Table Of ContentSpringer Texts in Statistics
Advisors:
Stephen Fienberg Ingram Olkin
Springer Texts in Statistics
Advisors:
Stephen Fienberg Ingram Olkin
Springer Texts in Statistics
Alfred: Elements of Statistics for the Life and Social Sciences
Berger: An Introduction to Probability and Stochastic Processes
Blom: Probability and Statistics: Theory and Applications
Chow and Teicher: Probability Theory: Independence,
Interchangeability, Martingales, Second Edition
Christensen: Plane Answers to Complex Questions: The Theory
of Linear Models
Christensen: Linear Models for Multivariate, Time Series,
and Spatial Data
Christensen: Log-Linear Models
Creighton: A First Course in Probability Models
and Statistical Inference
du Toit, Steyn and Stumpf: Graphical Exploratory Data Analysis
Edwards: Introduction to Graphical Modelling
Finkelstein and Levin: Statistics for Lawyers
Jobson: Applied Multivariate Data Analysis, Volume I: Regression
and Experimental Design
Jobson: Applied Multivariate Data Analysis, Volume II: Categorical
and Multivariate Methods
Kalbfleisch: Probability and Statistical Inference,
Volume I: Probability, Second Edition
Kalbfleisch: Probability and Statistical Inference,
Volume II: Statistical Inference, Second Edition
Karr: Probability
Keyfitz: Applied Mathematical Demography, Second Edition
Kiefer: Introduction to Statistical Inference
Kokoska and Nevison: Statistical Tables and Formulae
Lindman: Analysis of Variance in Experimental Design
Madansky: Prescriptions for Working Statisticians
Mueller: Basic Principles of Structural Equation Modelling
McPherson: Statistics in Scientific Investigation: Its Basis,
Application, and Interpretation
Nguyen and Rogers: Fundamentals of Mathematical Statistics:
Volume I: Probability for Statistics
Nguyen and Rogers: Fundamentals of Mathematical Statistics:
Volume II: Statistical Inference
Noether: Introduction to Statistics: The Nonparametric Way
Peters: Counting for Something: Statistical Principles
and Personalities
Continued at end of book
David Edwards
Introduction to
Graphical Modelling
Springer Science+B usiness Media, LLC
David Edwards
Statistics Department
N ovo Nordisk A/S
Krogshoejvej 55
DK-2880 Bagsvaerd
Denmark
Editorial Board
Stephen Fienberg lngram Olkin
Department of Statistics Department of Statistics
Carnegie Mellon University Stanford University
Pittsburgh, PA 15213, USA Stanford, CA 94305, USA
Library of Congress Cataloging-in-Publication Data
Edwards, David, 1949-
lntroduction to graphical modelling 1 David Edwards.
p. cm. - (Springer texts in statistics)
Includes bibliographical references (p. ) and index.
ISBN 978-1-4684-0483-8
1. Graphical modeling (Statistics) 1. Title. Il. Series.
QA279.E34 1995
519.5'38-dc20 95-3757
Printed on acid-free paper.
© 1995 Springer Science+Business Media New York
Originally published by Springer-Ve r1ag New York, Inc. in 1995
Softcover reprint of the hardcover 1s t edition 1995
Ali rights reserved. This work may not be translated or copied in whole or in part with
out the written permission of the publisher (Springer Science+Business Media, LLC)
except for brief excerpts in connection with reviews
or scholarly analysis. Use in connection with any form of information storage and
retrieval, electronic adaptation, computer software, or by similar or dissimilar method
ology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that such
names, as understood by the Trade Marks and Merchandise Marks Act, may accord
ingly be used freely by anyone.
Production managed by Frank Ganz; manufacturing supervised by J effrey Taub.
Photocomposed pages prepared using the author's LATEX file.
9 8 7 6 5 4 3 2 1
Additional material to this book can be downloaded from http://extras.springer.com
ISBN 978-1-4684-0483-8 ISBN 978-1-4684-0481-4 (eBook)
DOI 10.1007/978-1-4684-0481-4
I would rather discover a single causal relationship than be king of Persia.
Democritus.
Preface
Graphical modelling is a form of multivariate analysis that uses graphs to
represent models. Although its roots can be traced back to path analysis
(Wright, 1921) and statistical physics (Gibbs, 1902), its modern form is
of recent origin. Key papers in the modern development include Darroch,
Lauritzen, and Speed (1980), and Lauritzen and Wermuth (1989).
