Biomedical Engineering Reference
In-Depth Information
can accommodate time-dependent covariates (which play a key role in our
application), we focus on model (2).
The proportional means/rates models, (1) and (2), can be considered
recurrent event analogs of the Cox
5
proportional hazards model, for ap-
plication to recurrent event data. Model (2) also has a close connection to
the Andersen-Gill
2
model, which can be written as:
d
i
(t)E[dN
i
(t)jF
i
(t)] = d
0
(t) expf
0
Z
i
(t)g;
(3)
whereF
i
(t) can be thought of as the event history for subject i at time t.
One could refer to (3) and (2) as conditional and marginal models, respec-
tively. Practitioners may prefer the latter for at least two reasons. First,
it is often dicult to captureF
i
(t) through the covariate vector; e.g., by
including N
i
(t) or various other related functions ofF
i
(t) as elements
in Z
i
(t). Second, consider a study with one covariate, Z
i
(t) = Z
i
, which
takes the value 1 for `treated' subjects and 0 for those receiving placebo.
Compare two models, a marginal model,
E[dN
i
(t)jZ
i
] = d
0
(t) expf
0
Z
i
g;
(4)
and a conditional model,
E[dN
i
(t)jZ
i
; N
i
(t)] = d
0
(t) expf
0
Z
i
+
N
N
i
(t)g:
(5)
The quantity expf
0
gfrom (4) can be interpreted as the ratio of the event
rate, treated versus placebo (reference) subjects; that is,
E[dN
i
(t)jZ
i
= 1]
E[dN
i
(t)jZ
i
= 0]
= expf
0
g:
The quantity expf
0
gfrom model (5) would be making the same compari-
son, but restricting attention to subjects with the same number of previous
events; i.e.,
E[dN
i
(t)jZ
i
= 1; N
i
(t) = m]
E[dN
i
(t)jZ
i
= 0; N
i
(t) = m]
= expf
0
g:
If Z
i
aects the event rate, E[dN
i
(t)], then it will also aect the N
i
(t)
and, provided events within-subject are positively correlated, conditioning
on the previous number of events will attenuate the estimated marginal
eect of Z
i
; i.e.,j
0
j<j
0
j.
The estimate of the parameter of interest in the proportional rates
model,
0
, is the solution to the estimating equation,
n
Z
X
fZ
i
(t)Z(t; )gdN
i
(t) = 0;
(6)
0
i=1
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