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Contents
Preface ............................................................ vii
1 An introduction to survival and event history analysis .............. 1
1.1 Survival analysis: basic concepts and examples . . . . .............. 2
1.1.1 What makes survival special: censoring and truncation . . . . . 3
1.1.2 Survival function and hazard rate . . ..................... 5
1.1.3 Regression and frailty models . . . . . ..................... 7
1.1.4 Thepast............................................ 9
1.1.5 Some illustrative examples ............................ 9
1.2 Event history analysis: models and examples . . . . . . .............. 16
1.2.1 Recurrent event data . ................................. 17
1.2.2 Multistate models . ................................... 18
1.3 Datathatdonotinvolvetime................................. 24
1.4 Counting processes ......................................... 25
1.4.1 What is a counting process? . . . . . . ..................... 25
1.4.2 Survival times and counting processes . . . . .............. 28
1.4.3 Event histories and counting processes . . . . .............. 32
1.5 Modeling event history data . ................................. 33
1.5.1 The multiplicative intensity model . ..................... 34
1.5.2 Regression models . . . ................................ 34
1.5.3 Frailty models and first passage time models . . . .......... 35
1.5.4 Independent or dependent data? . . . ..................... 36
1.6 Exercises ................................................. 37
2 Stochastic processes in event history analysis ...................... 41
2.1 Stochastic processes in discrete time ........................... 43
2.1.1 Martingales in discrete time ........................... 43
2.1.2 Variationprocesses................................... 44
2.1.3 Stopping times and transformations ..................... 45
2.1.4 The Doob decomposition. . ............................ 47
2.2 Processes in continuous time . ................................ 48
xixii Contents
2.2.1 Martingales in continuous time ......................... 48
2.2.2 Stochastic integrals . . . ................................ 50
2.2.3 The Doob-Meyer decomposition . . ..................... 52
2.2.4 The Poisson process . . ................................ 52
2.2.5 Counting processes ................................... 53
2.2.6 Stochastic integrals for counting process martingales . . . . . . 55
2.2.7 The innovation theorem . . . ............................ 56
2.2.8 Independent censoring . . . . ............................ 57
2.3 Processes with continuous sample paths . . . ..................... 61
2.3.1 The Wiener process and Gaussian martingales . .......... 61
2.3.2 Asymptotic theory for martingales: intuitive discussion . . . . 62
2.3.3 Asymptotic theory for martingales: mathematical
formulation ......................................... 63
2.4 Exercises ................................................. 66
3 Nonparametric analysis of survival and event history data .......... 69
3.1 TheNelson-Aalenestimator.................................. 70
3.1.1 Thesurvivaldatasituation............................. 71
3.1.2 The multiplicative intensity model . ..................... 76
3.1.3 Handling of ties ..................................... 83
3.1.4 Smoothing the Nelson-Aalen estimator .................. 85
3.1.5 The estimator and its small sample properties . . . .......... 87
3.1.6 Large sample properties . . . ............................ 89
3.2 The Kaplan-Meier estimator ................................. 90
3.2.1 The estimator and confidence intervals . . . . .............. 90
3.2.2 Handling tied survival times ........................... 94
3.2.3 Median and mean survival times ........................ 95
3.2.4 Product-integral representation ......................... 97
3.2.5 Excess mortality and relative survival . . . . . .............. 99
3.2.6 Martingale representation and statistical properties . . . . . . . . 103
3.3 Nonparametric tests . . .......................................104
3.3.1 The two-sample case . ................................105
3.3.2 Extension to more than two samples ....................109
3.3.3 Stratifiedtests .......................................110
3.3.4 Handling of tied observations ..........................111
3.3.5 Asymptotics . .......................................112
3.4 The empirical transition matrix . . . ............................114
3.4.1 Competing risks and cumulative incidence functions . . . . . . . 114
3.4.2 An illness-death model . . . ............................117
3.4.3 The general case . . . . . ................................120
3.4.4 Martingale representation and large sample properties . . . . . 123
3.4.5 Estimation of (co)variances . . . . . . . .....................124
3.5 Exercises .................................................126Contents xiii
4 Regression models ..............................................131
4.1 Relative risk regression ......................................133
4.1.1 Partial likelihood and inference for regression coefficients . . 134
