List of Tables

List of Figures

Notation

Preface

I Introduction

Introduction

Survival concepts

Types of censoring

Right censoring

Interval and left censoring

Some special cases of interval censoring

Doubly interval censoring

Truncation

Ignoring interval censoring

Independent noninformative censoring

Independent noninformative right censoring

Independent noninformative interval censoring

Frequentist inference

Likelihood for interval-censored data

Maximum likelihood theory

Data sets and research questions

Homograft study

Breast cancer study

AIDS clinical trial

Sensory shelf life study

Survey on mobile phone purchases

Mastitis study

Signal Tandmobielr study

Censored data in R and SAS

R

SAS

Inference for right-censored data

Estimation of the survival function

Nonparametric maximum likelihood estimation

R solution

SAS solution

Comparison of two survival distributions

Review of signi_cance tests

R solution

SAS solution

Regression models

The proportional hazards model

Model description and estimation

Model checking

R solution

SAS solution

The accelerated failure time model

Model description and estimation

Model checking

R solution

SAS solution

II Frequentist methods for interval-censored data

Estimating the survival distribution

Nonparametric maximum likelihood

Estimation

Asymptotic results

R solution

SAS solution

Parametric modelling

Estimation

Model selection

Goodness of _t

R solution

SAS solution

Smoothing methods

Logspline density estimation

A smooth approximation to the density

Maximum likelihood estimation

R solution

Classical Gaussian mixture model

Penalized Gaussian mixture model

R solution

Concluding remarks

Comparison of two or more survival distributions

Nonparametric comparison of survival curves

The weighted log-rank test: derivation

The weighted log-rank test: linear form

The weighted log-rank test: derived from the linear

transformation model

The weighted log-rank test: the G family

The weighted log-rank test: significance testing

R solution

SAS solution

Sample size calculation

Concluding remarks

The proportional hazards model

Parametric approaches

Maximum likelihood estimation

R solution

SAS solution

Towards semiparametric approaches

The piecewise exponential baseline survival model

Model description and estimation

R solution

SAS solution

The SemiNonParametric approach

Model description and estimation

SAS solution

Spline-based smoothing approaches

Two spline-based smoothing approaches

R solution

SAS solution

Semiparametric approaches

Finkelstein's approach

Farrington's approach

The iterative convex minorant algorithm

The grouped proportional hazards model

Practical applications

Two examples

R solution

SAS solution

Multiple imputation approach

Data augmentation algorithm

Multiple imputation for interval-censored survival times

R solution

SAS solution

Model checking

Checking the PH model

R solution

SAS solution

Sample size calculation

Concluding remarks

The accelerated failure time model

The parametric model

Maximum likelihood estimation

R solution

SAS solution

The penalized Gaussian mixture model

Penalized maximum likelihood estimation

R solution

The SemiNonParametric approach

SAS solution

Model checking

Sample size calculation

Computational approach

SAS solution

Concluding remarks

Bivariate survival times

Nonparametric estimation of the bivariate survival function

The NPMLE of a bivariate survival function

R solution

SAS solution

Parametric models

Model description and estimation

R solution

SAS solution

Copula models

Background

Estimation procedures

R solution

Flexible survival models

The penalized Gaussian mixture model

SAS solution

Estimation of the association parameter

Measures of association

Estimating measures of association

R solution

SAS solution

Concluding remarks

Additional topics

Doubly interval-censored data

Background

R solution

Regression models for clustered data

Frailty models

R solution

SAS solution

A marginal approach to correlated survival times

Independence working model

SAS solution

A biplot for interval-censored data

The classical biplot

Extension to interval-censored observations

R solution

Concluding remarks

III Bayesian methods for interval-censored data

Bayesian concepts

Bayesian inference

Parametric versus nonparametric Bayesian approaches

Bayesian data augmentation

Markov chain Monte Carlo

Credible regions and contour probabilities

Selecting and checking the model

Sensitivity analysis

Nonparametric Bayesian inference

Bayesian nonparametric modelling of the hazard function

Bayesian nonparametric modelling of the distribution function

Bayesian software

WinBUGS and OpenBUGS

JAGS

R software

SAS procedures

Stan software

Applications for right-censored data

Parametric models

BUGS solution

SAS solution

Nonparametric Bayesian estimation of a survival curve

R solution

Semiparametric Bayesian survival analysis

BUGS solution

Concluding remarks

Bayesian estimation of the survival distribution for interval-censored observations

