  The Hazard Ratio (HR) is often reported as the main causal effect when
studying survival data. Despite its popularity, the HR suffers from an unclear
causal interpretation. As already pointed out in the literature, there is a
built-in selection bias in the HR, because similarly to the truncation by death
problem, the HR conditions on post-treatment survival. A recently proposed
alternative, inspired by the Survivor Average Causal Effect (SACE), is the
causal HR, defined as the ratio between hazards across treatment groups among
the study participants that would have survived regardless of their treatment
assignment. We discuss the challenge in identifying the causal HR and present a
sensitivity analysis identification approach in randomized controlled trials
utilizing a working frailty model. We further extend our framework to adjust
for potential confounders using inverse probability of treatment weighting. We
present a Cox-based and a flexible non-parametric kernel-based estimation under
right censoring. We study the finite-sample properties of the proposed
estimation method through simulations. We illustrate the utility of our
framework using two real-data examples.
