Biomedical Engineering Reference
In-Depth Information
intensities of these two events are incorporated in terms of three-state model
(Andersen et al., 1993). All animals are assumed to be without tumor at the
start of study. Some of the animals are transferred to the tumor state, some go
through the tumor state and result in the death state, and others still remain
with no tumors. This would be configured with the well-known illness-death
model: No tumor ! Tumor ! Death.
For interval-censored data, Commenges (2002) reviewed the inference pro-
cedure for multi-state models. Joly et al. (2002) applied a penalized likeli-
hood approach for smooth estimates under an illness-death model, and Com-
menges and Joly (2004) suggested a nonparametric estimator for a more com-
plicated data structure. Recently, the application of a multi-state model to
semi-competing risk has been treated (Xu et al., 2010; Barrett et al., 2011).
The goal of this study was to estimate the effect of treatment on tumor onset
time and death time with adjustment of possible correlations between these
two times when tumor onset time is not exactly observed. We construct the
models and likelihood in Section 5.2. Section 5.3 provides an estimation pro-
cedure at two different frailty distributions. In Section 5.4 we describe a new
R package \CSD," Section 5.5 gives the application of real data, and related
concluding remarks appear in Section 5.6.
5.2
Model
We consider a study that involves n independent subjects and every subject
may experience tumor onset whose time is denoted as T, but is not observed.
Instead, we know whether or not tumor onset occurred before a censoring
time, C. Denote a tumor indicator as = 1 if T < C and = 0, otherwise.
For censoring time, there are two possible cases: (i) natural death time, C
1
and
(ii) sacrifice time, C
2
. Thus, the observable censoring time is defined as C =
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