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