Nonparametric Hypothesis Testing

Rank and Permutation Methods with Applications in R

Stefano Bonnini, Livio Corain, Marco Marozzi, Luigi Salmaso

• Publisher: Wiley
• Year: 2014
• Print ISBN: 9781119952374
• eBook ISBN: 9781118763483
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

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

1.1 Introduction

Many real phenomena can be represented by numerical random variables. Considering a given population and a random sample of it, for forecasting or improving the effectiveness of inferential techniques related to estimation and testing of hypothesis, it would be useful to know the functional form of the distribution of the data. Sometimes, the central interest of the statistical analysis is focused only on the symmetry or on the location of the distribution itself. Another very common statistical problem consists of comparing two independent populations in terms of central tendency. In the simpler cases the object of the analysis is a univariate population, but in some real applications we are in the presence of many variables and multivariate datasets.

The methods presented in this chapter consist of rank or permutation procedures for the tests of the hypotheses cited above. Section 1.2 is an introduction to rank and permutation tests. In Section 1.3, devoted to one-sample tests, the Kolmogorov procedure for testing whether the data are distributed according to an hypothesized cumulative distribution function (CDF), and the permutation test on the symmetry of the distribution are taken into account. Section 1.4 deals with multivariate one-sample tests, and introduces the multivariate location problem and the multivariate test on symmetry. In Section 1.5 the univariate two-sample location problem is discussed. Section 1.6 considers the multivariate extension of the location problem for two independent populations and presents some solutions for it.

In the one-sample problems the data are a random sample of numerical data

from the unknown population under study. In the two-sample problems the numerical sample data fom the jth unknown population are

, with j = 1, 2 and n₁ + n₂ = n. In the multivariate extensions, in the presence of q component variables, for the one-sample problem, the observation related to the ith statistical unit is denoted by

and, for the two-sample problem, the observation related to the ith statistical unit in the jth group is denoted by

.

1.2 Nonparametric tests

Traditional parametric testing methods are based on the assumption that data are generated by well-known distributions, characterized by one or more unknown population parame...

Table of contents

• Cover
• Series
• Title Page
• Copyright
• Dedicated
• Presentation of the book
• Preface
• Notation and abbreviations
• 1 One- and two-sample location problems, tests for symmetry and tests on a single distribution
• 2 Comparing variability and distributions
• 3 Comparing more than two samples
• 4 Paired samples and repeated measures
• 5 Tests for categorical data
• 6 Testing for correlation and concordance
• 7 Tests for heterogeneity
• 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