Exposure–risk modeling for time-to-event data 115
effect compartment model in which
c
∗
i
(t) =
Z
t
0
g(t − τ, β
i
)e
i
(τ)dτ (5.16)
where we use e
i
(t) as general notation for the observed exposure. An example
for g(t, β) is
g(t, θ) = K
in
exp(−Kt), (5.17)
in w hich K
in
and K are unknown parameters, when the exposure is con-
stant, i.e., e(t) = E, c
∗
i
(t) = EK
in
(1 − e xp(−Kt)/K). This section e xplores
approaches to fitting TTE models when c
∗
(t) is k nown except for a finite
number of parameters to be estimated.
5.3.1 Semiparametric models
In some specific cases, one can estimate the exp osure dynamic represented by
(5.16) with an empirical model approximation while fitting the TTE model.
For example, if time t is discrete, taking values 0, 1, 2, 3, ..., (5.16) can be
approximated ...