Coarse structural nested mean models are used to estimate treatment effects from longitudinal observational data. Coarse structural nested mean models lead to a large class of estimators. It turns out that estimates and standard errors may differ considerably within this class. We prove that, under additional assumptions, there exists an explicit solution for the optimal estimator within the class of coarse structural nested mean models. Moreover, we show that even if the additional assumptions do not hold, this optimal estimator is doubly-robust: it is consistent and asymptotically normal not only if the model for treatment initiation is correct, but also if a certain outcome-regression model is correct. We compare the optimal estimator to some naive choices within the class of coarse structural nested mean models in a simulation study. Furthermore, we apply the optimal and naive estimators to study how the CD4 count increase due to one year of antiretroviral treatment (ART) depends on the time between HIV infection and ART initiation in recently infected HIV infected patients. Both in the simulation study and in the application, the use of optimal estimators leads to substantial increases in precision.
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