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The generalized g-formula can be used to estimate the probability of survivalunder a sustained treatment strategy. When treatment strategies aredeterministic, estimators derived from the so-called efficient influencefunction (EIF) for the g-formula will be doubly robust to modelmisspecification. In recent years, several practical applications havemotivated estimation of the g-formula under non-deterministic treatmentstrategies where treatment assignment at each time point depends on theobserved treatment process. In this case, EIF-based estimators may or may notbe doubly robust. In this paper, we provide sufficient conditions to ensureexistence of doubly robust estimators for intervention treatment distributionsthat depend on the observed treatment process for point treatmentinterventions, and give a class of intervention treatment distributionsdependent on the observed treatment process that guarantee model doubly andmultiply robust estimators in longitudinal settings. Motivated by anapplication to pre-exposure prophylaxis (PrEP) initiation studies, we propose anew treatment intervention dependent on the observed treatment process. We showthere exist 1) estimators that are doubly and multiply robust to modelmisspecification, and 2) estimators that when used with machine learningalgorithms can attain fast convergence rates for our proposed intervention.Theoretical results are confirmed via simulation studies.

Intervention treatment distributions that depend on the observed treatment process and model double robustness in causal survival analysis