4.1.2 Estimation of cumulative hazards and survival probabilities . 141
4.1.3 Martingale residual processes and model check . ..........142
4.1.4 Stratified models . . . . . ................................148
4.1.5 Large sample properties of
 β β β ..........................149
4.1.6 Large sample properties of estimators of cumulative
hazards and survival functions . . . . .....................152
4.2 Additive regression models . . ................................154
4.2.1 Estimation in the additive hazard model . . . ..............157
4.2.2 Interpreting changes over time . . . ......................163
4.2.3 Martingale tests and a generalized log-rank test . ..........164
4.2.4 Martingale residual processes and model check . ..........167
4.2.5 Combining the Cox and the additive models . . . . ..........171
4.2.6 Adjusted monotone survival curves for comparing groups . . 172
4.2.7 Adjusted Kaplan-Meier curves under dependent censoring . . 175
4.2.8 Excess mortality models and the relative survival function . . 179
4.2.9 Estimation of Markov transition probabilities . . . ..........181
4.3 Nested case-control studies . . ................................190
4.3.1 A general framework for nested-case control sampling . . . . . 192
4.3.2 Two important nested case-control designs . ..............194
4.3.3 Counting process formulation of nested case-control
sampling ...........................................195
4.3.4 Relative risk regression for nested case-control data . . . . . . . 196
4.3.5 Additive regression for nested case-control data: results . . . . 200
4.3.6 Additive regression for nested case-control data: theory . . . . 202
4.4 Exercises .................................................203
5 Parametric counting process models ..............................207
5.1 Likelihood inference . .......................................208
5.1.1 Parametric models for survival times . . . . . . ..............208
5.1.2 Likelihood for censored survival times . . . . ..............209
5.1.3 Likelihood for counting process models . . . ..............210
5.1.4 The maximum likelihood estimator and related tests . . . . . . . 213
5.1.5 Some applications ...................................214
5.2 Parametric regression models .................................223
5.2.1 Poisson regression ...................................223
5.3 Proof of large sample properties . . ............................226
5.4 Exercises .................................................228
6 Unobserved heterogeneity: The odd effects of frailty ...............231
6.1 What is randomness in survival models? . . .....................233
6.2 The proportional frailty model ................................234
6.2.1 Basic properties . . . . . ................................234xiv Contents
6.2.2 The Gamma frailty distribution .........................235
6.2.3 The PVF family of frailty distributions ..................238
6.2.4 L´ evy-type frailty distributions ..........................242
6.3 Hazard and frailty of survivors . . . ............................243
6.3.1 Results for the PVF distribution ........................243
6.3.2 Cure models . .......................................244
6.3.3 Asymptotic distribution of survivors ....................245
6.4 Parametric models derived from frailty distributions . . . ..........246
6.4.1 A model based on Gamma frailty: the Burr distribution . . . . 246
6.4.2 A model based on PVF frailty . . . . . .....................247
6.4.3 The Weibull distribution derived from stable frailty .......248
6.4.4 Frailty and estimation ................................249
6.5 The effect of frailty on hazard ratio . . . . . . .....................250
6.5.1 Decreasing relative risk and crossover . .................250
6.5.2 The effect of discontinuing treatment ...................253
6.5.3 Practical implications of artifacts . .....................255
6.5.4 Frailty models yielding proportional hazards . . . ..........257
6.6 Competing risks and false protectivity .........................260
6.7 A frailty model for the speed of a process . .....................262
6.8 Frailty and association between individuals . . . . . . ..............264
6.9 Case study: A frailty model for testicular cancer . . ..............265
6.10 Exercises .................................................268
7 Multivariate frailty models .....................................271
7.1 Censoring in the multivariate case .............................272
7.1.1 Censoring for recurrent event data . .....................273
7.1.2 Censoring for clustered survival data ....................274
7.2 Shared frailty models .......................................275
7.2.1 Joint distribution .....................................276
7.2.2 Likelihood . . . .......................................276
7.2.3 Empirical Bayes estimate of individual frailty . . ..........278
7.2.4 Gamma distributed frailty . ............................279
7.2.5 Other frailty distributions suitable for the shared frailty
model . . . ...........................................284
7.3 Frailty and counting processes ................................286
