Biomedical Engineering Reference
In-Depth Information
window on either side, then assume the events happen exactly at 4 weeks)
and those without progressions were right-censored, and performed the usual
logrank test.
Next, we consider methods that can be used with irregular assessments.
13.3.3
Likelihood for Grouped Continuous Model
In this subsection we discuss the likelihood associated with the grouped con-
tinuous model (see, for example,
Fay, 1996). The associated model is also
called the linear transformation model (see Kalbfleisch and Prentice, 2002, p.
241).
Suppose there exists some unknown increasing function of event time, h(),
and the event time Xi
i
is modeled as
= z
0
i
+
i
;
h(X
i
)
where
i
F for all i and F is some known distribution. Because F is known,
it is convenient to rewrite h(t) = F
1
f1 S
0
(t)g, where S
0
it is a completely
unspecified survival function. From the model, we can write the survival dis-
tribution at t for a subject with treatment indicator zi
i
as
S(t; z
0
i
;S
0
) = 1 F
F
1
[1 S
0
(t)] z
0
i
:
(13.1)
We mention two important special cases. When F is the extreme minimum
value distribution (i.e., F(t) = 1 exp(e
t
) ), then Equation (13.1) reduces to
S
0
(t)
exp(z
0
i
)
, so that this particular linear transformation model is another
way of representing the proportional hazards model (see also Kalbfleisch and
Prentice, 2002, p. 241). Also, when F is the logistic distribution, then the
model reduces to the proportional odds model.
If the censoring is CIA, then we can write the likelihood as
Y
fS(`
i
; z
0
i
;S
0
) S(r
i
; z
0
i
;S
0
)g:
L(;S
0
) =
i=1
For a score statistic, we maximize the likelihood under the null hypothesis of
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