Survival and Event History Analysis -- Preface -- 1 An introduction to survival and event history analysis -- 1.1 Survival analysis: basic concepts and examples -- 1.1.1 What makes survival special: censoring and truncation -- 1.1.2 Survival function and hazard rate -- 1.1.3 Regression and frailty models -- 1.1.4 The past -- 1.1.5 Some illustrative examples -- 1.2 Event history analysis: models and examples -- 1.2.1 Recurrent event data -- 1.2.2 Multistate models -- 1.3 Data that do not involve time -- 1.4 Counting processes -- 1.4.1 What is a counting process? -- 1.4.2 Survival times and counting processes -- 1.4.3 Event histories and counting processes -- 1.5 Modeling event history data -- 1.5.1 The multiplicative intensity model -- 1.5.2 Regression models -- 1.5.3 Frailty models and first passage time models -- 1.5.4 Independent or dependent data? -- 1.6 Exercises -- 2 Stochastic processes in event history analysis -- 2.1 Stochastic processes in discrete time -- 2.1.1 Martingales in discrete time -- 2.1.2 Variation processes -- 2.1.3 Stopping times and transformations -- 2.1.4 The Doob decomposition -- 2.2 Processes in continuous time -- 2.2.1 Martingales in continuous time -- 2.2.2 Stochastic integrals -- 2.2.3 The Doob-Meyer decomposition -- 2.2.4 The Poisson process -- 2.2.5 Counting processes -- 2.2.6 Stochastic integrals for counting process martingales -- 2.2.7 The innovation theorem -- 2.2.8 Independent censoring -- 2.3 Processes with continuous sample paths -- 2.3.1 The Wiener process and Gaussian martingales -- 2.3.2 Asymptotic theory for martingales: intuitive discussion -- 2.3.3 Asymptotic theory for martingales: mathematical formulation -- 2.4 Exercises -- 3 Nonparametric analysis of survival and event history data -- 3.1 The Nelson-Aalen estimator -- 3.1.1 The survival data situation -- 3.1.2 The multiplicative intensity model.

3.1.3 Handling of ties -- 3.1.4 Smoothing the Nelson-Aalen estimator -- 3.1.5 The estimator and its small sample properties -- 3.1.6 Large sample properties -- 3.2 The Kaplan-Meier estimator -- 3.2.1 The estimator and confidence intervals -- 3.2.2 Handling tied survival times -- 3.2.3 Median and mean survival times -- 3.2.4 Product-integral representation -- 3.2.5 Excess mortality and relative survival -- 3.2.6 Martingale representation and statistical properties -- 3.3 Nonparametric tests -- 3.3.1 The two-sample case -- 3.3.2 Extension to more than two samples -- 3.3.3 Stratified tests -- 3.3.4 Handling of tied observations -- 3.3.5 Asymptotics -- 3.4 The empirical transition matrix -- 3.4.1 Competing risks and cumulative incidence functions -- 3.4.2 An illness-death model -- 3.4.3 The general case -- 3.4.4 Martingale representation and large sample properties -- 3.4.5 Estimation of (co)variances -- 3.5 Exercises -- 4 Regression models -- 4.1 Relative risk regression -- 4.1.1 Partial likelihood and inference for regression coefficients -- 4.1.2 Estimation of cumulative hazards and survival probabilities -- 4.1.3 Martingale residual processes and model check -- 4.1.4 Stratified models -- 4.1.5 Large sample properties of "0362 -- 4.1.6 Large sample properties of estimators of cumulative hazards and survival functions -- 4.2 Additive regression models -- 4.2.1 Estimation in the additive hazard model -- 4.2.2 Interpreting changes over time -- 4.2.3 Martingale tests and a generalized log-rank test -- 4.2.4 Martingale residual processes and model check -- 4.2.5 Combining the Cox and the additive models -- 4.2.6 Adjusted monotone survival curves for comparing groups -- 4.2.7 Adjusted Kaplan-Meier curves under dependent censoring -- 4.2.8 Excess mortality models and the relative survival function -- 4.2.9 Estimation of Markov transition probabilities.

