Conditional and marginal hazard ratios: accounting for unmeasured confounding in interpretation and estimation

Conditional proportional hazard Cox regression models are routinely used in the analysis of time-to-event outcomes, with the association between one variable of interest and the outcome commonly reported as a hazard ratio (HR). However, the interpretation of HRs is strongly dependent on the underlying assumptions of the model, which can make causal interpretation prone to inaccuracy. The non-collapsibility of the model contributes heavily to such confusion, as it disqualifies the same model from holding conditionally and marginally, even when the study design is a randomized clinical trial (RCT). With the goal of facilitating causal interpretation of the results, alternative Cox-type models, such as marginal structural models, have been proposed. These models are collapsible, and therefore their parameters can be identified in the absence of unmeasured confounders. In this document, we provide a probabilistic interpretation of both conditional and marginal HRs; highlight the differences in the underlying models, and point out the subtleties in their interpretation. We discuss some limitations related to the real-world estimation of HRs. In particular, we explore the limitations of propensity‑score techniques when estimating conditional hazard ratios in the presence of unmeasured confounders. A synthetic dataset is used to illustrate the problem. In a practical application, we emulate a study in which the target was to compare the effectiveness of two procedures for treating carotid artery stenosis patients: transfemoral carotid artery stenting (TF-CAS), and a more recent procedure that was approved for use in 2015, transcarotid artery revascularization (TCAR).