Duchateau's book "the Frailty Model"

是 否 +2 论坛币

k人 参与回答

经管之家送您一份

应届毕业生专属福利!

求职就业群

赵安豆老师微信：zhaoandou666

经管之家联合CDA

送您一个全额奖学金名额~ !

立即领取

感谢您参与论坛问题回答

经管之家送您两个论坛币！

+2 论坛币

book.zip (6.51 MB, 需要: 20 个论坛币)

2011-11-14 23:18:26 上传

Duchateau and Janssen
需要: 20 个论坛币  [购买]

Survival analysis techniques are used in a variety of disciplines including human
and veterinary medicine, epidemiology, engineering, biology and economy.
Proportional hazards models and accelerated failure time models are
classical models that are frequently used for the analysis of univariate (censored)
survival data.

In this book the focus is on frailty models, but similarities as well as
differences between frailty models and copula models are highlighted. Frailty
models provide a nice way to capture and to describe the dependence of
observations within a cluster and/or the heterogeneity between clusters.



Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Glossary of Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Survival analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Survival likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 Proportional hazards models . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.3 Accelerated failure time models . . . . . . . . . . . . . . . . . . . . . 26
1.4.4 The loglinear model representation . . . . . . . . . . . . . . . . . . 30
1.5 Semantics and history of the term frailty . . . . . . . . . . . . . . . . . . . 32
2 Parametric proportional hazards models with gamma
frailty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1 The parametric proportional hazards model with frailty term . 44
2.2 Maximising the marginal likelihood: the frequentist approach . 45
2.3 Extension of the marginal likelihood approach to
interval-censored data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Posterior densities: the Bayesian approach . . . . . . . . . . . . . . . . . . 65
2.4.1 The Metropolis algorithm in practice for the
parametric gamma frailty model . . . . . . . . . . . . . . . . . . . . . 65
2.4.2 ∗ Theoretical foundations of the Metropolis algorithm . . . 74
2.5 Further extensions and references . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3 Alternatives for the frailty model . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1 The fixed effects model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.1 The model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xi
xii Contents
3.1.2 ∗ Asymptotic efficiency of fixed effects model parameter
estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2 The stratified model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3 The copula model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.1 Notation and definitions for the conditional, joint, and
population survival functions . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.2 Definition of the copula model . . . . . . . . . . . . . . . . . . . . . . 95
3.3.3 The Clayton copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3.4 The Clayton copula versus the gamma frailty model . . . 99
3.4 The marginal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4.1 Defining the marginal model . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4.2 ∗ Consistency of parameter estimates from marginal
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.3 Variance of parameter estimates adjusted for
correlation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5 Population hazards from conditional models . . . . . . . . . . . . . . . . 111
3.5.1 Population versus conditional hazard from frailty
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.5.2 Population versus conditional hazard ratio from frailty
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.6 Further extensions and references . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Frailty distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1 General characteristics of frailty distributions . . . . . . . . . . . . . . . 118
4.1.1 Joint survival function and the Laplace transform . . . . . 119
4.1.2 Population survival function and the copula . . . . . . . . . . 120
4.1.3 Conditional frailty density changes over time . . . . . . . . . . 122
4.1.4 Measures of dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2 The gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.2.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 130
4.2.2 Joint and population survival function . . . . . . . . . . . . . . . 131
4.2.3 Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.4 Copula form representation . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2.5 Dependence measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.2.6 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2.7 ∗ Estimation of the cross ratio function: some theoretical
considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.3 The inverse Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 150
4.3.2 Joint and population survival function . . . . . . . . . . . . . . . 152
4.3.3 Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.4 Copula form representation . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.5 Dependence measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.3.6 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

扫码加我 拉你入群

请注明：姓名-公司-职位

以便审核进群资格，未注明则拒绝

分享0 收藏1 回帖

关键词：Frailty model Hate mode chat including medicine survival between biology

Contents xiii
4.4 The positive stable distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.4.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 164
4.4.2 Joint and population survival function . . . . . . . . . . . . . . . 167
4.4.3 Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.4.4 Copula form representation . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.4.5 Dependence measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.4.6 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.5 The power variance function distribution . . . . . . . . . . . . . . . . . . . 177
4.5.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 177
4.5.2 Joint and population survival function . . . . . . . . . . . . . . . 181
4.5.3 Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.5.4 Copula form representation . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.5.5 Dependence measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.5.6 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.6 The compound Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . 190
4.6.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 190
4.6.2 Joint and population survival functions . . . . . . . . . . . . . . 192
4.6.3 Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.7 The lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.8 Further extensions and references . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5 The semiparametric frailty model . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.1 The EM algorithm approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.1.1 Description of the EM algorithm . . . . . . . . . . . . . . . . . . . . 199
5.1.2 Expectation and maximisation for the gamma frailty
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.1.3 Why the EM algorithm works for the gamma frailty
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.2 The penalised partial likelihood approach . . . . . . . . . . . . . . . . . . . 210
5.2.1 The penalised partial likelihood for the normal
random effects density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
5.2.2 The penalised partial likelihood for the gamma frailty
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.2.3 Performance of the penalised partial likelihood
estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.2.4 Robustness of the frailty distribution assumption . . . . . . 228
5.3 Bayesian analysis for the semiparametric gamma frailty
model through Gibbs sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.3.1 The frailty model with a gamma process prior for the
cumulative baseline hazard for grouped data . . . . . . . . . . 234
5.3.2 The frailty model with a gamma process prior for the
cumulative baseline hazard for observed event times . . . 239
5.3.3 The normal frailty model based on Poisson likelihood . . 244
5.3.4 Sampling techniques used for semiparametric frailty
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
xiv Contents
5.3.5 Gibbs sampling, a special case of the Metropolis–
Hastings algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
5.4 Further extensions and references . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6 Multifrailty and multilevel models . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.1 Multifrailty models with one clustering level . . . . . . . . . . . . . . . . 260
6.1.1 Bayesian analysis based on Laplacian integration . . . . . . 260
6.1.2 Frequentist approach using Laplacian integration . . . . . . 268
6.2 Multilevel frailty models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
6.2.1 Maximising the marginal likelihood with penalised
splines for the baseline hazard . . . . . . . . . . . . . . . . . . . . . . 277
6.2.2 The Bayesian approach for multilevel frailty models
using Gibbs sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
6.3 Further extensions and references . . . . . . . . . . . . . . . . . . . . . . . . . . 286
7 Extensions of the frailty model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.1 Censoring and truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.2 Correlated frailty models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.3 Joint modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
7.4 The accelerated failure time model . . . . . . . . . . . . . . . . . . . . . . . . . 292

感谢分享！

In the linear model setting, longitudinal data structure provides an opportunity to single out the unobserved heterogeneity. Common approaches like GLMM have been well developed to handle the unobserved heterogeneity issue. However, the problem becomes complex when the observations are of the time-to-event type: linear models prove biased under the censoring situation.

Frailty model opens your eyes on how to deal with longitudinal survivals, where the commen survival approaches (such as Cox ph models and AFT models) are not able to handle the within-patient associations well. This within-patient associations are caused by sharing the same unobserved heterogeneity.