Semiparametric estimation of structural nested mean models with irregularly spaced longitudinal observations

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 prefixed time points for all subjects, which, however, may be 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 semiparametric efficiency theory and locally efficient estimators for continuous‐time SNMMs. This task is nontrivial due to the restrictions from the NUC assumption imposed on the SNMM parameter. In the presence of ignorable censoring, we show that the complete‐case estimator is optimal among a class of weighting estimators including the inverse probability of censoring weighting estimator, and it achieves a double robustness feature in that it is consistent if at least one of the models for the potential outcome mean function and the treatment process is correctly specified. The new framework allows us to conduct causal analysis respecting the underlying continuous‐time nature of data processes. The simulation study shows that the proposed estimator outperforms existing approaches. 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.