  Most of the work on the structural nested model and g-estimation for causal
inference in longitudinal data assumes a discrete-time underlying data
generating process. However, in some observational studies, it is more
reasonable to assume that the data are generated from a continuous-time process
and are only observable at discrete time points. When these circumstances
arise, the sequential randomization assumption in the observed discrete-time
data, which is essential in justifying discrete-time g-estimation, may not be
reasonable. Under a deterministic model, we discuss other useful assumptions
that guarantee the consistency of discrete-time g-estimation. In more general
cases, when those assumptions are violated, we propose a controlling-the-future
method that performs at least as well as g-estimation in most scenarios and
which provides consistent estimation in some cases where g-estimation is
severely inconsistent. We apply the methods discussed in this paper to
simulated data, as well as to a data set collected following a massive flood in
Bangladesh, estimating the effect of diarrhea on children's height. Results
from different methods are compared in both simulation and the real
application.
