Nonparametric Hypothesis Testing Rank and Permutation Methods with 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.

Stefano Bonnini, Assistant Professor of Statistics, Faculty of Economics, Department of Economics, University of Ferrara, Italy.

Livio Corain, Assistant Professor of Statistics, Faculty of Engineering, Department of Management and Engineering, University of Padova, Italy.

Marco Marozzi, Associate Professor of Statistics, Faculty of Economics, Department of Economics and Statistics, University of Calabria, Italy.

Luigi Salmaso, Full Professor of Statistics, Faculty of Engineering, University of Padova, Italy.

Presentation of the book xi

Preface xiii

Notation and abbreviations xvii

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

1.1 Introduction 1

1.2 Nonparametric 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 variability 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 Cram´er–von Mises test 59

References 61

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 120

References 122

5 Tests 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 or 2 × 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

7 Tests 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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