Biomedical Engineering Reference
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ditionally independent given Xi. i . Based on observations f i ;C i ;X i g i=1 , Shen
(2000) develops a random-sieve likelihood-based method to make inference
on and the error variance 2
without specifying the error distribution X;
in fact, an asymptotically ecient estimator of is constructed. This model
has close connections to survival analysis as a variety of survival models can
be written in the form h(Yi) i ) = T X i + i for a monotone transformation
h of the survival time Yi. i . With i following the extreme-value distribution
F(x) = 1 e e x , the above model is simply the Cox PH model, where the
function h determines the baseline hazard. When F(x) = e x =(1 + e x ), that is,
the logistic distribution, one gets the proportional odds model. Such models
are also known as semiparametric linear transformation models and have been
studied in the context of current status data by other authors.
Sun and Sun (2005) deal with the analysis of current status data under
semiparametric linear transformation models for which they propose a general
inference procedure based on estimating functions. They allow time-dependent
covariates Z(t) and model the conditional survival function of the failure time
T as S Z (t) = g(h(t)+ T Z(t)) for a known continuous strictly decreasing func-
tion g and an unknown function h. This is precisely an extension of the mod-
els in the previous paragraph to the time-dependent covariate setting; with
time-independent covariates, setting g(t) = e e t gives the Cox PH model and
setting g to be the logistic distribution gives the proportional odds model. As
in the previous paragraph, i = 1fT i C i g is recorded. Sun and Sun (2005)
use counting process-based ideas to construct estimates in situations where C
is independent of (T;Z) and also when T and C are conditionally independent
given Z. A related paper by Zhang et al. (2005) deals with regression analysis
of interval-censored failure time data with linear transformation models. Ma
and Kosorok (2005) consider a more general problem where a continuous out-
come U is modeled as H(U) = T Z + h(W) + e, with H being an unknown
monotone transformation, h an unknown smooth function, e has a known dis-
tribution function, and Z 2R d ;W 2Rare covariates. The observed data are
 
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