Clustered 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. TOC:Introduction.- Parametric Proportional Hazards Models With Gamma Frailty.- Alternatives for the Frailty Model.- Frailty Distributions.- The Semiparametric Frailty Model.- Multi-Frailty and Multilevel Models.- Extensions of the Frailty Model. 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. Preface vii Glossary of Definitions and Notation xv Introduction 1(42) Goals 1(1) Outline 2(1) Examples 3(14) Survival analysis 17(15) Survival likelihood 18(2) Proportional hazards models 20(6) Accelerated failure time models 26(4) The loglinear model representation 30(2) Semantics and history of the term frailty 32(11) Parametric proportional hazards models with gamma frailty 43(34) The parametric proportional hazards model with frailty term 44(1) Maximising the marginal likelihood: the frequentist approach 45(16) Extension of the marginal likelihood approach to interval-censored data 61(4) Posterior densities: the Bayesian approach 65(10) The Metropolis algorithm in practice for the parametric gamma frailty model 65(9) Theoretical foundations of the Metropolis algorithm 74(1) Further extensions and references 75(2) Alternatives for the frailty model 77(40) The fixed effects model 78(9) The model specification 78(6) Asymptotic efficiency of fixed effects model parameter estimates 84(3) The stratified model 87(6) The copula model 93(11) Notation and definitions for the conditional, joint, and population survival functions 93(2) Definition of the copula model 95(2) The Clayton copula 97(2) The Clayton copula versus the gamma frailty model 99(5) The marginal model 104(7) Defining the marginal model 104(1) Consistency of parameter estimates from marginal model 105(2) Variance of parameter estimates adjusted for correlation structure 107(4) Population hazards from conditional models 111(5) Population versus conditional hazard from frailty models 111(3) Population versus conditional hazard ratio from frailty models 114(2) Further extensions and references 116(1) Frailty distributions 117(82) General characteristics of frailty distributions 118(12) Joint survival function and the Laplace transform 119(1) Population survival function and the copula 120(2) Conditional frailty density changes over time 122(1) Measures of dependence 123(7) The gamma distribution 130(20) Definitions and basic properties 130(1) Joint and population survival function 131(3) Updating 134(3) Copula form representation 137(1) Dependence measures 138(3) Diagnostics 141(6) Estimation of the cross ratio function: some theoretical considerations 147(3) The inverse Gaussian distribution 150(14) Definitions and basic properties 150(2) Joint and population survival function 152(6) Updating 158(1) Copula form representation 158(3) Dependence measures 161(3) Diagnostics 164(1) The positive stable distribution 164(13) Definitions and basic properties 164(3) Joint and population survival function 167(4) Updating 171(1) Copula form representation 171(2) Dependence measures 173(3) Diagnostics 176(1) The power variance function distribution 177(13) Definitions and basic properties 177(4) Joint and population survival function 181(3) Updating 184(1) Copula form representation 185(1) Dependence measures 186(3) Diagnostics 189(1) The compound Poisson distribution 190(5) Definitions and basic properties 190(2) Joint and population survival functions 192(1) Updating 193(2) The lognormal distribution 195(1) Further extensions and references 196(3) The semiparametric frailty model 199(60) The EM algorithm approach 199(11) Description of the EM algorithm 199(1) Expectation and maximisation for the gamma frailty model 200(7) Why the EM algorithm works for the gamma frailty model 207(3) The penalised partial likelihood approach 210(23) The penalised partial likelihood for the normal random effects density 210(4) The penalised partial likelihood for the gamma frailty distribution 214(7) Performance of the penalised partial likelihood estimates 221(7) Robustness of the frailty distribution assumption 228(5) Bayesian analysis for the semiparametric gamma frailty model through Gibbs sampling 233(25) The frailty model with a gamma process prior for the cumulative baseline hazard for grouped data 234(5) The frailty model with a gamma process prior for the cumulative baseline hazard for observed event times 239(5) The normal frailty model based on Poisson likelihood 244(6) Sampling techniques used for semiparametric frailty models 250(7) Gibbs sampling, a special case of the Metropolis-Hastings algorithm 257(1) Further extensions and references 258(1) Multifrailty and multilevel models 259(28) Multifrailty models with one clustering level 260(17) Bayesian analysis based on Laplacian integration 260(8) Frequentist approach using Laplacian integration 268(9) Multilevel frailty models 277(9) Maximising the marginal likelihood with penalised splines for the baseline hazard 277(2) The Bayesian approach for multilevel frailty models using Gibbs sampling 279(7) Further extensions and references 286(1) Extensions of the frailty model 287(8) Censoring and truncation 287(1) Correlated frailty models 288(2) Joint modelling 290(2) The accelerated failure time model 292(3) References 295(13) Applications and Examples Index 308(1) Author Index 309(5) Subject Index 314