7.4 Hierarchical multivariate frailty models . . .....................288
7.4.1 A multivariate model based on L´ evy-type distributions . . . . . 289
7.4.2 A multivariate stable model ............................290
7.4.3 The PVF distribution with m = 1 .......................290
7.4.4 A trivariate model ....................................290
7.4.5 A simple genetic model . . . ............................291
7.5 Case study: A hierarchical frailty model for testicular cancer . . . . . . 293
7.6 Random effects models for transformed times . . . . . ..............296
7.6.1 Likelihood function . . ................................296
7.6.2 General case . .......................................298Contents xv
7.6.3 Comparing frailty and random effects models . . ..........299
7.7 Exercises .................................................299
8 Marginal and dynamic models for recurrent events and clustered
survival data...................................................301
8.1 Intensity models and rate models . . ............................302
8.1.1 Dynamic covariates . . ................................304
8.1.2 Connecting intensity and rate models in the additive case . . . 305
8.2 Nonparametric statistical analysis . ............................308
8.2.1 A marginal Nelson-Aalen estimator for clustered survival
data................................................308
8.2.2 A dynamic Nelson-Aalen estimator for recurrent event data . 309
8.3 Regression analysis of recurrent events and clustered survival data . 311
8.3.1 Relative risk models . . ................................313
8.3.2 Additive models . . . . . ................................315
8.4 Dynamic path analysis of recurrent event data . . . . . ..............324
8.4.1 General considerations . . . . ............................325
8.5 Contrasting dynamic and frailty models . . . .....................331
8.6 Dynamic models – theoretical considerations . . . . . ..............333
8.6.1 A dynamic view of the frailty model for Poisson processes . 333
8.6.2 General view on the connection between dynamic
and frailty models ....................................334
8.6.3 Are dynamic models well defined? . .....................336
8.7 Case study: Protection from natural infections
with enterotoxigenic Escherichia coli ..........................340
8.8 Exercises .................................................346
9 Causality ......................................................347
9.1 Statistics and causality . . . . . . ................................347
9.1.1 Schools of statistical causality . . . . .....................349
9.1.2 Some philosophical aspects . . . . . . . .....................351
9.1.3 Traditional approaches to causality in epidemiology . . . . . . . 353
9.1.4 The great theory still missing? . . . . .....................353
9.2 Graphical models for event history analysis .....................354
9.2.1 Time-dependent covariates ............................356
9.3 Local characteristics - dynamic model . . . . .....................361
9.3.1 Dynamic path analysis – a general view . . . ..............363
9.3.2 Direct and indirect effects – a general concept . . ..........365
9.4 Granger-Schweder causality and local dependence . ..............367
9.4.1 Local dependence . . . . ................................367
9.4.2 A general definition of Granger-Schweder causality . . . . . . . 370
9.4.3 Statistical analysis of local dependence . . . . ..............371
9.5 Counterfactual causality . . . . . ................................373
9.5.1 Standard survival analysis and counterfactuals . . ..........376
9.5.2 Censored and missing data ............................377xvi Contents
9.5.3 Dynamic treatment regimes ............................378
9.5.4 Marginal versus joint modeling ........................380
9.6 Marginal modeling .........................................380
9.6.1 Marginal structural models ............................380
9.6.2 G-computation: A Markov modeling approach . . ..........382
9.7 Joint modeling . ............................................383
9.7.1 Joint modeling as an alternative to marginal structural
models . . ...........................................384
9.7.2 Modeling dynamic systems . . . . . . . .....................385
9.8 Exercises .................................................385
10 First passage time models: Understanding the shape of the hazard
rate ...........................................................387
10.1 First hitting time; phase type distributions . .....................389
10.1.1 Finite birth-death process with absorbing state . . ..........389
10.1.2 First hitting time as the time to event ....................390
10.1.3 The risk distribution of survivors .......................392
10.1.4 Reversibility and progressive models . . . . . . ..............393
10.2 Quasi-stationary distributions ................................395
10.2.1 Infinite birth-death process (infinite random walk) . . . . . . . . . 397
10.2.2 Interpretation........................................398
10.3 Wiener process models . . . . . . ................................399
