Biomedical Engineering Reference
In-Depth Information
where 0 is a vector of 0's and the risk-weighted covariate mean is given by:
P
n
i=1
I(C
i
> t)Z
i
(t) expf
T
Z
i
(t)g
P
n
i=1
I(C
i
> t) expf
T
Z
i
(t)g
:
We can re-express the left side of (6) in a perhaps more familiar form as
Z(t; ) =
X
n
X
N
i
fZ
i
(T
ij
)Z(T
ij
; )g;
(7)
i=1
j=1
without the stochastic integral. In the univariate survival setting, where
time until a single event is studied and subjects cannot experience multiple
events (i.e., N
i
1), the left side of (7) reduces to the partial likelihood
6
score equation,
X
n
i
fZ
i
(T
ij
)Z(T
ij
; )g;
i=1
where
i
= I(T
i
< C
i
). The correspondence between the Cox score equa-
tion and (7) makes sense in light of the close connection between the propor-
tional hazards and proportional rates models. As such, standard software
(e.g., PROC PHREG in SAS; coxph in R) can be used to t the propor-
tional rates model, as described in Allison
1
and Therneau and Hamilton
23
.
The model of current interest is given by:
E[dN
i
(t)jZ
i
] = d
0
(t) expf(t)Z
i
g;
(8)
the non-proportional rates model, which allows the covariate eects to
depend on time. Since the software cannot tell the dierence between (t)Z
i
and Z
i
(t), we can estimate (t) in (8) by tting (2) and adding time-
dependent elements to Z
i
which reect the nature of the hypothesized time-
dependence of the eects suspected of being non-proportional. For example,
returning to (4), a time-dependent treatment eect could be specied by
the model,
E[dN
i
(t)jZ
i
] = d
0
(t) expf(
0
+ t)Z
i
g;
(9)
b
b
where estimators
and
could be computed by tting the model,
E[dN
i
(t)jZ
i
] = d
0
(t) expf
0
Z
i
+ Z
i
tg:
(10)
Model (9) allows the eect of Z
i
on the event rate to change exponentially
with time. Naturally, other functional forms are possible. Depending on
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