Recursive Partitioning in the Health Sciences

228 Pages·1999·4.02 MB·English

by  Heping Zhang, Burton Singer (auth.)

Statistics for Biology and Health
Series Editors
K. Dietz, M. Gail, K. Krickeberg, B. Singer
Springer Science+Business Media, LLC
Statistics for Biology and Health
KleinlMoeschberger: Survival Analysis: Techniques for Censored and Truncated Data.
Kleinbaum: Logistic Regression: A Self-Learning Text.
Kleinbaum: Survival Analysis: A Self-Learning Text.
Lange: Mathematical and Statistical Methods for Genetic Analysis.
MantonlSinger/Suzman: Forecasting the Health of Elderly Populations.
Salsburg: The Use of Restricted Significance Tests in Clinical Trials.
Zhang/Singer: Recursive Partitioning in the Health Sciences.
Heping Zhang Burton Singer
Recursive Partitioning in
the Health Sciences
With 60 Illustrations
" Springer
Heping Zhang Burton Singer
Department of Epidemiology Office of Population Research
and Public Health Princeton University
School of Medicine Notestein Hali
Yale University 21 Prospect A venue
60 College Street Princeton, NJ 08544
New Haven, CT 06520-8034 USA
USA
Series Editors
K. Dietz M. Gail
Institut fUr Medizinische Biometrie National Cancer Institute
Universităt Tiibingen Rockville, MD 20892
Westbahnhofstr. 55 USA
D 72070 Tiibingen
Germany
K. Krickeberg B. Singer
3 Rue de L'Estrapade Office of Population Research
75005 Paris Princeton University
France Princeton, NJ 08544
USA
Library of Congress Cataloging-in-Publication Data
Zhang, Heping.
Recursive partitioning in the health sceinces / Heping Zhang,
Burton Singer.
p. cm. - (Statistics for biology and health)
IncIudes bibliographical references and index.
ISBN 978-1-4757-3029-6 ISBN 978-1-4757-3027-2 (eBook)
DOI 10.1007/978-1-4757-3027-2
1. Medicine-Research-Statistical methods. 1. Singer, Burton.
II. TitIe. III. Series.
R853.S7 Z48 1999
61O.7'27-ddc21 98-44699
Printed on acid-free paper.
© 1999 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1999
Softcover reprint of the hardcover 1s t edition 1999
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Photocomposed copy prepared from the authors' LaTeX files.
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ISBN 978-1-4757-3029-6 SPIN 10699055
Dedicated to Julan, Jeffrey, and Leon
Preface
Multiple complex pathways, characterized by interrelated events and con
ditions, represent routes to many illnesses, diseases, and ultimately death.
Although there are substantial data and plausibility arguments supporting
many conditions as contributory components of pathways to illness and
disease end points, we have, historically, lacked an effective methodology
for identifying the structure of the full pathways. Regression methods, with
strong linearity assumptions and data-based constraints on the extent and
order of interaction terms, have traditionally been the strategies of choice
for relating outcomes to potentially complex explanatory pathways. How
ever, nonlinear relationships among candidate explanatory variables are
a generic feature that must be dealt with in any characterization of how
health outcomes come about. Thus, the purpose of this book is to demon
strate the effectiveness of a relatively recently developed methodology
recursive partitioning-as a response to this challenge. We also compare
and contrast what is learned via recursive partitioning with results ob
tained on the same data sets using more traditional methods. This serves
to highlight exactly where--and for what kinds of questions-recursive
partitioning-based strategies have a decisive advantage over classical re
gression techniques.
This book is suitable for three broad groups of readers: (1) biomedical re
searchers, clinicians, public health practitioners including epidemiologists,
health service researchers, environmental policy advisers; (2) consulting
statisticians who can use the recursive partitioning technique as a guide
in providing effective and insightful solutions to clients' problems; and (3)
statisticians interested in methodological and theoretical issues. The book
viii Preface
provides an up-to-date summary of the methodological and theoretical un
derpinnings of recursive partitioning. More interestingly, it presents a host
of unsolved problems whose solutions would advance the rigorous under
pinnings of statistics in general.
From the perspective of the first two groups of readers, we demonstrate
with real applications the sequential interplay between automated produc
tion of multiple well-fitting trees and scientific judgment leading to respec
ification of variables, more refined trees subject to context-specific con
straints (on splitting and pruning, for example), and ultimately selection
of the most interpretable and useful tree(s). The sections marked with as
terisks can be skipped for application-oriented readers.
We show a more conventional regression analysis-having the same ob
jective as the recursive partitioning analysis-side by side with the newer
methodology. In each example, we highlight the scientific insight derived
from the recursive partitioning strategy that is not readily revealed by more
conventional methods. The interfacing of automated output and scientific
judgment is illustrated with both conventional and recursive partitioning
analysis.
Theoretically oriented statisticians will find a substantial listing of chal
lenging theoretical problems whose solutions would provide much deeper
insight than heretofore about the scope and limits of recursive partitioning
as such and multivariate adaptive splines in particular.
