Efficient Inference for Incremental Causal Effects of Time to Treatment

We consider continuous time to treatment initiation. This can commonly occur in preventive medicine, such as disease screening and vaccination; it can also occur with non-fatal health conditions such as HIV infection without the onset of AIDS. While traditional causal inference focused on `when to treat’ and its effects, we consider the incremental causal effect when the intensity of time to treatment initiation is intervened upon. We derive the efficient influence function for this estimand and develop an estimation framework that accommodates flexible machine learning methods while achieving fast convergence rates. Valid confidence bands are obtained leveraging empirical process theory. We illustrate our approach via simulation, and apply it to cervical cancer screening data to study the incremental effect of time to subsequent HPV testing on cervical intraepithelial neoplasia detection.