Doubly Robust Estimation under Covariate-induced Dependent Left Truncation

In prevalent cohort studies with follow-up, the time-to-event outcome is subject to left truncation when only subjects with event time greater than enrollment time are included. In such studies, subjects with early event times tend not to be captured, leading to selection bias if simply ignoring left truncation. Conventional methods adjusting for left truncation tend to rely on the (quasi-)independence assumption that the truncation time and the event time are "independent" on the observed region. This assumption is subject to failure when there is dependence between the truncation time and the event time possibly induced by measured covariates. Inverse probability of truncation weighting leveraging covariate information can be used in this case, but it is sensitive to misspecification of the truncation model. In this work, we first apply the semiparametric theory to find the efficient influence curve of the expectation of an arbitrary transformed survival time in the presence of covariate-induced dependent left truncation. We then use it to further construct estimators that are shown to enjoy double-robustness properties: 1) model double-robustness, that is, they are consistent and asymptotically normal (CAN) when the estimators for the nuisance parameters are both asymptotically linear and one of the two estimators is consistent, but not necessarily both; 2) rate double-robustness, that is, they are CAN when both of the nuisance parameters are consistently estimated and the error product rate under the two nuisance models is faster than root-n. Simulation studies demonstrate the finite sample performance of the estimators.