Permutation Methods: A Distance Function Approach

359 Pages·2001·6.964 MB·English

by  Paul W. Mielke Jr., Kenneth J. Berry (auth.)

Springer Series in Statistics
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Springer Series in Statistics
Advisors:
P. Bickel, P. Diggle, S. Fienberg K. Krickeberg,
I. Olkin, N. Wermuth, S. Zeger
Springer Science+Business Media, LLC
Paul W. Mielke, Jr.
Kenneth J. Berry
Permutation Methods
A Distance Function Approach
Springer
Paul W. Mielke, Jr. Kenneth J. Berry
Department of Statistics Department of Sociology
Colorado State University Colorado State University
Fort Collins, Colorado 80523 Fort Collins, Colorado 80523
E-mail: [e-mail] E-mail: [e-mail]
Library of Congress Cataloging-in-Publication Data
Mielke, Paul W.
Permutation methods: a distance function approach 1 Paul W. Mielke, Jr., Kenneth 1. Berry.
p. cm. - (Springer series in statistics)
Includes bibliographical references and index.
ISBN 978-1-4757-3451-5 ISBN 978-1-4757-3449-2 (eBook)
DOI 10.1007/978-1-4757-3449-2
1. Statistical hypothesis testing. 2. Resampling (Statistics) I. Berry, Kenneth J.
II. Title. III. Series.
QA277 .M53 2001
519.5'6--dc21 00-067920
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 200 I.
Softcover reprint of the harcover I st edition 200 I
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To our families.
Preface
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. These Euclidean distance-based
regression methods offer robust alternatives to the classical method of lin
ear model analyses involving the assumption of normality and ordinary
sum of least square deviations (OLS) regression linked with tests based on
squared Euclidean distance. In addition, consideration is given to a num
ber of permutation tests for (1) discrete and continuous goodness-of-fit,
viii Preface
(2) independence in multidimensional contingency tables, and (3) discrete
and continuous multisample homogeneity. Examples indicate some favor
able characteristics of seldom used tests.
Following a brief introduction in Chapter 1, Chapters 2, 3, and 4 provide
the motivation and description of univariate and multivariate permutation
tests based on distance functions for completely randomized and random
ized block designs. Applications are provided. Chapter 5 describes the linear
model methods based on the linkage between regression and permutation
tests, along with recently developed linear and nonlinear model prediction
techniques. Chapters 6, 7, and 8 include the goodness-of-fit, contingency ta
ble, and multisample homogeneity tests, respectively. Appendix A contains
an annotated listing of the computer programs used in the book, organized
by chapter.
Paul Mielke is indebted to the following former University of Minnesota
faculty members: his advisor Richard B. McHugh for introducing him to
permutation tests, Jacob E. Bearman and Eugene A. Johnson for moti
vating the examination of various problems from differing points of view,
and also to Constance van Eeden and 1. Richard Savage for motivating his
interest in nonparametric methods. He wishes to thank two of his Colorado
State University students, Benjamin S. Duran and Earl S. Johnson, for
stimulating his long term interest in alternative permutation methods. Fi
nally, he wishes to thank his Colorado State University colleagues Franklin
A. Graybill, Lewis O. Grant, William M. Gray, Hariharan K. lyer, David
C. Bowden, Peter J. Brockwell, Yi-Ching Yao, Mohammed M. Siddiqui,
Jagdish N. Srivastava, and James S. Williams, who have provided him
with motivation and various suggestions pertaining to this topic over the
years. Kenneth Berry is indebted to the former University of Oregon faculty
members Walter T. Martin, mentor and advisor, and William S. Robinson
who first introduced him to nonparametric statistical methods. Colorado
State University colleagues Jeffrey 1. Eighmy, R. Brooke Jacobsen, Michael
G. Lacy, and Thomas W. Martin were always there to listen, advise, and
encourage.
Acknowledgments. The authors thank the American Meteorological So
ciety for permission to reproduce excerpts from Weather and Forecasting
and the Journal of Applied Meteorology, Sage Publications, Inc. to repro
duce excerpts from Educational and Psychological Measurement, the Amer
ican Psychological Association for permission to reproduce excerpts from
Psychological Bulletin, the American Educational Research Association for
permission to reproduce excerpts from the Journal of Educational and Be
havioral Statistics, and the editors and publishers to reproduce excerpts
from Psychological Reports and Perceptual and Motor Skills.
The authors also wish to thank the following reviewers for their help
ful comments: Mayer Alvo, University of Ottawa; Bradley J. Biggerstaff,
Centers for Disease Control and Prevention; Brian S. Cade, U.S. Geolog
ical Survey; Hariharan K. lyer, Colorado State University; Bryan F. J.
Preface ix
Manly, WEST, Inc.; and Raymond K. W. Wong, Alberta Environment.
At Springer-Verlag New York, Inc., we thank our editor, John Kimmel,
for guiding the project throughout. We are grateful for the efforts of the
production editor, Antonio D. Orrantia, and the copy editor, Hal Henglein.
We wish to thank Roberta Mielke for reading the entire manuscript and
correcting our errors. Finally, we alone are responsible for any shortcomings
or inaccuracies.
Paul W. Mielke, Jr.
Kenneth J. Berry
Contents
Preface vii
1 Introduction 1
2 Description of MRPP 9
2.1 General Formulation of MRPP ..... . 12
2.1.1 Examples of MRPP ....... . 13
2.2 Choice of Weights and Distance Functions 18
2.3 Probability of an Observed 5 ..... . 21
2.3.1 Resampling Approximation .. . 22
2.3.2 Pearson Type III Approximation 22
2.3.3 Approximation Comparisons .. 26
2.3.4 Group Weights . . . . . . . . . . 27
2.3.5 Within-Group Agreement Measure 28
2.4 Exact and Approximate P-Values. 29
2.5 MRPP with an Excess Group . . . . . . . 32
2.6 Detection of Multiple Clumping ..... . 36
2.7 Detection of Evenly Spaced Location Patterns. 41
2.8 Dependence of MRPP on v ....... . 42
2.9 Permutation Version of One-Way AN OVA 46
2.10 Euclidean and Hotelling Commensuration 49
2.11 Power Comparisons ............ . 53
2.11.1 The Normal Probability Distribution. 59
2.11.2 The Cauchy Probability Distribution. 61