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Part I Data examples and some mathematical background
1 Data examples ............................................. 3
2 An informal introduction to hazard-based analyses ........ 9
2.1 Whysurvivalanalysisishazard-based. ..................... 10
2.1.1 Survival multistate model, hazard, and survival
probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Estimation: The hazard remains ‘undisturbed’ by
censoring. ........................................ 13
2.2 Consequences of survival analysis being based on hazards . . . . . 17
2.2.1 Countingprocessesand martingales ................. 17
2.2.2 Left-truncationand right-censoring .................. 20
2.2.3 Competingrisks................................... 23
2.2.4 Time-inhomogeneous Markov multistate models . . . . . . . 26
2.3 Approximate inference in practice based on large sample results 34
2.4 Exercises ............................................... 36
Part II Competing risks
3 Multistate modelling of competing risks ................... 41
3.1 Thecompetingrisksmultistatemodel...................... 42
3.2 Simulatingcompetingrisksdata........................... 45
3.3 Thelatentfailuretimemodel ............................. 50
3.4 Exercises ............................................... 53
IX
X Contents
4 Nonparametric estimation ................................. 55
4.1 The Nelson-Aalen and Aalen-Johansen estimators . . . . . . . . . . . 55
4.2 Analysisof simulatedcompetingrisksdata ................. 62
4.3 Analysisof hospitaldata ................................. 76
4.3.1 Latentfailuretimeapproach........................ 80
4.4 Analysisof pregnancyoutcomedata ....................... 82
4.5 A special case: No effect on the competing cause-specific hazard 85
4.6 Exercises ............................................... 87
5 Proportional hazards models .............................. 89
5.1 Introduction ............................................ 90
5.2 Proportionalcause-specifichazardsmodels ................. 93
5.2.1 Estimation ....................................... 93
5.2.2 Examples ........................................ 98
5.3 Proportionalsubdistributionhazardsmodel.................122
5.3.1 Thesubdistributionprocess ........................123
5.3.2 Estimation .......................................126
5.3.3 Examples ........................................133
5.3.4 Proportional subdistribution hazards analysis of all
cumulativeincidencefunctions ......................139
5.3.5 Left-truncation....................................140
5.3.6 Simulating proportional subdistribution hazards data . . 141
5.4 Theleastfalseparameter.................................144
5.5 Goodness-of-fitmethods..................................146
5.6 Beyondproportionalhazards..............................150
5.7 Exercises ...............................................152
6 Nonparametric hypothesis testing .........................155
6.1 Exercises ...............................................158
7 Further topics in competing risks ..........................159
7.1 Morethantwo competingrisks............................159
7.2 Frequentlyaskedquestions ...............................160
7.3 Discretetimemethodsfor competingrisks..................164
Part III Multistate models
8 Multistate models and their connection to competing risks 169
8.1 Time-inhomogeneous Markov processes with finite state space . 170
8.2 Nestedseriesof competingrisksexperiments................172
8.3 Exercises ...............................................175
Contents XI
9 Nonparametric estimation .................................177
9.1 The Nelson-Aalen and Aalen-Johansen estimators . . . . . . . . . . . 177
9.2 Examples...............................................182
9.2.1 Impact of hospital-acquired pneumonia on length of
stayand mortalityinintensivecareunits ............182
9.2.2 Impact of ventilation on length of intensive care unit
stay .............................................192
9.3 Exercises ...............................................193
10 Proportional transition hazards models ....................197
10.1 Model formulation and practical implementation . . . . . . . . . . . . 197
10.2 Examples...............................................202
10.2.1 Hospital-acquiredpneumonia .......................202
10.2.2 Ventilationinintensivecareunit ....................206
10.3 Exercises ...............................................208
11 Time-dependent covariates and multistate models.........211
11.1 A simple joint model for time-dependent covariates and
time-to-eventendpoints ..................................212
11.1.1 Transient states and time-dependent covariates . . . . . . . . 212
11.1.2 A proportional hazards model with time-dependent
covariates ........................................215
11.1.3 Example .........................................216
11.2 Time-dependent covariates and competing risks . . . . . . . . . . . . . 217
11.2.1 Cause-specifichazards .............................218
11.2.2 Subdistributionhazard.............................219
11.2.3 Example .........................................220
11.3 Further topics in the analysis of time-dependent covariates . . . . 222
11.4 Exercises ...............................................225
12 Further topics in multistate modelling .....................227
A A nonparametric bootstrap for multistate models ..........231
References.....................................................233
Index ..........................................................241
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