Zhang / Singer Recursive Partitioning and Applications

2. Auflage 2010
ISBN: 978-1-4419-6824-1
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 262 Seiten, eBook

Reihe: Springer Series in Statistics

Multiple complex pathways, characterized by interrelated events and c- ditions, represent routes to many illnesses, diseases, and ultimately death. Although there are substantial data and plausibility arguments suppo- ing many conditions as contributory components of pathways to illness and disease end points, we have, historically, lacked an e?ective method- ogy for identifying the structure of the full pathways. Regression methods, with strong linearity assumptions and data-basedconstraints onthe extent and order of interaction terms, have traditionally been the strategies of choice for relating outcomes to potentially complex explanatory pathways. However, nonlinear relationships among candidate explanatory variables are a generic feature that must be dealt with in any characterization of how health outcomes come about. It is noteworthy that similar challenges arise from data analyses in Economics, Finance, Engineering, etc. Thus, the purpose of this book is to demonstrate the e?ectiveness of a relatively recently developed methodology-recursive partitioning-as a response to this challenge. We also compare and contrast what is learned via rec- sive partitioning with results obtained 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 regression techniques.

Heping Zhang is Professor of Public Health, Statistics, and Child Study, and director of the Collaborative Center for Statistics in Science, at Yale University. He is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics, a Myrto Lefkopoulou Distinguished Lecturer Awarded by Harvard School of Public Health, and a Medallion lecturer selected by the Institute of Mathematical Statistics. Burton Singer is Courtesy Professor in the Emerging Pathogens Institute at University of Florida, and previously Charles and Marie Robertson Professor of Public and International Affairs at Princeton University. He is a member of the National Academy of Sciences and Institute of Medicine of the National Academies, and a Fellow of the American Statistical Association.

Zielgruppe

Research

Autoren/Hrsg.

Zhang, Heping

Singer, Burton H.

