Semiparametric efficient estimation of structural nested mean models with irregularly spaced observations
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Structural Nested Mean Models (SNMMs) are useful for causal inference of treatment effects in longitudinal observational studies, accounting for time-varying confounding. Most of existing estimation of SNMMs assumes that the data are collected at finite, discrete, and pre-fixed time points for all subjects. However, the variables and processes are likely to be observed at irregularly-spaced time points in practice, and it is more realistic to assume that the data are generated from continuous-time processes. We establish the semiparametric efficiency theory for continuous-time SNMMs with irregularly-spaced observations under a martingale condition of no unmeasured confounding. We construct locally efficient estimators of the causal effect parameters, which achieve the semiparametric efficiency bound. In the presence of dependent censoring, we propose an inverse probability of censoring weighted estimator. The resulting estimator is doubly robust, in the sense that if either the model for the treatment process is correctly specified or the potential outcome mean model is correctly specified, but not necessarily both. We establish the asymptotic distribution of the proposed estimator, which allows for doubly robust inference. Simulation studies demonstrate that the proposed estimator is superior to the current competitors in finite samples. We apply the proposed estimator to estimate the effect of the effect of highly active antiretroviral therapy on the CD4 count at year 2 in HIV-positive patients with early and acute infection.
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