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Semiparametric efficient estimation of structural nested mean models with irregularly spaced 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 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.