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 leading to selection bias. For estimation of the distribution of the time to event, 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 violated when there is dependence between the truncation time and the event time, possibly induced by measured covariates. Inverse probability of truncation weighting can be used in this case, but it is sensitive to misspecification of the truncation model. In this work, we apply semiparametric theory to find the efficient influence curve of the expectation of an arbitrarily transformed survival time in the presence of covariate-induced dependent left truncation. We then use it to construct estimators that are shown to enjoy double-robustness properties. Our work represents the first attempt to construct doubly robust estimators in the presence of left truncation, which does not fall under the established framework of coarsened data where doubly robust approaches were developed. We provide technical conditions for the asymptotic properties that appear to not have been carefully examined in the literature for time-to-event data, and study the estimators via extensive simulation. We apply the estimators to two datasets from practice, with different right-censoring patterns.