
Exposure–risk modeling for time-to-event data 145
0 1 2 3 4 5
0.1 0.2 0.3 0.4 0.5 0.6
t
h(t)
h(t)
Estimated(shape=5)
Estimated(shape=20)
FIGURE 5.4
True and estimated exposure effects with placebo adjustment: constant base-
line hazard with gamma frailty (shape = 5 and 20).
with time-varying exposures. Consider the Cox regre ssion model
h(t, β) = h
0
(t) exp(X
T
β). (5.68)
The survival function of this mode l is
S(t|X) = exp[−H
0
(t) exp(X
T
β)]. (5.69)
When H
0
(t) is k nown, and its inverse function c an be found, a sample T
following the Cox model can be generated as (Bender e t al., 2005)
T = H
−1
0
[−log (U ) exp(−X
T
β)]. (5.70)