4.3 Nested case-control studies -- 4.3.1 A general framework for nested-case control sampling -- 4.3.2 Two important nested case-control designs -- 4.3.3 Counting process formulation of nested case-control sampling -- 4.3.4 Relative risk regression for nested case-control data -- 4.3.5 Additive regression for nested case-control data: results -- 4.3.6 Additive regression for nested case-control data: theory -- 4.4 Exercises -- 5 Parametric counting process models -- 5.1 Likelihood inference -- 5.1.1 Parametric models for survival times -- 5.1.2 Likelihood for censored survival times -- 5.1.3 Likelihood for counting process models -- 5.1.4 The maximum likelihood estimator and related tests -- 5.1.5 Some applications -- 5.2 Parametric regression models -- 5.2.1 Poisson regression -- 5.3 Proof of large sample properties -- 5.4 Exercises -- 6 Unobserved heterogeneity: The odd effects of frailty -- 6.1 What is randomness in survival models? -- 6.2 The proportional frailty model -- 6.2.1 Basic properties -- 6.2.2 The Gamma frailty distribution -- 6.2.3 The PVF family of frailty distributions -- 6.2.4 Lévy-type frailty distributions -- 6.3 Hazard and frailty of survivors -- 6.3.1 Results for the PVF distribution -- 6.3.2 Cure models -- 6.3.3 Asymptotic distribution of survivors -- 6.4 Parametric models derived from frailty distributions -- 6.4.1 A model based on Gamma frailty: the Burr distribution -- 6.4.2 A model based on PVF frailty -- 6.4.3 The Weibull distribution derived from stable frailty -- 6.4.4 Frailty and estimation -- 6.5 The effect of frailty on hazard ratio -- 6.5.1 Decreasing relative risk and crossover -- 6.5.2 The effect of discontinuing treatment -- 6.5.3 Practical implications of artifacts -- 6.5.4 Frailty models yielding proportional hazards -- 6.6 Competing risks and false protectivity -- 6.7 A frailty model for the speed of a process.

6.8 Frailty and association between individuals -- 6.9 Case study: A frailty model for testicular cancer -- 6.10 Exercises -- 7 Multivariate frailty models -- 7.1 Censoring in the multivariate case -- 7.1.1 Censoring for recurrent event data -- 7.1.2 Censoring for clustered survival data -- 7.2 Shared frailty models -- 7.2.1 Joint distribution -- 7.2.2 Likelihood -- 7.2.3 Empirical Bayes estimate of individual frailty -- 7.2.4 Gamma distributed frailty -- 7.2.5 Other frailty distributions suitable for the shared frailty model -- 7.3 Frailty and counting processes -- 7.4 Hierarchical multivariate frailty models -- 7.4.1 A multivariate model based on Lévy-type distributions -- 7.4.2 A multivariate stable model -- 7.4.3 The PVF distribution with m=1 -- 7.4.4 A trivariate model -- 7.4.5 A simple genetic model -- 7.5 Case study: A hierarchical frailty model for testicular cancer -- 7.6 Random effects models for transformed times -- 7.6.1 Likelihood function -- 7.6.2 General case -- 7.6.3 Comparing frailty and random effects models -- 7.7 Exercises -- 8 Marginal and dynamic models for recurrent events and clustered survival data -- 8.1 Intensity models and rate models -- 8.1.1 Dynamic covariates -- 8.1.2 Connecting intensity and rate models in the additive case -- Leaving out a fixed covariate -- Leaving out the dynamic covariate N(t-) -- 8.2 Nonparametric statistical analysis -- 8.2.1 A marginal Nelson-Aalen estimator for clustered survival data -- 8.2.2 A dynamic Nelson-Aalen estimator for recurrent event data -- 8.3 Regression analysis of recurrent events and clustered survival data -- 8.3.1 Relative risk models -- A marginal Cox model -- A dynamic Cox model -- 8.3.2 Additive models -- Residual processes -- Testing the martingale property -- Variance estimators for rate functions -- Embedding a rate model in an intensity model.

Testing for influence: The hat matrix -- 8.4 Dynamic path analysis of recurrent event data -- 8.4.1 General considerations -- 8.5 Contrasting dynamic and frailty models -- 8.6 Dynamic models -- theoretical considerations -- 8.6.1 A dynamic view of the frailty model for Poisson processes -- 8.6.2 General view on the connection between dynamic and frailty models -- 8.6.3 Are dynamic models well defined? -- 8.7 Case study: Protection from natural infections with enterotoxigenic Escherichia coli -- 8.8 Exercises -- 9 Causality -- 9.1 Statistics and causality -- 9.1.1 Schools of statistical causality -- 9.1.2 Some philosophical aspects -- 9.1.3 Traditional approaches to causality in epidemiology -- 9.1.4 The great theory still missing? -- 9.2 Graphical models for event history analysis -- 9.2.1 Time-dependent covariates -- 9.3 Local characteristics - dynamic model -- 9.3.1 Dynamic path analysis -- a general view -- 9.3.2 Direct and indirect effects -- a general concept -- 9.4 Granger-Schweder causality and local dependence -- 9.4.1 Local dependence -- 9.4.2 A general definition of Granger-Schweder causality -- 9.4.3 Statistical analysis of local dependence -- 9.5 Counterfactual causality -- 9.5.1 Standard survival analysis and counterfactuals -- 9.5.2 Censored and missing data -- 9.5.3 Dynamic treatment regimes -- 9.5.4 Marginal versus joint modeling -- 9.6 Marginal modeling -- 9.6.1 Marginal structural models -- 9.6.2 G-computation: A Markov modeling approach -- 9.7 Joint modeling -- 9.7.1 Joint modeling as an alternative to marginal structural models -- 9.7.2 Modeling dynamic systems -- 9.8 Exercises -- 10 First passage time models: Understanding the shape of the hazard rate -- 10.1 First hitting time -- phase type distributions -- 10.1.1 Finite birth-death process with absorbing state -- 10.1.2 First hitting time as the time to event.

10.1.3 The risk distribution of survivors.