  Structural Nested Mean Models (SNMMs) are useful for causal inference of
treatment effects in longitudinal observational studies. Most existing works
assume that the data are collected at pre-fixed time points for all subjects,
which, however, is restrictive in practice. To deal with irregularly spaced
observations, we assume a class of continuous-time SNMMs and a martingale
condition of no unmeasured confounding (NUC) to identify the causal parameters.
We develop the first semiparametric efficiency theory and locally efficient
estimators for continuous-time SNMMs. This task is non-trivial due to the
restrictions from the NUC assumption imposed on the SNMM parameter. In the
presence of dependent censoring, we propose an inverse probability of censoring
weighting estimator, which achieves a multiple robustness feature in that it is
unbiased if either the model for the treatment process or the potential outcome
mean function is correctly specified, regardless whether the censoring model is
correctly specified. The new framework allows us to conduct causal analysis
respecting the underlying continuous-time nature of the data processes. We
estimate the effect of time to initiate highly active antiretroviral therapy on
the CD4 count at year 2 from the observational Acute Infection and Early
Disease Research Program database.
