Event History Analysis with R

234 Pages·2012·2.539 MB·English

by  Göran Broström

The R Series
Statistics
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With an emphasis on social science applications, Event History t Event History Analysis

Analysis with R presents an introduction to survival and event H
history analysis using real-life examples. Keeping mathematical
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with R
details to a minimum, the book covers key topics, including both s
discrete and continuous time data, parametric proportional hazards, t
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and accelerated failure times.
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• Introduces parametric proportional hazards models with
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baseline distributions like the Weibull, Gompertz, Lognormal,
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and Piecewise constant hazard distributions, in addition to
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traditional Cox regression
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• Presents mathematical details as well as technical material in an
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appendix
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• Includes real examples with applications in demography,
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econometrics, and epidemiology

• Provides a dedicated R package, eha, containing special
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treatments, including making cuts in the Lexis diagram, creating
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communal covariates, and creating period statistics
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A much-needed primer, Event History Analysis with R is a didactically

excellent resource for students and practitioners of applied event R
history and survival analysis.
B
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ö Göran Broström
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K11534
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Event History Analysis
with R
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Chapman & Hall/CRC
The R Series
Series Editors
John M. Chambers Torsten Hothorn
Department of Statistics Institut für Statistik
Stanford University Ludwig-Maximilians-Universität
Stanford, California, USA München, Germany
Duncan Temple Lang Hadley Wickham
Department of Statistics Department of Statistics
University of California, Davis Rice University
Davis, California, USA Houston, Texas, USA
Aims and Scope
This book series reflects the recent rapid growth in the development and application of R, the
programming language and software environment for statistical computing and graphics. R is
now widely used in academic research, education, and industry. It is constantly growing, with
new versions of the core software released regularly and more than 2,600 packages available.
It is difficult for the documentation to keep pace with the expansion of the software, and this
vital book series provides a forum for the publication of books covering many aspects of the
development and application of R.
The scope of the series is wide, covering three main threads:
• Applications of R to specific disciplines such as biology, epidemiology, genetics,
engineering, finance, and the social sciences.
• Using R for the study of topics of statistical methodology, such as linear and mixed
modeling, time series, Bayesian methods, and missing data.
• The development of R, including programming, building packages, and graphics.
The books will appeal to programmers and developers of R software, as well as applied
statisticians and data analysts in many fields. The books will feature detailed worked
examples and R code fully integrated into the text, ensuring their usefulness to researchers,
practitioners and students.
Published Titles
Customer and Business Analytics: Applied Data Mining for Business Decision
Making Using R, Robert E. Krider and Daniel S. Putler
Event History Analysis with R, Göran Broström
Programming Graphical User Interfaces with R, John Verzani and Michael Lawrence
R Graphics, Second Edition, Paul Murrell
Statistical Computing in C++ and R, Randall L. Eubank and Ana Kupresanin
K11534_FM.indd 2 2/27/12 3:03 PM
The R Series
Event History Analysis
with R
Göran Broström
Professor of Statistics
Centre for Population Studies
Umeå University
Umeå, Sweden
K11534_FM.indd 3 2/27/12 3:03 PM
CRC Press
Taylor & Francis Group
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© 2012 by Taylor & Francis Group, LLC
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Version Date: 20120106
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Contents
List of Figures ix
List of Tables xiii
Preface xv
1 Event History and Survival Data 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Survival Data . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Right Censoring . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Left Truncation . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Event History Data . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 More Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Single Sample Data 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Continuous Time Model Descriptions . . . . . . . . . . . . . 17
2.3 Discrete Time Models . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Nonparametric Estimators . . . . . . . . . . . . . . . . . . . 23
2.5 Doing it in R . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Cox Regression 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 ProportionalHazards . . . . . . . . . . . . . . . . . . . . . . 31
3.3 The Log-Rank Test . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 ProportionalHazards in Continuous Time . . . . . . . . . . 39
3.5 Estimation of the Baseline Hazard . . . . . . . . . . . . . . . 42
3.6 Explanatory Variables . . . . . . . . . . . . . . . . . . . . . . 42
3.7 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8 Interpretation of Parameter Estimates . . . . . . . . . . . . . 47
3.9 ProportionalHazards in Discrete Time . . . . . . . . . . . . 47
3.10 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.11 Male Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
