach to nonparametric statistical analysis and includes comprehensive coverage of both established and recently developed methods. This ‘problem oriented’ approach makes the book useful also for non-statisticians. All the considered problems are real problems faced by the authors in their activities of academic counseling or found in the literature in their teaching and research activities. Sometimes data are exactly the same as in the original problem (and the data source is cited) but in most cases data are simulated and not real.

All R codes are commented and made available through the book’s website www.wiley.com/go/hypothesis_testing, where data used throughout the book may also be downloaded. Part of the material, including R codes, presented in the book is new and part is taken from existing publications from the literature and/or from websites of different authors providing suitable R codes. We fully recognize the authorship of each R code and a comprehensive list of useful websites is reported in Appendix D.

The book is mainly addressed to university students, in particular for undergraduate and postgraduate studies (i.e., PhD courses, Masters, etc.), statisticians and non-statisticians experts in empirical sciences, and it can also be used by practitioners with a basic knowledge in statistics interested in the same applications described in the book or in similar problems, or consultants/experts in statistics.

Chapter 1 deals with one-sample and two-sample location problems, tests for symmetry and tests on a single distribution. First of all an introduction to rank based testing procedures and to permutation testing procedures (including nonparametric combination methodology useful for multivariate or multiple tests) is presented. Then in this chapter, according to the number of response variables and to the number of samples, we distinguish four kinds of methods: univariate one-sample tests, multivariate one-sample tests, univariate two-sample tests and multivariate two-sample tests. In the first category the Kolmogorov–Smirnov test and the permutation test for symmetry are considered; in the second group of procedures the multivariate rank test for central tendency and the multivariate extension of the permutation test on symmetry are presented; among the procedures included in the third family of solutions the Wilcoxon test and the permutation test on central tendency are described; finally the multivariate extensions of the two-sample test on central tendency both with the rank based and permutation approach are discussed.

Chapter 2 presents some tests for comparing variabilities and distributions. For problems of variability comparisons the Ansari–Bradley test, the permutation Pan test and the permutation O’Brien test are considered. For jointly comparing central tendency and variability the Lapage test and the Cucconi test are presented. For problems related to comparisons of distributions the Kolmogorov–Smirnov and the Cramer–von Mises proposals are taken into account.

Chapter 3 is dedicated to multisample tests. For the one-way analysis of variance (ANOVA) layout the following methods are presented: the Kruskal–Wallis test, the permutation one-way ANOVA, the Mack–Wolfe test and the permutation test for umbrella alternatives. As regards the two-way ANOVA layout, the considered procedures are the Friedman test, the permutation test for related samples, the Page test for ordered alternatives and the permutation two-way ANOVA. Multiple comparison procedures for the Kruskal–Wallis test and for a permutation test are also considered. For multivariate and multisample problems a rank based and a permutation approach are presented.

Chapter 4 concerns problems for paired samples and repeated measures. For the two-sample test with paired data the Wilcoxon signed rank proposal and the permutation test for two dependent samples are discussed. For repeated measures problems the Friedman rank based test and a permutation test are considered.

Chapter 5 deals with tests for categorical data. Among one-sample problems the binomial test on one proportion, the McNemar test for paired data with binary variables and its multivariate extension are illustrated. Then two-sample tests for proportion comparisons and in general tests for 2 × 2 contingency tables are discussed. In particular the Fisher exact test and the permutation test for comparison of proportions are examined. The considered solutions for general problems related to R × C contingency tables are: the Anderson–Darling type permutation test, the permutation test on moments, and the chi-square permutation test.

Chapter 6 studies correlation and concordance. First, the statistical relationship between two variables is considered. The Spearman test and the Kendall test for independence are presented. Secondly, the problem of whether a set of criteria or a group of judges is concordant in ranking some objects is addressed. The Kendall–Babington Smith test and a permutation test for concordance are presented.

Finally, Chapter 7 contains a wide range of application problems and methodological solutions concerning comparisons of heterogeneity for categorical variables. The definition of statistical heterogeneity, the description of the testing problem of dominance in heterogeneity (two-sample one-sided test on heterogeneity), its two-sided and multisample extensions and the related permutation solutions are included.

We would like to express our thanks to Fortunato Pesarin, for stimulating discussions and helpful comments, to Rosa Arboretti, Eleonora Carrozzo and Iulia Cichi for helping with some R codes and in finding suitable reference literature. We also wish to thank Kathryn Sharples, Richard Davies and the John Wiley & Sons group in Chichester for their valuable publishing suggestions.

We welcome any suggestions for the improvement of the book and would be very pleased if the book provides users with new insights to the analysis of their data.

Stefano Bonnini Department of Economics and ManagementUniversity of Ferrara

Marco Marozzi Department of Economics, Statistics and FinanceUniversity of Calabria

Livio Corain and Luigi Salmaso Department of Management and EngineeringUniversity of Padova