When a subject can have more than one episode over follow-up (e.g.
repeated infections or hospitalizations), the time of each subsequent
episode can be generated conditional on the time of the previous one,
following Leemis (1987). Two assumptions
about how risk behaves after an episode give two different
processes.
Renewal process
Under a renewal process, risk resets after each episode: the time to
the next episode is a fresh, independent draw from the same (baseline,
proportional hazards, or accelerated failure time) distribution, added
to the time of the previous episode: \[
T_{i+1} = T_i + H^{-1}\left(\frac{-\log(U)}{HR}\right) \] or,
under an accelerated failure time effect, \[
T_{i+1} = T_i + \frac{H^{-1}(-\log(U))}{AFT} \] This is
implemented by renewhr(s_object, hr, prevtime) and
renewaft(s_object, aft, prevtime), which take the time of
the previous episode prevtime and generate the time of the
next one.
Non-homogeneous Poisson process
Under a non-homogeneous Poisson process, risk does not reset after
each episode: a single cumulative hazard \(H(t)\) accumulates over calendar time from
the start of follow-up, and consecutive episode times satisfy \(H(T_{i+1}) - H(T_i) \sim
\text{Exponential}(1)\), so \[ T_{i+1}
= H^{-1}\left(H(T_i) - \frac{\log(U)}{HR}\right) \] or, under an
accelerated failure time effect, \[ T_{i+1} =
\frac{H^{-1}\left(H(AFT \cdot T_i) - \log(U)\right)}{AFT} \] This
is implemented by nhpphr(s_object, hr, prevtime) and
nhppaft(s_object, aft, prevtime).
The difference between the two processes only shows up when the
baseline hazard is not constant. With an increasing baseline hazard, for
example, a renewal process keeps generating episodes at the same average
pace, since every episode resets the risk back to its starting point.
Under a non-homogeneous Poisson process, risk keeps accumulating over
calendar time and never resets, so later episodes follow each other
increasingly quickly:
# Weibull baseline with an increasing hazard (shape > 1)
s_obj <- s_weibull(scale = 1, shape = 2)
n <- 5000
hr <- rep(1, n)
# First episode, common to both processes
t1 <- rsurvhr(s_obj, hr)
# Renewal process: risk resets at each episode
t2 <- renewhr(s_obj, hr, t1)
t3 <- renewhr(s_obj, hr, t2)
c(gap1 = mean(t1), gap2 = mean(t2 - t1), gap3 = mean(t3 - t2))
#> gap1 gap2 gap3
#> 0.8928635 0.8815392 0.8948339
# Non-homogeneous Poisson process: risk keeps accumulating, never resets
p2 <- nhpphr(s_obj, hr, t1)
p3 <- nhpphr(s_obj, hr, p2)
c(gap1 = mean(t1), gap2 = mean(p2 - t1), gap3 = mean(p3 - p2))
#> gap1 gap2 gap3
#> 0.8928635 0.4388354 0.3335675
The average gap between episodes stays roughly constant under the
renewal process, while it shrinks with each successive episode under the
non-homogeneous Poisson process.
The function 
