Biomedical Engineering Reference
In-Depth Information
lihood function as follows:
Y
F(R
i
jx
i
)
i1
[F(R
i
jx
i
) F(L
i
jx
i
)]
i2
[1 F(L
i
jx
i
)]
i3
;
L =
(6.1)
i=1
where
i1
,
i2
, and
i3
are the censoring indicators for subject i denoting
left-, interval-, and right-censoring, respectively. These censoring indicators
are subject to the constraint
i1
+
i2
+
i3
= 1.
Semiparametric regression models are increasingly popular in dealing with
survival data due to their great modeling flexibility. Here we focus on the
three commonly used semiparametric models: the proportional hazards (PH)
model, the proportional odds (PO) model, and the Probit model, (Lin and
Wang, 2010) in which the CDFs of the failure time of interest are given by
1 expf
0
(t) exp (x
0
)g;
F(tjx)
=
(6.2)
expf(t) + x
0
g
1 + expf(t) + x
0
g
;
F(tjx)
=
(6.3)
f(t) + x
0
g;
F(tjx)
=
(6.4)
where
0
(t) is the baseline cumulative hazard function in the PH model and
(t) is interpreted as the log baseline odds function in the PO model and the
transformed baseline CDF with the probit link in the Probit model (Lin and
Wang, 2010). The connection and difference between these three models are
clearly described in the following linear transformation models,
H(T) = x
0
+ ; G;
in which H is a strictly increasing unknown function and G is a known distri-
bution. All the PH, PO, and Probit models are special cases of the family of
linear transformation models.
By taking G to be the CDFs of an extreme value random variable, a logistic
random variable, and a standard normal random variable, one obtains the
PH model, the PO model, and the Probit model, respectively. The unknown
function H takes the form of log(
0
) in model (6.2), and in models (6.3)
and (6.4), respectively.
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