ABSTRACT
Department of Statistics, University of Connecticut, Storrs, Connecticut, USA
Ming-Hui Chen
Department of Statistics, University of Connecticut, Storrs, Connecticut, USA
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.2 Bayesian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.2.2 Cox Model with Time-Varying Regression Coefficients . 171
7.2.3 Piecewise Exponential Model for h0 . . . . . . . . . . . . . . . . . . . . . 171
7.2.4 Gamma Process Prior Model for H0 . . . . . . . . . . . . . . . . . . . . 172
7.2.5 Autoregressive Prior Model for β . . . . . . . . . . . . . . . . . . . . . . . 173
7.2.6 Dynamic Model for h0 and β . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3 Posterior Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.3.1 Sample Event Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.3.2 Sample Baseline Hazards and Regression Coefficients . . 178
168 Interval-Censored Time-to-Event Data: Methods and Applications
7.3.2.1 Update Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.3.2.2 Birth Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.3.2.3 Death Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.4 Bayesian Model Comparison Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.4.1 Logarithm of Pseudo-Marginal Likelihood . . . . . . . . . . . . . . 181
7.4.2 Deviance Information Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.5 “Dynsurv” Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.6 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.7 Analysis of the Breast Cancer Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Interval-censored data arise when the onset time of a certain event is not
observed exactly but only known to be either between two observed time
points or after an observed time. The latter case, known as right-censored,
can be viewed as interval-censored with the right end of the interval being
infinity. Interval-censored data are common in biomedical research and epi-
demiological studies, where each subject is assessed at a sequence of visits,
instead of being monitored continuously, during the whole follow-up period.
The Cox proportional hazards model (Cox, 1972) has been extended to work
with interval-censored data. Existing methods include the rank-based Monte
Carlo approach (Satten, 1996), the Markov chain Monte Carlo (MCMC) EM
approach (Goggins et al., 1998), the iterative convex minorant (ICM) approach
(Pan, 1999), the multiple imputations (Pan, 2000), and the EM-based missing
data approach (Goetghebeur and Ryan, 2000).
