## The Frailty ModelClustered survival data are encountered in many scientific disciplines including human and veterinary medicine, biology, epidemiology, public health and demography. Frailty models provide a powerful tool to analyse clustered survival data. In contrast to the large number of research publications on frailty models, relatively few statistical software packages contain frailty models. It is demanding for statistical practitioners and graduate students to grasp a good knowledge on frailty models from the existing literature. This book provides an in-depth discussion and explanation of the basics of frailty model methodology for such readers. The discussion includes parametric and semiparametric frailty models and accelerated failure time models. Common techniques to fit frailty models include the EM-algorithm, penalised likelihood techniques, Laplacian integration and Bayesian techniques. More advanced frailty models for hierarchical data are also included. Real-life examples are used to demonstrate how particular frailty models can be fitted and how the results should be interpreted. The programs to fit all the worked-out examples in the book are available from the Springer website with most of the programs developed in the freeware packages R and Winbugs. The book starts with a brief overview of some basic concepts in classical survival analysis, collecting what is needed for the reading on the more complex frailty models. |

### What people are saying - Write a review

### Contents

1 | |

2 | |

3 | |

14 Survival analysis | 17 |

141 Survival likelihood | 18 |

142 Proportional hazards models | 20 |

143 Accelerated failure time models | 26 |

144 The loglinear model representation | 30 |

435 Dependence measures | 161 |

436 Diagnostics | 164 |

442 Joint and population survival function | 167 |

443 Updating | 171 |

445 Dependence measures | 173 |

446 Diagnostics | 176 |

45 The power variance function distribution | 177 |

452 Joint and population survival function | 181 |

15 Semantics and history of the term frailty | 32 |

Parametric proportional hazards models with gamma frailty | 43 |

21 The parametric proportional hazards model with frailty term | 44 |

the frequentist approach | 45 |

23 Extension of the marginal likelihood approach to intervalcensored data | 61 |

the Bayesian approach | 65 |

242 Theoretical foundations of the Metropolis algorithm | 74 |

25 Further extensions and references | 75 |

Alternatives for the frailty model | 77 |

31 The ﬁxed effects model | 78 |

312 Asymptotic efficiency of fixed effects model parameter estimates | 84 |

32 The stratified model | 87 |

33 The copula model | 93 |

332 Definition of the copula model | 95 |

333 The Clayton copula | 97 |

334 The Clayton copula versus the gamma frailty model | 99 |

34 The marginal model | 104 |

342 Consistency of parameter estimates from marginal model | 105 |

343 Variance of parameter estimates adjusted for correlation structure | 107 |

35 Population hazards from conditional models | 111 |

352 Population versus conditional hazard ratio from frailty models | 114 |

36 Further extensions and references | 116 |

Frailty distributions | 117 |

41 General characteristics of frailty distributions | 118 |

411 Joint survival function and the Laplace transform | 119 |

412 Population survival function and the copula | 120 |

413 Conditional frailty density changes over time | 122 |

414 Measures of dependence | 123 |

42 The gamma distribution | 130 |

422 Joint and population survival function | 131 |

423 Updating | 134 |

424 Copula form representation | 137 |

425 Dependence measures | 138 |

426 Diagnostics | 141 |

some theoretical considerations | 147 |

43 The inverse Gaussian distribution | 150 |

432 Joint and population survival function | 152 |

433 Updating | 158 |

453 Updating | 184 |

454 Copula form representation | 185 |

455 Dependence measures | 186 |

456 Diagnostics | 189 |

46 The compound Poisson distribution | 190 |

462 Joint and population survival functions | 192 |

463 Updating | 193 |

47 The lognormal distribution | 195 |

48 Further extensions and references | 196 |

The semiparametric frailty model | 199 |

512 Expectation and maximisation for the gamma frailty model | 200 |

513 Why the EM algorithm works for the gamma frailty model | 207 |

52 The penalised partial likelihood approach | 210 |

522 The penalised partial likelihood for the gamma frailty distribution | 214 |

523 Performance of the penalised partial likelihood estimates | 221 |

524 Robustness of the frailty distribution assumption | 228 |

53 Bayesian analysis for the semiparametric gamma frailty model through Gibbs sampling | 233 |

531 The frailty model with a gamma process prior for the cumulative baseline hazard for grouped data | 234 |

532 The frailty model with a gamma process prior for the cumulative baseline hazard for observed event times | 239 |

533 The normal frailty model based on Poisson likelihood | 244 |

534 Sampling techniques used for semiparametric frailty models | 250 |

535 Gibbs sampling a special case of the MetropolisHastings algorithm | 257 |

54 Further extensions and references | 258 |

Multifrailty and multilevel models | 259 |

61 Multifrailty models with one clustering level | 260 |

612 Frequentist approach using Laplacian integration | 268 |

62 Multilevel frailty models | 277 |

622 The Bayesian approach for multilevel frailty models using Gibbs sampling | 279 |

63 Further extensions and references | 286 |

Extensions of the frailty model | 287 |

72 Correlated frailty models | 288 |

73 Joint modelling | 290 |

74 The accelerated failure time model | 292 |

295 | |

308 | |

309 | |

314 | |