Biomedical Engineering Reference
In-Depth Information
A few methods have been proposed for maximizing L S (p) with respect
to p. Among them, three commonly used algorithms are the self-consistency
algorithm given in Turnbull (1976), the iterative convex minorant (ICM) algo-
rithm introduced by Groeneboom and Wellner (1992) and further studied by
Jongbloed (1998), and the EM-ICM algorithm proposed in Wellner and Zhan
(1997). The first one is essentially an application of the EM algorithm, while
the ICM algorithm can be seen as an optimized version of the well-known
pool-adjacent-violator algorithm for the isotonic regression (Robertson et al.
(1988)). As suggested by the name, the EM-ICM algorithm is a hybrid algo-
rithm that combines the first two approaches. All of the above algorithms are
iterative and, in fact, there is no closed form for S except in some special cases.
For detailed description and comparison of the three algorithms and others,
the readers are referred to Sun (2006) and Zhang and Sun (2010b) as well
as the references mentioned above. From these references, one can also find
some discussion on estimation of a hazard function based on interval-censored
data and other references on the topic. Note that although it may not be of
primary interest in many applications, the estimated hazard function could
provide some insight information into the shape of a survival function.
The discussion above has assumed that the main interest is on nonpara-
metric estimation of a marginal survival function. Sometimes there may exist
some other variables such as covariates in addition to the failure time vari-
able, and one may be interested in nonparametrically estimating a conditional
survival function given the variables. A common and simple example is to es-
timate a survival function conditional on some treatments and to perform
some graphical treatment comparison. Of course, for this situation or the
case where the variable takes only discrete values, the estimation is straight-
forward. One recent reference on this was given by Dehghan and Duchesne
(2011), which discussed the problem with a continuous covariate. To estimate
the conditional survival function, they generalized the self-consistency algo-
 
Search WWH ::




Custom Search