Bayesian parametric modelling

JAGS solution

SAS solution

Bayesian smoothing methods

Classical Gaussian mixture

R solution

Penalized Gaussian mixture

Nonparametric Bayesian estimation

The Dirichlet Process prior approach

R solution

The Dirichlet Process Mixture approach

R solution

Concluding remarks

The Bayesian proportional hazards model

The parametric PH model

JAGS solution

SAS solution

The PH model with exible baseline hazard

A Bayesian PH model with a smooth baseline hazard

R solution

A PH model with piecewise constant baseline hazard

R solution

The semiparametric PH model

Concluding remarks

The Bayesian accelerated failure time model

The Bayesian parametric AFT model

JAGS solution

SAS solution

AFT model with a classical Gaussian mixture as an error distribution

R solution

AFT model with a penalized Gaussian mixture as an error distribution

R solution

A Bayesian semiparametric AFT model

R solution

Concluding remarks

Additional topics

Hierarchical models

Parametric shared frailty models

JAGS solution

SAS solution

Flexible shared frailty models

R solution

Semiparametric shared frailty models

Multivariate models

Parametric bivariate models

JAGS solution

SAS solution

Bivariate copula models

Flexible bivariate models

R solution

Semiparametric bivariate models

R solution

The multivariate case

Doubly interval censoring

Parametric modelling of univariate DI-censored data

JAGS solution

Flexible modelling of univariate DI-censored data

R solution

Semiparametric modelling of univariate DI-censored data

R solution

Modelling of multivariate DI-censored data

Concluding remarks

IV Concluding part

Omitted topics and outlook

Omitted topics

Competing risks and multi-state models

Survival models with a cured subgroup

Multilevel models

Informative censoring

Interval-censored covariates

Joint longitudinal and survival models

Spatial-temporal models

Time points measured with error

Quantile regression

Outlook

V Appendices

A Data sets

A Homograft study

A AIDS clinical trial

A Survey on mobile phone purchases

A Mastitis study

A Signal Tandmobiel R study

B Distributions

B Log-normal LN(; _)

B Log-logistic LL(; _)

B Weibull W(; _)

B Exponential E(_)

B Rayleigh R(_)

B Gamma(; _)

B R solution

B SAS solution

B BUGS solution

B R and BUGS parametrizations

C Prior distributions

C Beta prior: Beta(_; _)

C Dirichlet prior: Dir (_)

C Gamma prior: G(_; _)

C Inverse gamma prior: IG(_; _)

C Wishart prior: Wishart(R; k)

C Inverse Wishart prior: Wishart(R; k)

C Link between Beta, Dirichlet and Dirichlet Process prior

D Description of selected R packages

D The icensBKL package

D The Icens package

D The interval package

D The survival package

D The logspline package

D The smoothSurv package

D The mixAK package

D The bayesSurv package

D The DPpackage package

D Other packages

E Description of selected SAS procedures

E PROC LIFEREG

E PROC RELIABILITY

E PROC ICLIFETEST

E PROC ICPHREG

F Technical details

F The Iterative Convex Minorant (ICM) algorithm

F Regions of possible support for bivariate interval-censored data

F The algorithm of Gentleman and Vandal ()

F The algorithm of Bogaerts and Lesa_re ()

F The height map algorithm of Maathuis ()

F Splines

F Polynomial _tting

F Polynomial splines

F Natural cubic splines

F Truncated power series

F B-splines

F M-splines and I-splines

F Penalized splines (P-splines)

References

Author Index

Subject Index