This book deals with nonparametric statistical solutions for hypotheses testing problems and codes for the softwareenvironment 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 appli- cability 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 prob- lem 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 Rcode and a comprehensive list of useful websites is reported in Appendix D.The book is mainly addressed to university students, in particular for under- graduate and postgraduate studies(i.e., PhD courses, Masters,etc.),statisticians and non-statisticiansexperts inempirical 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 l 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,multivariateone-sample tests,univariate two-sample tests and multivariate twosample 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 Wilcoxontest and the permutation test oncentral 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.

Presentation of the book xi

Preface xii

Notation and abbreviations xvii

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

1.1Introduction 1

1.2Nonparametric tests 2

      1.2.1 Rank tests 2

      1.2.2 Permutation tests and combination based tests 3

1.3 Univariate one-sample tests 5

      1.3.1 The Kolmogorov goodness-of-fit test 6

      1.3.2 A univariate permutation test for symmetry 10

1.4 Multivariate one-sample tests 15

      1.4.1 Multivariate rank test for central tendency 15

      1.4.2 Multivariate permutation test for symmetry 18

      1.5 Univariate two-sample tests 20

      1.5.1 The Wilcoxon (Mann-Whitney) test 21

      1.5.2 Permutation test on central tendency 27

1.6 Multivariate two-sample tests 29

      1.6.1 Multivariate tests based on rank 29

      1.6.2 Multivariate permutation test on central tendency 34

References 37

2 Comparing variability and distributions 38

2.1 Introduction 38

2.2 Comparing variability 39

      2.2.1 The Ansari-Bradley test 40

      2.2.2 The permutation Pan test 43

      2.2.3 The permutation O'Brien test 46

2.3 Jointly comparing central tendency and variabili 49

      2.3.1 The Lepage test 50

      2.3.2 The Cucconi test 52

2.4 Comparing distributions 56

      2.4.1 The Kolmogorov-Smirnov test 56

      2.4.2 The Cramér-von Mises test 59

References 6l

3 Comparing more than two samples 65

3.1 Introduction 65

3.2 One-way ANOVA layout 66

      3.2.1 The Kruskal-Wallis test 67

      3.2.2 Permutation ANOVA in the presence of one factor 73

      3.2.3 The Mack-Wolfe test for umbrella alternatives 76

      3.2.4 Permutation test for umbrella alternatives 83

3.3 Two-way ANOVA layout 87

3.3.1 The Friedman rank test for unreplicated block design 87

3.3.2 Permutation test for related samples 89

3.3.3 The Page test for ordered alternatives 91

3.3.4 Permutation analysis of variance in the presence of two factors 93

3.4 Pairwise multiple comparisons 95

      3.4.1 Rank-based multiple comparisons for the Kruskal-Wallis test 96

      3.4.2 Permutation tests for multiple comparisons 98

3.5 Multivariate multisample tests 99

      3.5.1 A multivariate multisample rank-based test 99

      3.5.2 A multivariate multisample permutation test 103

      References 105

4 Paired samples and repeated measures 107

4.1 Introduction 107

4.2 Two-sample problems with paired data 108

      4.2.1 The Wilcoxon signed rank test 108

      4.2.2 A permutation test for paired samples 114

4.3 Repeated measures tests 116

      4.3.1 Friedman rank test for repeated measures 117

      4.3.2 A permutation test for repeated measures References 120

5Tests for categorical data 124

5.1 Introduction 124

5.2 One-sample tests 125

      5.2.1 Binomial test on one proportion 125

      5.2.2 The McNemar test for paired data (or bivariate responses) with binary variables 128

      5.2.3 Multivariate extension of the McNemar test 131

5.3 Two-sample tests on proportions or2×2 contingency tables 134

      5.3.1 The Fisher exact test 135

      5.3.2 A permutation test for comparing two proportions 138

5.4 Tests for R× C contingency tables 139

5.4.1 The Anderson-Darling permutation test for R×C contingency tables 140

5.4.2 Permutation test on moments 145

5.4.3 The chi-square permutation test 148

References 151

6 Testing for correlation and concordance 153

6.1 Introduction 153

6.2 Measuring correlation 154

6.3 Tests for independence 156

      6.3.1 The Spearman test 157

      6.3.2 The Kendall test 160

6.4 Tests for concordance 166

      6.4.1 The Kendall-Babington Smith test 167

      6.4.2 A permutation test for concordance 172

References 174

7Tests for heterogeneity 176

7.1 Introduction 176

7.2 Statistical heterogeneity 177

7.3 Dominance in heterogeneity 178

      7.3.1 Geographical heterogeneity 180

      7.3.2 Market segmentation 184

7.4 Two-sided and multisample test 188

      7.4.1 Customer satisfaction 189

      7.4.2 Heterogeneity as a measure of uncertainty 191

      7.4.3 Ethnic heterogeneity 194

      7.4.4 Reliability analysis 196

References 197

Appendix A Selected critical values for the null distribution of the peak-known Mack-Wolfe statistic 201

Appendix B Selected critical values for the null distribution of the peak-unknown Mack-Wolfe statistic 203

Appendix C Selected upper-tail probabilities for the null distribution of the Page L statistic 206

Appendix D R functions and codes 213

Index 219

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