vi
4 Poisson Regression 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The PoissonDistribution . . . . . . . . . . . . . . . . . . . . 57
4.3 The Connection to Cox Regression . . . . . . . . . . . . . . . 59
4.4 The Connection to the Piecewise Constant Hazards Model . 62
4.5 Tabular Lifetime Data . . . . . . . . . . . . . . . . . . . . . . 62
5 More on Cox Regression 67
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Time-Varying Covariates . . . . . . . . . . . . . . . . . . . . 67
5.3 Communal covariates . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Tied Event Times . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Sampling of Risk Sets . . . . . . . . . . . . . . . . . . . . . . 75
5.7 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.8 Checking Model Assumptions . . . . . . . . . . . . . . . . . 80
5.9 Fixed Study Period Survival . . . . . . . . . . . . . . . . . . 83
5.10 Left- or Right-CensoredData . . . . . . . . . . . . . . . . . . 84
6 Parametric Models 85
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 ProportionalHazards Models . . . . . . . . . . . . . . . . . . 85
6.3 Accelerated Failure Time Models . . . . . . . . . . . . . . . . 112
6.4 ProportionalHazards or AFT Model? . . . . . . . . . . . . . 115
6.5 Discrete Time Models . . . . . . . . . . . . . . . . . . . . . . 116
7 Multivariate Survival Models 127
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Frailty Models . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3 Parametric Frailty Models . . . . . . . . . . . . . . . . . . . 134
7.4 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8 Competing Risks Models 139
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.2 Some Mathematics . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.4 Meaningful Probabilities . . . . . . . . . . . . . . . . . . . . 140
8.5 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.6 R Code for Competing Risks . . . . . . . . . . . . . . . . . . 144
9 Causality and Matching 147
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2 Philosophical Aspects of Causality . . . . . . . . . . . . . . . 147
9.3 Causal Inference . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.4 Aalen’s Additive Hazards Model . . . . . . . . . . . . . . . . 150
9.5 Dynamic Path Analysis . . . . . . . . . . . . . . . . . . . . . 152
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9.6 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
A Basic Statistical Concepts 159
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A.2 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . 159
A.3 Asymptotic theory . . . . . . . . . . . . . . . . . . . . . . . . 161
A.4 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . 163
B Survival Distributions 165
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.2 Relevant Distributions in R . . . . . . . . . . . . . . . . . . 165
B.3 ParametricProportionalHazardsandAcceleratedFailureTime
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
C A Brief Introduction to R 177
C.1 R in General . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.2 Some Standard R Functions . . . . . . . . . . . . . . . . . . 182
C.3 Writing Functions . . . . . . . . . . . . . . . . . . . . . . . . 187
C.4 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
C.5 Probability Functions . . . . . . . . . . . . . . . . . . . . . . 194
C.6 Help in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
C.7 Functions in eha and survival . . . . . . . . . . . . . . . . 197
C.8 Reading Data into R . . . . . . . . . . . . . . . . . . . . . . 202
D Survival Packages in R 205
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
D.2 eha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
D.3 survival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
D.4 Other Packages . . . . . . . . . . . . . . . . . . . . . . . . . 207
Bibliography 209
Index 213
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List of Figures
1.1 Survival data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Right censoring. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 A Lexis diagram . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Marital fertility . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 The illness-death model. . . . . . . . . . . . . . . . . . . . . . 12
2.1 Life table and survival function . . . . . . . . . . . . . . . . . 19
2.2 Interpretation of the density function . . . . . . . . . . . . . . 20
2.3 Interpretation of the hazard function . . . . . . . . . . . . . . 21
2.4 The geometric distribution. . . . . . . . . . . . . . . . . . . . 24
2.5 A simple survival data set.. . . . . . . . . . . . . . . . . . . . 24
2.6 Preliminaries for estimating the hazard function. . . . . . . . 25
2.7 Nonparametric estimation of the hazard function. . . . . . . . 26
2.8 The Nelson–Aalen estimator. . . . . . . . . . . . . . . . . . . 27
2.9 The Kaplan–Meier estimator . . . . . . . . . . . . . . . . . . 27
2.10 Male mortality, Nelson–Aalen and Kaplan–Meier plots . . . . 28
2.11 Male mortality, Weibull fit. . . . . . . . . . . . . . . . . . . . 30
3.1 Proportionalhazard functions . . . . . . . . . . . . . . . . . . 32
3.2 Proportionalhazard functions, log scale . . . . . . . . . . . . 33
3.3 Two-sample data . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Old age mortality, women vs. men, cumulative hazards. . . . 37
3.5 Old age mortality by birthplace, cumulative hazards. . . . . . 39
3.6 Effect of proportional hazards . . . . . . . . . . . . . . . . . . 40
3.7 Estimated cumulative baseline hazard function. . . . . . . . . 52
3.8 Effects, interaction with centering. . . . . . . . . . . . . . . . 55
3.9 Effects, interaction without centering . . . . . . . . . . . . . . 55
4.1 The Poisson cdf . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Number of children beyond one for married women . . . . . . 59
4.3 Theoretical Poisson distribution . . . . . . . . . . . . . . . . 59
4.4 Hazard functions for males and females, model based . . . . . 65
4.5 Hazard functions for males and females, raw data . . . . . . . 66
5.1 A time-varying covariate . . . . . . . . . . . . . . . . . . . . . 68
5.2 Log rye price deviations from trend . . . . . . . . . . . . . . . 69
5.3 Nelson–Aalen plot with the aid of the function risksets . . 73
ix