10.3.1 The inverse Gaussian hitting time distribution . . ..........400
10.3.2 Comparison of hazard rates ............................402
10.3.3 The distribution of survivors ..........................404
10.3.4 Quasi-stationary distributions for the Wiener process
withabsorption......................................405
10.3.5 Wiener process with a random initial value . ..............407
10.3.6 Wiener process with lower absorbing and upper reflecting
barriers . ...........................................408
10.3.7 Wiener process with randomized drift . . . . ..............408
10.3.8 Analyzing the effect of covariates for the randomized
Wiener process . . . . . . ................................410
10.4 Diffusion process models . . . . ................................416
10.4.1 The Kolmogorov equations and a formula for the hazard
rate ...............................................418
10.4.2 An equation for the quasi-stationary distribution ..........419
10.4.3 The Ornstein-Uhlenbeck process . . .....................421
10.5 Exercises .................................................424
11 Diffusion and L´ evy process models for dynamic frailty .............425
11.1 Population versus individual survival . . . . . .....................426
11.2 Diffusion models for the hazard . . ............................428
11.2.1 A simple Wiener process model . . . .....................428Contents xvii
11.2.2 The hazard rate as the square of an Ornstein-Uhlenbeck
process ............................................430
11.2.3 More general diffusion processes . . .....................431
11.3 Models based on L´ evyprocesses..............................432
11.4 L´ evy processes and subordinators . ............................433
11.4.1 Laplace exponent . . . . ................................433
11.4.2 Compound Poisson processes and the PVF process . . . . . . . 434
11.4.3 Other examples of subordinators . . .....................435
11.4.4 L´ evy measure .......................................436
11.5 A L´ evy process model for the hazard . . . . . .....................438
11.5.1 Population survival . . ................................440
11.5.2 The distribution of h conditional on no event . . . ..........440
11.5.3 Standard frailty models . . . ............................441
11.5.4 Movingaverage .....................................441
11.5.5 Accelerated failure times . ............................443
11.6 ResultsforthePVFprocesses ................................444
11.6.1 Distribution of survivors for the PVF processes . ..........445
11.6.2 Moving average and the PVF process . . . . . ..............446
11.7 Parameterization and estimation ..............................448
11.8 Limit results and quasi-stationary distributions ..................450
11.8.1 Limits for the PVF process ............................452
11.9 Exercises .................................................453
A Markov processes and the product-integral .......................457
A.1 Hazard, survival, and the product-integral . .....................458
A.2 Markov chains, transition intensities, and the Kolmogorov
equations . . . . . . ...........................................461
A.2.1 Discrete time-homogeneous Markov chains ..............463
A.2.2 Continuous time-homogeneous Markov chains . ..........465
A.2.3 The Kolmogorov equations for homogeneous Markov
chains . . ...........................................467
A.2.4 Inhomogeneous Markov chains and the product-integral . . . . 468
A.2.5 Common multistate models ............................471
A.3 Stationary and quasi-stationary distributions . . . . . . ..............475
A.3.1 The stationary distribution of a discrete Markov chain .....475
A.3.2 The quasi-stationary distribution of a Markov chain
withanabsorbingstate ...............................477
A.4 Diffusion processes and stochastic differential equations . . . . . . . . . . 479
A.4.1 The Wiener process . . ................................480
A.4.2 Stochastic differential equations . . . .....................482
A.4.3 The Ornstein-Uhlenbeck process . . .....................484
A.4.4 The infinitesimal generator and the Kolmogorov equations
for a diffusion process ................................486
A.4.5 The Feynman-Kac formula ............................488
A.5 L´ evy processes and subordinators . ............................490xviii Contents
A.5.1 The L´ evy process . . . . ................................491
A.5.2 The Laplace exponent ................................493
B Vector-valued counting processes, martingales and stochastic
integrals ......................................................495
B.1 Counting processes, intensity processes and martingales ..........495
B.2 Stochastic integrals . . .......................................496
B.3 Martingale central limit theorem . . ............................497
References .........................................................499
Author index .......................................................521
Index .............................................................529

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