Singer, Burton

Inhaltsverzeichnis

1;Preface;8
2;Contents;12
3;1 Introduction;16
3.1;1.1 Examples Using CART;18
3.2;1.2 The Statistical Problem;21
3.3;1.3 Outline of the Methodology;22
4;2 A Practical Guide to TreeConstruction;24
4.1;2.1 The Elements of Tree Construction;26
4.2;2.2 Splitting a Node;27
4.3;2.3 Terminal Nodes;34
4.4;2.4 Download and Use of Software;35
5;3 Logistic Regression;38
5.1;3.1 Logistic Regression Models;38
5.2;3.2 A Logistic Regression Analysis;39
6;4 Classification Trees for a BinaryResponse;45
6.1;4.1 Node Impurity;45
6.2;4.2 Determination of Terminal Nodes;48
6.2.1;4.2.1 Misclassification Cost;48
6.2.2;4.2.2 Cost–Complexity;51
6.2.3;4.2.3 Nested Optimal Subtrees*;53
6.3;4.3 The Standard Error of Rcv*;56
6.4;4.4 Tree-Based Analysis of the Yale Pregnancy Outcome Study;57
6.5;4.5 An Alternative Pruning Approach;58
6.6;4.6 Localized Cross-Validation;63
6.7;4.7 Comparison Between Tree-Based and Logistic Regression Analyses;65
6.8;4.8 Missing Data;68
6.8.1;4.8.1 Missings Together Approach;69
6.8.2;4.8.2 Surrogate Splits;70
6.9;4.9 Tree Stability;71
6.10;4.10 Tree for Treatment Effectiveness;72
6.11;4.11 Implementation*;73
7;5 Examples Using Tree-Based Analysis;77
7.1;5.1 Risk-Factor Analysis in Epidemiology;77
7.1.1;5.1.1 Background;77
7.1.2;5.1.2 The Analysis;79
7.2;5.2 Customer Credit Assessment;85
8;6 Random and Deterministic Forests;92
8.1;6.1 Introduction to Random Forests;92
8.2;6.2 The Smallest Forest;94
8.3;6.3 Importance Score;97
8.3.1;6.3.1 Gini Importance;97
8.3.2;6.3.2 Depth Importance;97
8.3.3;6.3.3 Permutation Importance;97
8.3.4;6.3.4 Maximum Conditional Importance;99
8.4;6.4 Random Forests for Predictors with Uncertainties;102
8.5;6.5 Random Forests with Weighted Feature Selection;106
8.6;6.6 Deterministic Forests;106
8.7;6.7 A Note on Interaction;107
9;7 Analysis of Censored Data: Examples;109
9.1;7.1 Introduction;109
9.2;7.2 Tree-Based Analysis for the Western Collaborative Group Study Data;112
10;8 Analysis of Censored Data: Conceptsand Classical Methods;116
10.1;8.1 The Basics of Survival Analysis;116
10.1.1;8.1.1 Kaplan–Meier Curve;119
10.1.2;8.1.2 Log-Rank Test;121
10.2;8.2 Parametric Regression for Censored Data;116
10.2.1;8.2.1 Linear Regression with Censored Data*;124
10.2.2;8.2.2 Cox Proportional Hazard Regression;126
10.2.3;8.2.3 Reanalysis of the Western Collaborative Group Study Data;127
11;9 Analysis of Censored Data: Survival Trees and Random Forests;130
11.1;9.1 Splitting Criteria;130
11.1.1;9.1.1 Gordon and Olshen’s Rule*;130
11.1.2;9.1.2 Maximizing the Difference;133
11.1.3;9.1.3 Use of Likelihood Functions*;133
11.1.4;9.1.4 A Straightforward Extension;136
11.2;9.2 Pruning a Survival Tree;137
11.3;9.3 Random Survival Forests;138
11.4;9.4 Implementation;138
11.5;9.5 Survival Trees for the Western Collaborative Group Study Data;139
11.6;9.6 Combinations of Biomarkers Predictive of Later Life Mortality;140
12;10 Regression Trees and Adaptive Splinesfor a Continuous Response;143
12.1;10.1 Tree Representation of Spline Model and Analysis of Birth Weight;144
12.2;10.2 Regression Trees;146
12.3;10.3 The Profile of MARS Models;150
12.4;10.4 Modified MARS Forward Procedure;153
12.5;10.5 MARS Backward-Deletion Step;156
12.6;10.6 The Best Knot*;158
12.7;10.7 Restrictions on the Knot*;161
12.7.1;10.7.1 Minimum Span;161
12.7.2;10.7.2 Maximal Correlation;162
12.7.3;10.7.3 Patches to the MARS Forward Algorithm;164
12.8;10.8 Smoothing Adaptive Splines*;165
12.8.1;10.8.1 Smoothing the Linearly Truncated Basis Functions;166
12.8.2;10.8.2 Cubic Basis Functions;166
12.9;10.9 Numerical Examples;167
13;11 Analysis of Longitudinal Data;173
13.1;11.1 Infant Growth Curves;173
13.2;11.2 The Notation and a General Model;175
13.3;11.3 Mixed-Effects Models;176
13.4;11.4 Semiparametric Models;179
13.5;11.5 Adaptive Spline Models;180
13.5.1;11.5.1 Known Covariance Structure;181
13.5.2;11.5.2 Unknown Covariance Structure;182
13.5.3;11.5.3 A Simulated Example;185
13.5.4;11.5.4 Reanalyses of Two Published Data Sets;188
13.5.5;11.5.5 Analysis of Infant Growth Curves;197
13.5.6;11.5.6 Remarks;202
13.6;11.6 Regression Trees for Longitudinal Data;203
13.6.1;11.6.1 Example: HIV in San Francisco;205
14;12 Analysis of Multiple Discrete Responses;209
14.1;12.1 Parametric Methods for Binary Responses;211
14.1.1;12.1.1 Log-Linear Models;212
14.1.2;12.1.2 Marginal Models;213
14.1.3;12.1.3 Parameter Estimation*;215
14.1.4;12.1.4 Frailty Models;216
14.2;12.2 Classification Trees for Multiple Binary Responses;219
14.2.1;12.2.1 Within-Node Homogeneity;219
14.2.2;12.2.2 Terminal Nodes;220
14.2.3;12.2.3 Computational Issues*;221
14.2.4;12.2.4 Parameter Interpretation*;222
14.3;12.3 Application: Analysis of BROCS Data;223
14.3.1;12.3.1 Background;223
14.3.2;12.3.2 Tree Construction;223
14.3.3;12.3.3 Description of Numerical Results;225
14.3.4;12.3.4 Alternative Approaches;228
14.3.5;12.3.5 Predictive Performance;229
14.4;12.4 Ordinal and Longitudinal Responses;229
14.5;12.5 Analysis of the BROCS Data via Log-Linear Models;231
15;13 Appendix;236
15.1;13.1 The Script for Running RTREE Automatically;236
15.2;13.2 The Script for Running RTREE Manually;238
15.3;13.3 The .inf File;242
16;References;245
17;Index;264

A Practical Guide to Tree Construction.- Logistic Regression.- Classification Trees for a Binary Response.- Examples Using Tree-Based Analysis.- Random and Deterministic Forests.- Analysis of Censored Data: Examples.- Analysis of Censored Data: Concepts and Classical Methods.- Analysis of Censored Data: Survival Trees and Random Forests.- Regression Trees and Adaptive Splines for a Continuous Response.- Analysis of Longitudinal Data.- Analysis of Multiple Discrete Responses.

Über Autor(innen)

Heping Zhang is Professor of Public Health, Statistics, and Child Study, and director of the Collaborative Center for Statistics in Science, at Yale University. He is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics, a Myrto Lefkopoulou Distinguished Lecturer Awarded by Harvard School of Public Health, and a Medallion lecturer selected by the Institute of Mathematical Statistics.
Burton Singer is Courtesy Professor in the Emerging Pathogens Institute at University of Florida, and previously Charles and Marie Robertson Professor of Public and International Affairs at Princeton University. He is a member of the National Academy of Sciences and Institute of Medicine of the National Academies, and a Fellow of the American Statistical Association.