Nonparametric Hypothesis Testing: Rank and Permutation Methods with Applications in R

2 Comparing variability and distributions

2.1 Introduction

In this chapter, the data consist of two independent random samples, one sample drawn from each of two underlying populations. We would like to make inference about the possible presence of difference other than pure location ones between two populations. Section 2.1 introduces the problems to be addressed and the method presented in the successive sections.

Section 2.2 deals with nonparametric methods for detecting the difference in variability. Problems concerning comparison of variability are of interest in quality control, in agricultural production systems, in experimental education and in many other areas. Tests for the equality of variances are often of interest also as a preliminary to analysis of variance, dose–response modeling or discriminant analysis. Comparing scales is much harder than comparing locations. There are two reasons for this and these will be discussed in Section 2.2. In Section 2.2.1 we present the Ansari–Bradley rank test which is the most familiar rank test for comparing variability. This test requires that the medians of the populations underlying the samples are equal. In Sections 2.2.2 and 2.2.3 we present two permutation tests which do not require this assumption: the permutation Pan test, which is a version of the Levene test, is particularly useful when the distributions of the populations underlying the samples of interest are heavy-tailed and/or highly skewed; and the permutation ...