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Event History Analysis with R
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Event History Analysis with R
by Göran Broström
Event History Analysis with R
Preliminaries
Preface
Chapter 1 Event History and Survival Data
1.1 Introduction
1.2 Survival Data
1.3 Right Censoring
1.4 Left Truncation
1.5 Time Scales
1.5.1 The Lexis Diagram
1.6 Event History Data
1.7 More Data Sets
Figure 1.1
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 1.5
Chapter 2 Single Sample Data
2.1 Introduction
2.2 Continuous Time Model Descriptions
2.2.1 The Survival Function
2.2.2 The Density Function
2.2.3 The Hazard Function
2.2.4 The cumulative hazard function
2.3 Discrete Time Models
2.4 Nonparametric Estimators
2.4.1 The Hazard Atoms
2.4.2 The Nelson–Aalen Estimator
2.4.3 The Kaplan–Meier Estimator.
2.5 Doing it in R
2.5.1 Nonparametric Estimation
2.5.2 Parametric Estimation
Figure 2.1
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Table 2.1
Table 2.1
Chapter 3 Cox Regression
3.1 Introduction
3.2 Proportional Hazards
3.3 The Log-Rank Test
3.3.1 Two Samples
3.3.2 Several Samples
3.4 Proportional Hazards in Continuous Time
3.4.1 Proportional Hazards, Two Groups
3.4.2 Proportional Hazards, More Than Two Groups
3.4.3 The General Proportional Hazards Regression Model
3.5 Estimation of the Baseline Hazard
3.6 Explanatory Variables
3.6.1 Continuous Covariates
3.6.2 Factor Covariates
3.7 Interactions
3.7.1 Two Factors
3.7.2 One Factor and one Continuous Covariate
3.7.3 Two Continuous Covariates
3.8 Interpretation of Parameter Estimates
3.8.1 Continuous Covariate
3.8.2 Factor
3.9 Proportional Hazards in Discrete Time
3.9.1 Logistic Regression
3.10 Model Selection
3.10.1 Model Selection in General
3.11 Male Mortality
3.11.1 Likelihood Ratio Test
3.11.2 The Estimated Baseline Cumulative Hazard Function
3.11.3 Interaction
Figure 3.1
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Table 3.1
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Table 3.5
Table 3.6
Chapter 4 Poisson Regression
4.1 Introduction
4.2 The Poisson Distribution
4.3 The Connection to Cox Regression
4.4 The Connection to the Piecewise Constant Hazards Model
4.5 Tabular Lifetime Data
Figure 4.1
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Chapter 5 More on Cox Regression
