Biomedical Engineering Reference
In-Depth Information
4.4
Regression
Analysis
with
Linear
Transformation
Model
This section considers the fitting of linear transformation models to current
status data. The linear transformation model specifies that, given Z,
H
0
(T) = Z
0
+ ;
(4.3)
where H
0
is an unknown monotonically increasing function and is an error
term assumed to follow a known distribution free of Z (Sun and Sun (2005)).
The main advantage of linear transformation models is their flexibility because
they include many well-known regression models as special cases. For example,
one can get the proportional hazards model by taking F to be the extreme
value distribution and if follows the logistic distribution, then the model in
Equation (4.3) gives the proportional odds model.
Suppose that (t) is twice dierentiable; then it follows from the model in
Equation (4.3) that the survival function of T has the form
S(tjZ) = exp[(H
0
(t) + Z
0
)]:
Thus the likelihood can be written as
Y
f1 exp[(H
0
(C
i
) + Z
0
)]g
i
fexp[(H
0
(C
i
) + Z
0
)]g
1
i
:
L(;H
0
) =
i=1
The log-likelihood immediately follows;
X
f
i
log[1 exp((H
0
(C
i
) + Z
0
i
))] (1
i
)(H
0
(C
i
) + Z
0
i
)g:
l(;H
0
) =
i=1
Without loss of generality, we assume that C
1
;:::;C
n
are the order statis-
tics with C
1
C
2
::: C
n
. Because only the values of H
0
at the C
0
i
s matter
in the log-likelihood, in the following, for estimation of H
0
, we only consider
those H that are right-continuous increasing step functions with jumps at the
C
i
's and H(t) = 1 for all t < C
1
and H(t) = H(c
n
) for t > C
n
. The
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