The purpose of this book is to provide a concise, application-oriented in
troduction to graphical modelling. The theoretical coverage is informal,
and should be supplemented by other sources: the book by Whittaker
(1990) would be a natural choice. Readers primarily interested in dis
crete data should consult the introductory-level book by Christensen
(1990). Lauritzen (1992) provides a mathematically rigorous treatment:
this is the source to consult about results stated here without proof.
Applications of graphical modelling in a wide variety of areas are shown.
These analyses make use of l'v1IM, a command-driven PC-program de
signed for graphical modelling. A student version of l'v1IM is included
with the book, and a reference guide is included as an appendix.
My interest in graphical modelling started in 1978-1980 under the in
fluence of Terry Speed, who held a seminal lecture course on the topic
in Copenhagen in 1978. Subsequent participation in a study group on
graphical modelling, together with Steffen Lauritzen, Svend Kreiner,
Morten Frydenberg, Jens Henrik Badsberg, and Poul Svante Eriksen,
has served to stimulate and broaden my interest in the topic.
The first version of MlM was written in 1986-1987 at the Statistical Re
search Unit, Copenhagen University, when I was supported by a Dan
ish Social Science Research Council grant. I wish to thank all the peo
ple who helped in the development of l'v1IM, including: Brian Murphy,
viii Preface
for assistance and inspiration, and for kindly supplying the program
LOLITA which served as a starting point for MIl'v'I; Morten Frydenberg,
for crucial assistance with the modified iterative proportional scaling
(MIPS) algorithm; Steffen Lauritzen, Nanny Wermuth, Hanns-Georg
Leimer, Svend Kreiner, Jens Henrik Badsberg, and Joe Whittaker for
encouraging help; Brian Francis and Joe Whittaker for contributing the
SUSPEND and RETURN commands; Egon Hansen for help with the inter
active graphics; The Tjur for generously letting me use his module for
distribution functions; Svend Kreiner for helping me code Patefield's
(1981) algorithm; and Marta Honikova for programming the explicit
estimation and EH-selection procedures.
Finally, thanks are also due to Peter Smith, Philip Hougaard, and Helle
Lynggaard for helpful suggestions.
January 28, 1995 David Edwards
Roskilde
Contents
Preface vii
1 Introduction 1
1.1 Conditional Independence . 1
1.2 Graphs ........... 4
1.3 Data, Models, and Graphs 6
1.4 Simpson's Paradox. . 8
1.5 Overview of the Book 11
2 Discrete Models 13
2.1 Three-Way Tables 13
2.2 Example: Lizard Perching Behaviour. 18
2.3 Multi-Way Tables . . . . . . . . . . . 20
2.4 Example: rusk Factors for Coronary Heart Disease . 24
2.5 Example: Chromosome Mapping 27
2.6 Example: University Admissions 32
x Contents
3 Continuous Models 35
3.1 Graphical Gaussian Models 35
3.2 Example: Digoxin Clearance 42
3.3 Example: Anxiety and Anger 45
3.4 Example: Mathematics Marks 47
3.5 Regression Models. . . . . . . 51
3.6 Example: Determinants of Bone Mineral Content . 54
4 Mixed Models 61
4.1 CG-Distribution 61
4.2 Models with One Discrete and One Continuous Variable. 62
4.3 A Model with Two Discrete and Two Continuous Variables 64
4.4 Model Formulae . . . 65
4.5 Formulae and Graphs 66
4.6 Maximum Likelihood Estimation. 68
4.7 Deviance . . . . . . 70
4.8 A Simple Example . 72
4.9 The MANOVA Framework 75
4.10 Example: A Drug 'Irial Using Mice. 77
4.11 Example: A Drug 'Irial Using Rats. 80
4.12 Example: Lipids .......... . 83
4.13 Breaking Models into Smaller Ones 90
4.14 Decomposable Models 97
5 Hypothesis Testing 103
5.1 An Overview 103
5.2 X2-Tests. 104
5.3 F-Tests 106
5.4 Exact Conditional Tests. 108
5.5 Deviance-Based Tests 112
5.6 Permutation F-Test 117