We emphasize the development of narratives to summarize the formal
Boolean statements that define routes down the trees to terminal nodes.
Particularly with complex-by scientific necessity-trees, narrative output
facilitates understanding and interpretation of what has been provided by
automated techniques.
We illustrate the sensitivity of trees to variation in choosing misclassi
fication cost, where the variation is a consequence of divergent views by
clinicians of the costs associated with differing mistakes in prognosis.
The book of Breiman et al. (1984) is a classical work on the subject of
recursive partitioning. In Chapter 4, we reiterate the key ideas expressed
in that book and expand our discussions in different directions on the is
sues that arise from applications. Other chapters on survival trees, adaptive
splines, and classification trees for multiple discrete outcomes are new de
velopments since the work of Breiman et al.
Heping Zhang wishes to thank his colleagues and students, Joan Buen
consejo, Theodore Holford, James Leckman, Ju Li, Robert Makuch, Kath
leen Merikangas, Bradley Peterson, Norman Silliker, Daniel Zelterman, and
Hongyu Zhao among others, for their help with reading and commenting
on earlier drafts of this book. He is also grateful to Drs. Michael Bracken,
Dorit Carmelli, and Brian Leaderer for making their data sets available
to this book. This work was supported in part by NIH grant HD30712 to
Heping Zhang.
Contents
Preface vii
1 Introduction 1
1.1 Examples Using CART 2
1.2 The Statistical Problem 4
1.3 Outline of the Methodology 5
2 A Practical Guide to Tree Construction 7
2.1 The Elements of Tree Construction . 9
2.2 Splitting a Node ........ 10
2.3 Terminal Nodes . . . . . . . . . 15
2.4 Download and Use of Software 16
3 Logistic Regression 21
3.1 Logistic Regression Models 21
3.2 A Logistic Regression Analysis 22
4 Classification Trees for a Binary Response 29
4.1 Node Impurity ........... 29
4.2 Determination of Terminal Nodes. 32
..
4.2.1 Misclassification Cost 32
4.2.2 Cost Complexity ..... 35
4.2.3 Nested Optimal Subtrees· 37
4.3 The Standard Error of Rev * . . . 40
x Contents
4.4 Tree-Based Analysis of the Yale Pregnancy Outcome Study 41
4.5 An Alternative Pruning Approach ............. . 43
4.6 Localized Cross-Validation ...... . 47
4.7 Comparison Between Tree-Based and Logistic Regression
Analyses. . . . . . . . . . . . . . . . 49
4.8 Missing Data . . . . . . . . . . . . . 53
4.8.1 Missings Together Approach 53
4.8.2 Surrogate Splits 54
4.9 Tree Stability . . 55
4.10 Implementation* . . . . 56
5 Risk-Factor Analysis Using Tree-Based Stratification 61
5.1 Background. 61
5.2 The Analysis . . . . . . . . . . . . . . 63
6 Analysis of Censored Data: Examples 71
6.1 Introduction........................... 71
6.2 Tree-Based Analysis for the Western Collaborative Group
Study Data . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
7 Analysis of Censored Data:
Concepts and Classical Methods 79
7.1 The Basics of Survival Analysis 79
7.1.1 Kaplan-Meier Curve. . . 84
7.1.2 Log-Rank Test . . . . . . 85
7.2 Parametric Regression for Censored Data 87
7.2.1 Linear Regression with Censored Data* 87
7.2.2 Cox Proportional Hazard Regression. . 89
7.2.3 Reanalysis ofthe Western Collaborative Group Study
Data. . . . . . . . . . . . . . . . . . . . . . . . . .. 91
8 Analysis of Censored Data: Survival Trees 93
8.1 Splitting Criteria . . . . . . . . . . 93
8.1.1 Gordon and Olshen's Rule* . 93
8.1.2 Maximizing the Difference . . 96
8.1.3 Use of Likelihood Functions* 96
8.1.4 A Straightforward Extension 99
8.2 Pruning a Survival Tree ...... . 99
8.3 Implementation............ 100
8.4 Survival Trees for the Western Collaborative Group Study
Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 101
Contents xi
9 Regression Trees and Adaptive Splines for a Continuous
Response 105
9.1 Tree Representation of Spline Model and Analysis of Birth
Weight. . . . . . . . . . . . . 106
9.2 Regression Trees . . . . . . . . . . . 108
9.3 The Profile of MARS Models . . . . 112
9.4 Modified MARS Forward Procedure 115
9.5 MARS Backward-Deletion Step 118
9.6 The Best Knot* . . . . . . 120
9.7 Restrictions on the Knot* . . 123
9.7.1 Minimum Span. . . . 123
9.7.2 Maximal Correlation. 124
9.7.3 Patches to the MARS Forward Algorithm 127
9.8 Smoothing Adaptive Splines* . . . . . . . . . . . 127
9.8.1 Smoothing the Linearly Truncated Basis Functions. 128
9.8.2 Cubic Basis Functions 128
9.9 Numerical Examples ..... 129
10 Analysis of Longitudinal Data 137
10.1 Infant Growth Curves .... 137
10.2 The Notation and a General Model. 139
10.3 Mixed-Effects Models . 140
10.4 Semiparametric Models ...... . 143
10.5 Adaptive Spline Models . . . . . . . 144
10.5.1 Known Covariance Structure 145
10.5.2 Unknown Covariance Structure 146
10.5.3 A Simulated Example . . . . . 149
10.5.4 Reanalyses of Two Published Data Sets 152
10.5.5 Analysis of Infant Growth Curves 161
10.5.6 Remarks. . . . . . . . . . . . . . 166
10.6 Regression Trees for Longitudinal Data 167
10.6.1 Example: HIV in San Francisco. 169
11 Analysis of Multiple Discrete Responses 173
11.1 Parametric Methods for Binary Responses. 175
11.1.1 Log-Linear Models .. . 176
11.1.2 Marginal Models ... . 178
11.1.3 Parameter Estimation* 179
11.1.4 Frailty Models ..... 181
11.2 Classification Trees for Multiple Binary Responses 183
11.2.1 Within-Node Homogeneity 183
11.2.2 Terminal Nodes ..... . 184
11.2.3 Computational Issues* .. . 185
11.2.4 Parameter Interpretation* . 186
11.3 Application: Analysis of BROCS Data 187