Biomedical Engineering Reference
In-Depth Information
rithm given by Turnbull (1976) by adding covariate-dependent weights into
the self-consistency equation used in the algorithm.
In addition to directly estimating a conditional survival function, an alter-
native and general approach is to estimate the joint distribution of a failure
time variable, and other related variables. This latter problem has been dis-
cussed by many authors when right-censored data are available for the fail-
ure time variable of interest and the other variables are known or can be
completely observed. For interval-censored data, however, these other vari-
ables or covariates may suffer censoring or incompleteness too due to interval-
censoring. One example of such data is given by studies of some disease pro-
gression such as preventive HIV vaccine ecacy trials that involve some con-
tinuous marker variables. Among others, Hudgens et al. (2007) and Maathuis
and Wellner (2007) investigated the joint estimation problem with a continu-
ous marker variable. In particular, the former presented three nonparametric
estimates of the joint distribution when the marker variable may suffer miss-
ing, while the latter showed that the nonparametric maximum likelihood is
inconsistent in general. Corresponding to this, Groeneboom et al. (2011) pro-
posed a consistent estimate by maximizing a smoothed likelihood function for
the case of current status data.
The problem of nonparametric estimation of a survival function based on
interval-censored data can occur under more complex situations. For example,
Frydman and Szarek (2009) considered the problem under the framework of
a three state or illness-death Markov model where the intermediate transition
status may be missing. This can happen in, for example, cancer clinical trials
and tumorgenicity experiments. Chen et al. (2010) and Grin and Lagakos
(2010) also discussed the problem but under multivariate state progressive
disease models. In this situation, the patient is assumed to move only forward
from one state to another but not reversely, and they investigated nonpara-
metric estimation of the distribution of sojourn times between states.
 
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