Permutation Methods

The introduction of permutation tests by R. A. Fisher relaxed the paramet­ ric structure requirement of a test statistic. For example, the structure of the test statistic is no longer required if the assumption of normality is removed. The between-object distance function of classical test statis­ tics based on the assumption of normality is squared Euclidean distance. Because squared Euclidean distance is not a metric (i. e. , the triangle in­ equality is not satisfied), it is not at all surprising that classical tests are severely affected by an extreme measurement of a single object. A major purpose of this book is to take advantage of the relaxation of the struc­ ture of a statistic allowed by permutation tests. While a variety of distance functions are valid for permutation tests, a natural choice possessing many desirable properties is ordinary (i. e. , non-squared) Euclidean distance. Sim­ ulation studies show that permutation tests based on ordinary Euclidean distance are exceedingly robust in detecting location shifts of heavy-tailed distributions. These tests depend on a metric distance function and are reasonably powerful for a broad spectrum of univariate and multivariate distributions. Least sum of absolute deviations (LAD) regression linked with a per­ mutation test based on ordinary Euclidean distance yields a linear model analysis which controls for type I error.

From the reviews:

The authors describe permutation methods associated with completely randomized and randomized block experimental designs. Applications of the distance function permutation approach include binary and rank data. Type I error controlled regression analyses, including both prediction and linear models, are delineated. Because the regression analyses are based on a metric Euclidean distance function, special attention is paid to least sum of absolute deviations (LAD) regression. Recently developed goodness-of-fit tests, measures of agreement, multidimensional contingency table tests, and multisample univariate permutation methods are discussed and compared. The level of this monograph is that of upper division undergraduate and beginning graduate students.

--Mathematical Reviews

"It is only with the advent of powerful computing that permutation methods have become practical. The text by Mielke and Berry presents a good and accessible account of the subject to date with emphasis on the applications made easy with the developed software. … The text covers a wide variety of topics and provides an important set of tools. It is recommended for use by graduate students and practitioners having a broad knowledge of the statistical literature." (Mayer Alvo, Sankhya: Indian Journal of Statistics, Vol. 64 (B Part 3), 2002)

"This is a book on non-parametric statistics, in particular about extending permutation methods to several contexts, notably multivariate analysis and randomized block experimental designs. … The procedures favoured by the authors rely on general metric distance functions instead of the usual (non-metric) squared distance implied by the Normality assumption. … anybody interested in permutation methods should find some useful material in it." (Mario Cortina Borja, Journal of Applied Statistics, Vol. 30 (5), 2003)

"Permutation methods such as the authors suggest, are generalizations of theFisher-Pitman permutation test, which is nicely outlined … . this is a well-written book with an intent to generalize this seemly limited tool to a lot applications." (Shin Ta Liu, Technometrics, Vol. 44 (3), 2002)

"This monograph discusses univariate and multivariate permutation tests based on distance measures. … the book … is well written and thorough. The material in the book is easily accessible to upper-division undergraduates and beginning graduates. Permutation Methods: A Distance Function Approach is a welcome addition to the literature on permutation tests, and the computer package available with it is an added attractive feature." (Sreenivasa Rao Jammalamadaka, Journal of the American Statistical Association, March, 2003)

"This monograph presents a fresh approach to statistical ideas based on permutation procedures. The focus is not on elaborating a mathematical theory in a theorem-proof framework. Rather it is the authors’ goal to give an overview how permutation methodology may be applied to quite different statistical questions. Most of the issues are illustrated by real data examples. Computer programs are available at the authors’ web site. … The text can be understood without deep measure-theoretic background." (W. Stute, Zentralblatt MATH, Vol. 979, 2002)