5.1 Introduction
5.2 Time-Varying Covariates
5.3 Communal covariates
5.4 Tied Event Times
5.5 Stratification
5.6 Sampling of Risk Sets
5.7 Residuals
5.7.1 Martingale Residuals
5.8 Checking Model Assumptions
5.8.1 Proportionality
5.8.2 Log-Linearity
5.9 Fixed Study Period Survival
5.10 Left- or Right-Censored Data
Figure 5.1
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Table 5.1
Table 5.1
Table 5.2
Chapter 6 Parametric Models
6.1 Introduction
6.2 Proportional Hazards Models
6.2.1 The Weibull Model
6.2.2 The Lognormal Model
6.2.3 Comparing the Weibull and Lognormal Fits
6.2.4 The Piecewise Constant Hazards (PCH) Model
6.2.4.1 Testing the Proportionality Assumption with the PCH Model
6.2.5 Choosing the best parametric model
6.3 Accelerated Failure Time Models
6.3.1 The AFT Regression Model
6.3.2 Different Parametrizations
6.3.3 AFT Models in R
6.4 Proportional Hazards or AFT Model?
6.5 Discrete Time Models
Figure 6.1
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 6.10
Figure 6.11
Figure 6.12
Figure 6.13
Figure 6.14
Figure 6.15
Figure 6.16
Figure 6.17
Figure 6.18
Figure 6.19
Figure 6.20
Chapter 7 Multivariate Survival Models
7.1 Introduction
7.1.1 An Introductory Example
7.2 Frailty Models
7.2.1 The Simple Frailty Model
7.2.2 The Shared Frailty Model
7.3 Parametric Frailty Models
7.4 Stratification
Figure 7.1
Figure 7.1
Chapter 8 Competing Risks Models
8.1 Introduction
8.2 Some Mathematics
8.3 Estimation
8.4 Meaningful Probabilities
8.5 Regression
8.6 R Code for Competing Risks
Figure 8.1
Figure 8.1
Figure 8.2
Figure 8.3
Table 8.1
Table 8.1
Chapter 9 Causality and Matching
9.1 Introduction
9.2 Philosophical Aspects of Causality
9.3 Causal Inference
9.3.1 Graphical Models
9.3.2 Predictive causality
9.3.3 Counterfactuals
9.4 Aalen’s Additive Hazards Model
9.5 Dynamic Path Analysis
9.6 Matching
9.6.1 Paired Data
9.6.2 More than One Control
9.7 Conclusion
Figure 9.1
Figure 9.1
Figure 9.2
Figure 9.3
Appendix A Basic Statistical Concepts
A.1 Introduction
A.2 Statistical Inference
A.2.1 Point Estimation
A.2.2 Interval Estimation
A.2.3 Hypothesis Testing
A.2.3.1 The Log-Rank Test
A.3 Asymptotic theory
A.3.1 Partial likelihood
A.4 Model Selection
A.4.1 Nested Models
Table A.1
Table A.1
Appendix B Survival Distributions
B.1 Introduction
B.2 Relevant Distributions in R
B.2.1 The Exponential Distribution
B.2.2 The Piecewise Constant Hazard Distribution
B.2.3 The Weibull Distribution
B.2.3.1 Graphical Test of the Weibull Distribution
B.2.4 The Lognormal Distribution
B.2.5 The Loglogistic Distribution
B.2.6 The Gompertz Distribution
B.2.7 The Gompertz–Makeham Distribution
B.2.8 The Gamma Distribution
B.3 Parametric Proportional Hazards and Accelerated Failure Time Models
B.3.1 Introduction
B.3.2 The Proportional Hazards Model
B.3.2.1 Data and the Likelihood Function
B.3.3 The Shape-Scale Families
B.3.3.1 The Weibull Family of Distributions
B.3.3.2 The EV family of distributions
B.3.3.3 The Gompertz Family of Distributions
B.3.3.4 Other Families of Distributions
B.3.4 The Accelerated Failure Time Model
B.3.4.1 Data and the Likelihood Function
Figure B.1
Figure B.1
Figure B.2
Figure B.3
Figure B.4
Appendix C A Brief Introduction to R
C.1 R in General
C.1.1 R Objects
C.1.2 Expressions and Assignments
C.1.3 Objects
C.1.4 Vectors and Matrices
C.1.5 Lists
C.1.6 Data Frames
C.1.7 Factors
C.1.8 Operators
C.1.9 Recycling
C.1.10 Precedence
C.2 Some Standard R Functions
C.2.1 Sequences
C.2.2 Logical expression
C.2.3 Indexing
C.2.4 Vectors and Matrices
C.2.5 Conditional Execution
C.2.6 Loops
C.2.7 Vectorizing
C.3 Writing Functions
C.3.1 Calling Conventions
C.3.2 The Argument “...”
C.3.3 Writing Functions
C.3.4 Lazy Evaluation
C.3.5 Recursion
C.3.6 Vectorized Functions
C.3.7 Scoping Rules
C.4 Graphics
C.5 Probability Functions
C.5.1 Some Useful R Functions
C.5.1.1 Matching
C.5.1.2 General utility functions
C.6 Help in R
C.7 Functions in eha and survival
C.7.1 Checking the Integrity of Survival Data
C.8 Reading Data into R
C.8.1 Reading Data from ASCII Files
C.8.2 Reading Foreign Data Files
Figure C.1
Figure C.1
Figure C.2
Table C.1
Table C.1
Appendix D Survival Packages in R
D.1 Introduction
D.2 eha
D.3 survival
D.4 Other Packages
D.4.1 coxme
D.4.2 timereg
D.4.3 cmprsk
Bibliography
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