Biomedical Engineering Reference
In-Depth Information
out that one fruitful way to view mixed-case models is through the notion of
panel count data, which is described below.
Suppose that N = fN(t) : t 0g is a counting process with mean
function EN(t) = (t), K is an integer-valued random variable, and T =
T
k;j
; j = 1;:::;k;k = 1; 2;:::, is a triangular array of potential observation
times. It is assumed that N and (K;T) are independent, that K and T
are independent, and T
k;j1
T
k;j
for j = 1;:::;k, for every k; we inter-
pret T
k;0
as 0. Let X = (N
K
;T
K
;K) be the observed random vector for
an individual. Here, K is the number of times that the individual was ob-
served during a study; T
K;1
T
K;2
::: T
K;K
are the times when they
were observed and N
K
= fN
K;j
N(T
K;j
)g
j=1
are the observed counts at
those times. The above scenario specializes easily to the mixed-case interval-
censoring model, when the counting process is N(t) = 1(S t), S being
a positive random variable with distribution function F and independent of
(T;K). To understand the issues with mixed-case interval-censoring it is best
to restrict to Case 2 interval-censoring, where K is identically 2. For this case,
I use slightly different notation, denoting T
2;1
and T
2;2
by U and V , respec-
tively. With n individuals, our (i.i.d) data can be written as f
i
;U
i
;V
i
g
i=1
,
where
i
= (
(1
i
;
(2
i
;
(3
i
) and
(1
i
= 1(S
i
U
i
),
(2
i
= 1(U
i
< Si
i
V
i
),
and
(3
i
= 1(V
i
< Si
i
) 1
(1
i
(2
i
. Here, S
i
is the survival time of the
i-th individual. The likelihood function for Case 2 censoring is given by
Y
F(U
i
)
(1)
(F(V
i
) F(U
i
))
(2)
(1 F(V
i
))
(3)
L
n
=
;
i
i
i
i=1
and the corresponding log-likelihood by
n
o
X
(1
i
log F(U
i
) +
(2
i
log(F(V
i
) F(U
i
)) +
(3
i
log(1 F(V
i
))
l
n
=
:
i=1
Now, let t
0
0 < t
1
< t
2
::: < t
J
denote the ordered distinct observation
times. If (U;V ) has a continuous distribution, then, of course, J = 2n, but in
general this may not be the case. Now consider the rank function R on the set
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